diff --git a/doc/equations/digamma1.png b/doc/equations/digamma1.png index 295415c27..408c17be2 100644 Binary files a/doc/equations/digamma1.png and b/doc/equations/digamma1.png differ diff --git a/doc/equations/digamma1.svg b/doc/equations/digamma1.svg index 034d066d3..5feb7f95b 100644 --- a/doc/equations/digamma1.svg +++ b/doc/equations/digamma1.svg @@ -1,2 +1,2 @@ -ψ(x)=ddxln(Γ(x))=Γ′(x)Γ(x) \ No newline at end of file +ψ(x)=ddxln(Γ(x))=Γ′(x)Γ(x) \ No newline at end of file diff --git a/doc/equations/digamma2.svg b/doc/equations/digamma2.svg index b4db806c0..9b7ea8fd5 100644 --- a/doc/equations/digamma2.svg +++ b/doc/equations/digamma2.svg @@ -1,2 +1,2 @@ -ψ(x)=ln(x)+12x−∑n=1∞B2n2nx2n \ No newline at end of file +ψ(x)=ln(x)+12x−∑n=1∞B2n2nx2n \ No newline at end of file diff --git a/doc/equations/digamma3.png b/doc/equations/digamma3.png index cbf8f6e91..789e1021b 100644 Binary files a/doc/equations/digamma3.png and b/doc/equations/digamma3.png differ diff --git a/doc/equations/digamma3.svg b/doc/equations/digamma3.svg index 7ab0de08d..0200e6495 100644 --- a/doc/equations/digamma3.svg +++ b/doc/equations/digamma3.svg @@ -1,2 +1,2 @@ -ψ(x)=z−12x+g−12+ln(x+g−12)+P′(x)P(x)−Q′(x)Q(x)−1 \ No newline at end of file +ψ(x)=z−12x+g−12+ln(x+g−12)+P′(x)P(x)−Q′(x)Q(x)−1 \ No newline at end of file diff --git a/doc/equations/digamma4.mml b/doc/equations/digamma4.mml new file mode 100644 index 000000000..16e227469 --- /dev/null +++ b/doc/equations/digamma4.mml @@ -0,0 +1,47 @@ + +]> + +digamma4 + + + + + + ψ + + + n + + + = + + ∑ + + k + = + 1 + + + n + − + 1 + + + + 1 + k + + − + γ + + ; + + n + ∈ + ℕ + + + + \ No newline at end of file diff --git a/doc/equations/digamma4.png b/doc/equations/digamma4.png new file mode 100644 index 000000000..5de4146f7 Binary files /dev/null and b/doc/equations/digamma4.png differ diff --git a/doc/equations/digamma4.svg b/doc/equations/digamma4.svg new file mode 100644 index 000000000..05fac0afa --- /dev/null +++ b/doc/equations/digamma4.svg @@ -0,0 +1,2 @@ + +ψ(n)=∑k=1n−11k−γ;n∈ℕ \ No newline at end of file diff --git a/doc/equations/digamma5.mml b/doc/equations/digamma5.mml new file mode 100644 index 000000000..ad6c9b313 --- /dev/null +++ b/doc/equations/digamma5.mml @@ -0,0 +1,78 @@ + +]> + +digamma5 + + + + + + ψ + + + + 1 + 2 + + − + n + + + = + + ∑ + + k + = + 1 + + + n + − + 1 + + + + 1 + k + + + + + ∑ + + k + = + n + + + 2 + n + − + 1 + + + + 2 + k + + − + log + + + 4 + + + − + γ + + ; + + n + ∈ + ℕ + + + + \ No newline at end of file diff --git a/doc/equations/digamma5.png b/doc/equations/digamma5.png new file mode 100644 index 000000000..79c42400f Binary files /dev/null and b/doc/equations/digamma5.png differ diff --git a/doc/equations/digamma5.svg b/doc/equations/digamma5.svg new file mode 100644 index 000000000..8509f8828 --- /dev/null +++ b/doc/equations/digamma5.svg @@ -0,0 +1,2 @@ + +ψ(12−n)=∑k=1n−11k+∑k=n2n−12k−log(4)−γ;n∈ℕ \ No newline at end of file diff --git a/doc/equations/polygamma1.mml b/doc/equations/polygamma1.mml new file mode 100644 index 000000000..1b6e92ed8 --- /dev/null +++ b/doc/equations/polygamma1.mml @@ -0,0 +1,94 @@ + +]> + +polygamma1 + + + + + + + ψ + + + n + + + + + + x + + + = + + + + − + 1 + + + + n + + + 1 + + + n + ! + + ∑ + + k + = + 0 + + ∞ + + + 1 + + + + + k + + + x + + + + n + + + 1 + + + + + = + + + + ∂ + n + + ψ + + + x + + + + + + ∂ + n + + x + + + + + + \ No newline at end of file diff --git a/doc/equations/polygamma1.png b/doc/equations/polygamma1.png new file mode 100644 index 000000000..8eaaca1ac Binary files /dev/null and b/doc/equations/polygamma1.png differ diff --git a/doc/equations/polygamma1.svg b/doc/equations/polygamma1.svg new file mode 100644 index 000000000..737a855aa --- /dev/null +++ b/doc/equations/polygamma1.svg @@ -0,0 +1,2 @@ + +ψ(n)(x)=(−1)n+1n!∑k=0∞1(k+x)n+1=∂nψ(x)∂nx \ No newline at end of file diff --git a/doc/equations/polygamma2.mml b/doc/equations/polygamma2.mml new file mode 100644 index 000000000..60c9744fd --- /dev/null +++ b/doc/equations/polygamma2.mml @@ -0,0 +1,90 @@ + +]> + +polygamma2 + + + + + + + ψ + + + + n + + + + + + + 1 + − + x + + + = + + + + − + 1 + + + n + + + ψ + + + + n + + + + + + + x + + + + + + + + − + 1 + + + n + + π + + + + ∂ + n + + cot + + + π + x + + + + + ∂ + + x + n + + + + + + + \ No newline at end of file diff --git a/doc/equations/polygamma2.png b/doc/equations/polygamma2.png new file mode 100644 index 000000000..c91e5f274 Binary files /dev/null and b/doc/equations/polygamma2.png differ diff --git a/doc/equations/polygamma2.svg b/doc/equations/polygamma2.svg new file mode 100644 index 000000000..4a1c0d2c7 --- /dev/null +++ b/doc/equations/polygamma2.svg @@ -0,0 +1,2 @@ + +ψ(n)(1−x)=(−1)nψ(n)(x)+(−1)nπ∂ncot(πx)∂xn \ No newline at end of file diff --git a/doc/equations/polygamma3.mml b/doc/equations/polygamma3.mml new file mode 100644 index 000000000..3568763aa --- /dev/null +++ b/doc/equations/polygamma3.mml @@ -0,0 +1,95 @@ + +]> + +polygamma3 + + + + + + + + + ∂ + n + + cot + + + π + x + + + + + ∂ + + x + n + + + + = + + + + π + n + + + + + sin + + n + + + 1 + + + + + π + x + + + + + + + ∑ + + k + = + 0 + + + n + − + 1 + + + + C + + k + , + n + + + cos + + + k + + + + π + n + + + + + + \ No newline at end of file diff --git a/doc/equations/polygamma3.png b/doc/equations/polygamma3.png new file mode 100644 index 000000000..e3e38511f Binary files /dev/null and b/doc/equations/polygamma3.png differ diff --git a/doc/equations/polygamma3.svg b/doc/equations/polygamma3.svg new file mode 100644 index 000000000..3e05baaef --- /dev/null +++ b/doc/equations/polygamma3.svg @@ -0,0 +1,2 @@ + +∂ncot(πx)∂xn=πnsinn+1(πx)∑k=0n−1Ck,ncosk(πn) \ No newline at end of file diff --git a/doc/equations/polygamma4.mml b/doc/equations/polygamma4.mml new file mode 100644 index 000000000..2f7570b69 --- /dev/null +++ b/doc/equations/polygamma4.mml @@ -0,0 +1,113 @@ + +]> + +polygamma4 + + + + + + + ψ + + + + n + + + + + + + x + + + = + + + + + + − + 1 + + + + n + − + 1 + + + n + ! + + + x + + n + + + 1 + + + + + + + ∑ + + k + = + 0 + + ∞ + + + + + + + − + 1 + + + + k + + + n + + + 1 + + + + + k + + + n + + + ! + ζ + + + k + + + n + + + 1 + + + + x + k + + + + k + ! + + + + + + \ No newline at end of file diff --git a/doc/equations/polygamma4.png b/doc/equations/polygamma4.png new file mode 100644 index 000000000..9f013bf8a Binary files /dev/null and b/doc/equations/polygamma4.png differ diff --git a/doc/equations/polygamma4.svg b/doc/equations/polygamma4.svg new file mode 100644 index 000000000..5a2860a2e --- /dev/null +++ b/doc/equations/polygamma4.svg @@ -0,0 +1,2 @@ + +ψ(n)(x)=(−1)n−1n!xn+1+∑k=0∞(−1)k+n+1(k+n)!ζ(k+n+1)xkk! \ No newline at end of file diff --git a/doc/equations/polygamma5.mml b/doc/equations/polygamma5.mml new file mode 100644 index 000000000..1104b9560 --- /dev/null +++ b/doc/equations/polygamma5.mml @@ -0,0 +1,134 @@ + +]> + +polygamma5 + + + + + + + ψ + + + + n + + + + + + + x + + + ∝ + + + + + + − + 1 + + + + n + − + 1 + + + + + n + − + 1 + + + ! + + + n + + + 2 + x + + + + + 2 + + x + + n + + + 1 + + + + + − + + + + − + 1 + + + n + + + ∑ + + k + = + 1 + + ∞ + + + + + + 2 + k + + + n + − + 1 + + + ! + + + + + 2 + k + + + ! + + x + + 2 + k + + + n + + + + + + B + + 2 + k + + + + + + \ No newline at end of file diff --git a/doc/equations/polygamma5.png b/doc/equations/polygamma5.png new file mode 100644 index 000000000..aaf99373d Binary files /dev/null and b/doc/equations/polygamma5.png differ diff --git a/doc/equations/polygamma5.svg b/doc/equations/polygamma5.svg new file mode 100644 index 000000000..7bd89991e --- /dev/null +++ b/doc/equations/polygamma5.svg @@ -0,0 +1,2 @@ + +ψ(n)(x)∝(−1)n−1(n−1)!(n+2x)2xn+1−(−1)n∑k=1∞(2k+n−1)!(2k)!x2k+nB2k \ No newline at end of file diff --git a/doc/equations/polygamma6.mml b/doc/equations/polygamma6.mml new file mode 100644 index 000000000..47938c868 --- /dev/null +++ b/doc/equations/polygamma6.mml @@ -0,0 +1,88 @@ + +]> + +polygamma6 + + + + + + + ψ + + + + n + + + + + + + x + − + m + + + = + + ψ + + + + n + + + + + + + x + + + − + + + + − + 1 + + + n + + n + ! + + ∑ + + k + = + 1 + + m + + + 1 + + + + + x + − + k + + + + n + + + 1 + + + + + + + + \ No newline at end of file diff --git a/doc/equations/polygamma6.png b/doc/equations/polygamma6.png new file mode 100644 index 000000000..d634ae162 Binary files /dev/null and b/doc/equations/polygamma6.png differ diff --git a/doc/equations/polygamma6.svg b/doc/equations/polygamma6.svg new file mode 100644 index 000000000..0ce063c0a --- /dev/null +++ b/doc/equations/polygamma6.svg @@ -0,0 +1,2 @@ + +ψ(n)(x−m)=ψ(n)(x)−(−1)nn!∑k=1m1(x−k)n+1 \ No newline at end of file diff --git a/doc/equations/polygamma7.mml b/doc/equations/polygamma7.mml new file mode 100644 index 000000000..d3ea70c85 --- /dev/null +++ b/doc/equations/polygamma7.mml @@ -0,0 +1,107 @@ + +]> + +polygamma7 + + + + + + + ∂ + + ∂ + x + + + + + cos + + + k + + + + θ + + + + + + sin + n + + + + θ + + + + + = + + 1 + + + sin + + n + + + 1 + + + + + θ + + + + + + + + + + k + − + n + + + + cos + + n + + + 1 + + + + + θ + + + − + k + + cos + + n + − + 1 + + + + + θ + + + + + + + + + \ No newline at end of file diff --git a/doc/equations/polygamma7.png b/doc/equations/polygamma7.png new file mode 100644 index 000000000..36a0c03c8 Binary files /dev/null and b/doc/equations/polygamma7.png differ diff --git a/doc/equations/polygamma7.svg b/doc/equations/polygamma7.svg new file mode 100644 index 000000000..2f83cc0b4 --- /dev/null +++ b/doc/equations/polygamma7.svg @@ -0,0 +1,2 @@ + +∂∂xcosk(θ)sinn(θ)=1sinn+1(θ)((k−n)cosn+1(θ)−kcosn−1(θ)) \ No newline at end of file diff --git a/doc/equations/polygamma8.mml b/doc/equations/polygamma8.mml new file mode 100644 index 000000000..a76cb5f1c --- /dev/null +++ b/doc/equations/polygamma8.mml @@ -0,0 +1,52 @@ + +]> + +polygamma8 + + + + + + + ψ + + + n + + + + + + 1 + + + = + + + + − + 1 + + + + n + + + 1 + + + n + ! + ς + + + n + + + 1 + + + + + + \ No newline at end of file diff --git a/doc/equations/polygamma8.png b/doc/equations/polygamma8.png new file mode 100644 index 000000000..c39f4f668 Binary files /dev/null and b/doc/equations/polygamma8.png differ diff --git a/doc/equations/polygamma8.svg b/doc/equations/polygamma8.svg new file mode 100644 index 000000000..baa17122c --- /dev/null +++ b/doc/equations/polygamma8.svg @@ -0,0 +1,2 @@ + +ψ(n)(1)=(−1)n+1n!ς(n+1) \ No newline at end of file diff --git a/doc/equations/polygamma9.mml b/doc/equations/polygamma9.mml new file mode 100644 index 000000000..1fd598cfe --- /dev/null +++ b/doc/equations/polygamma9.mml @@ -0,0 +1,69 @@ + +]> + +polygamma9 + + + + + + + ψ + + + n + + + + + + + 1 + 2 + + + + = + + + + − + 1 + + + + n + + + 1 + + + n + ! + + + + 2 + + n + + + 1 + + + − + 1 + + + ς + + + n + + + 1 + + + + + + \ No newline at end of file diff --git a/doc/equations/polygamma9.png b/doc/equations/polygamma9.png new file mode 100644 index 000000000..505ba0e7d Binary files /dev/null and b/doc/equations/polygamma9.png differ diff --git a/doc/equations/polygamma9.svg b/doc/equations/polygamma9.svg new file mode 100644 index 000000000..1af965392 --- /dev/null +++ b/doc/equations/polygamma9.svg @@ -0,0 +1,2 @@ + +ψ(n)(12)=(−1)n+1n!(2n+1−1)ς(n+1) \ No newline at end of file diff --git a/doc/equations/trigamma1.mml b/doc/equations/trigamma1.mml new file mode 100644 index 000000000..89d09624e --- /dev/null +++ b/doc/equations/trigamma1.mml @@ -0,0 +1,69 @@ + +]> + +trigamma1 + + + + + + + ψ + + + 1 + + + + + + x + + + = + + ∑ + + k + = + 0 + + ∞ + + + 1 + + + + + k + + + x + + + 2 + + + + = + + + ∂ + ψ + + + x + + + + + ∂ + x + + + + + + \ No newline at end of file diff --git a/doc/equations/trigamma1.png b/doc/equations/trigamma1.png new file mode 100644 index 000000000..9e966e099 Binary files /dev/null and b/doc/equations/trigamma1.png differ diff --git a/doc/equations/trigamma1.svg b/doc/equations/trigamma1.svg new file mode 100644 index 000000000..9ff41b44c --- /dev/null +++ b/doc/equations/trigamma1.svg @@ -0,0 +1,2 @@ + +ψ(1)(x)=∑k=0∞1(k+x)2=∂ψ(x)∂x \ No newline at end of file diff --git a/doc/equations/trigamma2.mml b/doc/equations/trigamma2.mml new file mode 100644 index 000000000..37ac715c7 --- /dev/null +++ b/doc/equations/trigamma2.mml @@ -0,0 +1,65 @@ + +]> + +trigamma2 + + + + + + + ψ + + + 1 + + + + + + 1 + − + x + + + = + + + + π + 2 + + + + + sin + 2 + + + + π + x + + + + + − + + ψ + + + 1 + + + + + + x + + + + + + \ No newline at end of file diff --git a/doc/equations/trigamma2.png b/doc/equations/trigamma2.png new file mode 100644 index 000000000..45676ec18 Binary files /dev/null and b/doc/equations/trigamma2.png differ diff --git a/doc/equations/trigamma2.svg b/doc/equations/trigamma2.svg new file mode 100644 index 000000000..68f4d9510 --- /dev/null +++ b/doc/equations/trigamma2.svg @@ -0,0 +1,2 @@ + +ψ(1)(1−x)=π2sin2(πx)−ψ(1)(x) \ No newline at end of file diff --git a/doc/equations/trigamma3.mml b/doc/equations/trigamma3.mml new file mode 100644 index 000000000..934572d82 --- /dev/null +++ b/doc/equations/trigamma3.mml @@ -0,0 +1,54 @@ + +]> + +trigamma3 + + + + + + + ψ + + + 1 + + + + + + x + + + = + + ψ + + + 1 + + + + + + 1 + + + x + + + + + + 1 + + + x + 2 + + + + + + + \ No newline at end of file diff --git a/doc/equations/trigamma3.png b/doc/equations/trigamma3.png new file mode 100644 index 000000000..38aceee22 Binary files /dev/null and b/doc/equations/trigamma3.png differ diff --git a/doc/equations/trigamma3.svg b/doc/equations/trigamma3.svg new file mode 100644 index 000000000..121ab5517 --- /dev/null +++ b/doc/equations/trigamma3.svg @@ -0,0 +1,2 @@ + +ψ(1)(x)=ψ(1)(1+x)+1x2 \ No newline at end of file diff --git a/doc/equations/trigamma4.mml b/doc/equations/trigamma4.mml new file mode 100644 index 000000000..557d16d0e --- /dev/null +++ b/doc/equations/trigamma4.mml @@ -0,0 +1,49 @@ + +]> + +trigamma4 + + + + + + + ψ + + + 1 + + + + + + x + + + = + + + + + C + + + R + + + x + + + + + + + x + 2 + + + + + + \ No newline at end of file diff --git a/doc/equations/trigamma4.png b/doc/equations/trigamma4.png new file mode 100644 index 000000000..2a527a420 Binary files /dev/null and b/doc/equations/trigamma4.png differ diff --git a/doc/equations/trigamma4.svg b/doc/equations/trigamma4.svg new file mode 100644 index 000000000..1ccac5dd3 --- /dev/null +++ b/doc/equations/trigamma4.svg @@ -0,0 +1,2 @@ + +ψ(1)(x)=(C+R(x))x2 \ No newline at end of file diff --git a/doc/equations/trigamma5.mml b/doc/equations/trigamma5.mml new file mode 100644 index 000000000..4a8736435 --- /dev/null +++ b/doc/equations/trigamma5.mml @@ -0,0 +1,49 @@ + +]> + +trigamma5 + + + + + + + ψ + + + 1 + + + + + + x + + + = + + + + + 1 + + + R + + + + 1 + x + + + + + + + x + + + + + \ No newline at end of file diff --git a/doc/equations/trigamma5.png b/doc/equations/trigamma5.png new file mode 100644 index 000000000..671a4b324 Binary files /dev/null and b/doc/equations/trigamma5.png differ diff --git a/doc/equations/trigamma5.svg b/doc/equations/trigamma5.svg new file mode 100644 index 000000000..3813ab577 --- /dev/null +++ b/doc/equations/trigamma5.svg @@ -0,0 +1,2 @@ + +ψ(1)(x)=(1+R(1x))x \ No newline at end of file diff --git a/doc/equations/zeta1.svg b/doc/equations/zeta1.svg index 2e8015703..c9b001711 100644 --- a/doc/equations/zeta1.svg +++ b/doc/equations/zeta1.svg @@ -1,2 +1,2 @@ -ζ(s)=∑k=1∞1ks \ No newline at end of file +ζ(s)=∑k=1∞1ks \ No newline at end of file diff --git a/doc/equations/zeta2.svg b/doc/equations/zeta2.svg index bf5fc046c..cb9f3d59b 100644 --- a/doc/equations/zeta2.svg +++ b/doc/equations/zeta2.svg @@ -1,2 +1,2 @@ -ζ(s)=11−21−s∑n=0∞12n+1∑k=0n(−1)k(nk)(k+1)−s \ No newline at end of file +ζ(s)=11−21−s∑n=0∞12n+1∑k=0n(−1)k(nk)(k+1)−s \ No newline at end of file diff --git a/doc/equations/zeta3.svg b/doc/equations/zeta3.svg index e7e71543b..694ece17d 100644 --- a/doc/equations/zeta3.svg +++ b/doc/equations/zeta3.svg @@ -1,2 +1,2 @@ -ζ(1−s)=2sin(π1−s2)(2π−s)Γ(s)ζ(s) \ No newline at end of file +ζ(1−s)=2sin(π1−s2)(2π−s)Γ(s)ζ(s) \ No newline at end of file diff --git a/doc/equations/zeta4.svg b/doc/equations/zeta4.svg index 2b6ce6eca..10aa94d12 100644 --- a/doc/equations/zeta4.svg +++ b/doc/equations/zeta4.svg @@ -1,2 +1,2 @@ -ς(s)=C+R(1−s)−s1−s \ No newline at end of file +ς(s)=C+R(1−s)−s1−s \ No newline at end of file diff --git a/doc/equations/zeta5.svg b/doc/equations/zeta5.svg index 8dd13f008..704710fcd 100644 --- a/doc/equations/zeta5.svg +++ b/doc/equations/zeta5.svg @@ -1,2 +1,2 @@ -ς(s)=C+R(s−n)+1s−1 \ No newline at end of file +ς(s)=C+R(s−n)+1s−1 \ No newline at end of file diff --git a/doc/equations/zeta6.svg b/doc/equations/zeta6.svg index 03dc8c806..911760732 100644 --- a/doc/equations/zeta6.svg +++ b/doc/equations/zeta6.svg @@ -1,2 +1,2 @@ -ζ(s)=−1sn(1−21−s)∑j=02n−1ej(j+1)s+γ(s);ej=(−1)j(∑k=0j−nn!k!(n−k)!−2n)∧|γn(s)|<=18n|1−21−s| \ No newline at end of file +ζ(s)=−1sn(1−21−s)∑j=02n−1ej(j+1)s+γ(s);ej=(−1)j(∑k=0j−nn!k!(n−k)!−2n)∧|γn(s)|<=18n|1−21−s| \ No newline at end of file diff --git a/doc/equations/zeta7.mml b/doc/equations/zeta7.mml new file mode 100644 index 000000000..406f4209b --- /dev/null +++ b/doc/equations/zeta7.mml @@ -0,0 +1,53 @@ + +]> + +zeta7 + + + + + + ζ + + + − + n + + + = + + + + + − + 1 + + + n + + + n + + + 1 + + + + B + + n + + + 1 + + + + ; + + n + ∈ + ℕ + + + + \ No newline at end of file diff --git a/doc/equations/zeta7.png b/doc/equations/zeta7.png new file mode 100644 index 000000000..536b9ace6 Binary files /dev/null and b/doc/equations/zeta7.png differ diff --git a/doc/equations/zeta7.svg b/doc/equations/zeta7.svg new file mode 100644 index 000000000..601a95ddf --- /dev/null +++ b/doc/equations/zeta7.svg @@ -0,0 +1,2 @@ + +ζ(−n)=(−1)nn+1Bn+1;n∈ℕ \ No newline at end of file diff --git a/doc/equations/zeta8.mml b/doc/equations/zeta8.mml new file mode 100644 index 000000000..70cd0ce62 --- /dev/null +++ b/doc/equations/zeta8.mml @@ -0,0 +1,31 @@ + +]> + +zeta8 + + + + + + ζ + + + − + 2 + n + + + = + 0 + + ; + + n + ∈ + ℕ + + + + \ No newline at end of file diff --git a/doc/equations/zeta8.png b/doc/equations/zeta8.png new file mode 100644 index 000000000..b1ef089b7 Binary files /dev/null and b/doc/equations/zeta8.png differ diff --git a/doc/equations/zeta8.svg b/doc/equations/zeta8.svg new file mode 100644 index 000000000..c6965a607 --- /dev/null +++ b/doc/equations/zeta8.svg @@ -0,0 +1,2 @@ + +ζ(−2n)=0;n∈ℕ \ No newline at end of file diff --git a/doc/equations/zeta9.mml b/doc/equations/zeta9.mml new file mode 100644 index 000000000..a1e3e2e4d --- /dev/null +++ b/doc/equations/zeta9.mml @@ -0,0 +1,78 @@ + +]> + +zeta9 + + + + + + ζ + + + 2 + n + + + = + + + + + + − + 1 + + + + n + − + 1 + + + + 2 + + 2 + n + − + 1 + + + + π + + 2 + n + + + + + + + 2 + n + + + ! + + + + B + + 2 + n + + + + ; + + n + ∈ + ℕ + + + + \ No newline at end of file diff --git a/doc/equations/zeta9.png b/doc/equations/zeta9.png new file mode 100644 index 000000000..191c31247 Binary files /dev/null and b/doc/equations/zeta9.png differ diff --git a/doc/equations/zeta9.svg b/doc/equations/zeta9.svg new file mode 100644 index 000000000..fb39389e2 --- /dev/null +++ b/doc/equations/zeta9.svg @@ -0,0 +1,2 @@ + +ζ(2n)=(−1)n−122n−1π2n(2n)!B2n;n∈ℕ \ No newline at end of file diff --git a/doc/graphs/generate.sh b/doc/graphs/generate.sh index bff99777e..d9659141f 100755 --- a/doc/graphs/generate.sh +++ b/doc/graphs/generate.sh @@ -8,15 +8,15 @@ # Paths to tools come first, change these to match your system: # math2svg='m:\download\open\SVGMath-0.3.1\math2svg.py' -python=/cygdrive/c/Python26/python.exe -inkscape=/cygdrive/c/progra~1/Inkscape/inkscape +python='/cygdrive/c/program files/Python27/python.exe' +inkscape='/cygdrive/c/Program Files (x86)/Inkscape/inkscape.exe' # Image DPI: dpi=96 for svgfile in $*; do pngfile=$(basename $svgfile .svg).png echo Generating $pngfile - $inkscape -d $dpi -e $(cygpath -a -w $pngfile) $(cygpath -a -w $svgfile) + "$inkscape" -d $dpi -e $(cygpath -a -w $pngfile) $(cygpath -a -w $svgfile) done diff --git a/doc/graphs/polygamma2.png b/doc/graphs/polygamma2.png new file mode 100644 index 000000000..b292a7ee9 Binary files /dev/null and b/doc/graphs/polygamma2.png differ diff --git a/doc/graphs/polygamma2.svg b/doc/graphs/polygamma2.svg new file mode 100644 index 000000000..733112c61 --- /dev/null +++ b/doc/graphs/polygamma2.svg @@ -0,0 +1,70 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +0 +5 +0 +-5 + +0 +10 +20 +30 +40 +50 +0 +-10 +-20 +-30 +-40 +-50 + +polygamma(2, x) + +x + + + + + + + + + + +Polygamma + + + diff --git a/doc/graphs/polygamma3.png b/doc/graphs/polygamma3.png new file mode 100644 index 000000000..03f765920 Binary files /dev/null and b/doc/graphs/polygamma3.png differ diff --git a/doc/graphs/polygamma3.svg b/doc/graphs/polygamma3.svg new file mode 100644 index 000000000..79c57f591 --- /dev/null +++ b/doc/graphs/polygamma3.svg @@ -0,0 +1,67 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +0 +5 +0 +-5 + +0 +100 +200 +300 +400 +500 +600 +700 +800 + +polygamma(3, x) + +x + + + + + + + + + + +Polygamma + + + diff --git a/doc/graphs/sf_graphs.cpp b/doc/graphs/sf_graphs.cpp index 9bae8db8e..713665c60 100644 --- a/doc/graphs/sf_graphs.cpp +++ b/doc/graphs/sf_graphs.cpp @@ -220,10 +220,10 @@ double find_end_point(F f, double x0, double target, bool rising, double x_off = { boost::math::tools::eps_tolerance tol(50); boost::uintmax_t max_iter = 1000; - return boost::math::tools::bracket_and_solve_root( + return x_off + boost::math::tools::bracket_and_solve_root( location_finder(f, target, x_off), x0, - double(1.5), + 1.5, rising, tol, max_iter).first; @@ -248,10 +248,11 @@ int main() double (*f2i)(int, double); double (*f3)(double, double, double); double (*f4)(double, double, double, double); + double max_val; f = boost::math::zeta; - plot.add(f, 1 + find_end_point(f, 0.1, 40.0, false, 1.0), 10, ""); - plot.add(f, -20, 1 + find_end_point(f, -0.1, -40.0, false, 1.0), ""); + plot.add(f, find_end_point(f, 0.1, 40.0, false, 1.0), 10, ""); + plot.add(f, -20, find_end_point(f, -0.1, -40.0, false, 1.0), ""); plot.plot("Zeta Function Over [-20,10]", "zeta1.svg", "z", "zeta(z)"); plot.clear(); @@ -259,34 +260,34 @@ int main() plot.plot("Zeta Function Over [-14,0]", "zeta2.svg", "z", "zeta(z)"); f = boost::math::tgamma; - double max_val = f(6); + max_val = f(6); plot.clear(); plot.add(f, find_end_point(f, 0.1, max_val, false), 6, ""); - plot.add(f, -1 + find_end_point(f, 0.1, -max_val, true, -1), find_end_point(f, -0.1, -max_val, false), ""); - plot.add(f, -2 + find_end_point(f, 0.1, max_val, false, -2), -1 + find_end_point(f, -0.1, max_val, true, -1), ""); - plot.add(f, -3 + find_end_point(f, 0.1, -max_val, true, -3), -2 + find_end_point(f, -0.1, -max_val, false, -2), ""); - plot.add(f, -4 + find_end_point(f, 0.1, max_val, false, -4), -3 + find_end_point(f, -0.1, max_val, true, -3), ""); + plot.add(f, find_end_point(f, 0.1, -max_val, true, -1), find_end_point(f, -0.1, -max_val, false), ""); + plot.add(f, find_end_point(f, 0.1, max_val, false, -2), find_end_point(f, -0.1, max_val, true, -1), ""); + plot.add(f, find_end_point(f, 0.1, -max_val, true, -3), find_end_point(f, -0.1, -max_val, false, -2), ""); + plot.add(f, find_end_point(f, 0.1, max_val, false, -4), find_end_point(f, -0.1, max_val, true, -3), ""); plot.plot("tgamma", "tgamma.svg", "z", "tgamma(z)"); f = boost::math::lgamma; max_val = f(10); plot.clear(); plot.add(f, find_end_point(f, 0.1, max_val, false), 10, ""); - plot.add(f, -1 + find_end_point(f, 0.1, max_val, false, -1), find_end_point(f, -0.1, max_val, true), ""); - plot.add(f, -2 + find_end_point(f, 0.1, max_val, false, -2), -1 + find_end_point(f, -0.1, max_val, true, -1), ""); - plot.add(f, -3 + find_end_point(f, 0.1, max_val, false, -3), -2 + find_end_point(f, -0.1, max_val, true, -2), ""); - plot.add(f, -4 + find_end_point(f, 0.1, max_val, false, -4), -3 + find_end_point(f, -0.1, max_val, true, -3), ""); - plot.add(f, -5 + find_end_point(f, 0.1, max_val, false, -5), -4 + find_end_point(f, -0.1, max_val, true, -4), ""); + plot.add(f, find_end_point(f, 0.1, max_val, false, -1), find_end_point(f, -0.1, max_val, true), ""); + plot.add(f, find_end_point(f, 0.1, max_val, false, -2), find_end_point(f, -0.1, max_val, true, -1), ""); + plot.add(f, find_end_point(f, 0.1, max_val, false, -3), find_end_point(f, -0.1, max_val, true, -2), ""); + plot.add(f, find_end_point(f, 0.1, max_val, false, -4), find_end_point(f, -0.1, max_val, true, -3), ""); + plot.add(f, find_end_point(f, 0.1, max_val, false, -5), find_end_point(f, -0.1, max_val, true, -4), ""); plot.plot("lgamma", "lgamma.svg", "z", "lgamma(z)"); f = boost::math::digamma; max_val = 10; plot.clear(); plot.add(f, find_end_point(f, 0.1, -max_val, true), 10, ""); - plot.add(f, -1 + find_end_point(f, 0.1, -max_val, true, -1), find_end_point(f, -0.1, max_val, true), ""); - plot.add(f, -2 + find_end_point(f, 0.1, -max_val, true, -2), -1 + find_end_point(f, -0.1, max_val, true, -1), ""); - plot.add(f, -3 + find_end_point(f, 0.1, -max_val, true, -3), -2 + find_end_point(f, -0.1, max_val, true, -2), ""); - plot.add(f, -4 + find_end_point(f, 0.1, -max_val, true, -4), -3 + find_end_point(f, -0.1, max_val, true, -3), ""); + plot.add(f, find_end_point(f, 0.1, -max_val, true, -1), find_end_point(f, -0.1, max_val, true), ""); + plot.add(f, find_end_point(f, 0.1, -max_val, true, -2), find_end_point(f, -0.1, max_val, true, -1), ""); + plot.add(f, find_end_point(f, 0.1, -max_val, true, -3), find_end_point(f, -0.1, max_val, true, -2), ""); + plot.add(f, find_end_point(f, 0.1, -max_val, true, -4), find_end_point(f, -0.1, max_val, true, -3), ""); plot.plot("digamma", "digamma.svg", "z", "digamma(z)"); f = boost::math::erf; @@ -300,16 +301,16 @@ int main() f = boost::math::erf_inv; plot.clear(); - plot.add(f, -1 + find_end_point(f, 0.1, -3, true, -1), 1 + find_end_point(f, -0.1, 3, true, 1), ""); + plot.add(f, find_end_point(f, 0.1, -3, true, -1), find_end_point(f, -0.1, 3, true, 1), ""); plot.plot("erf_inv", "erf_inv.svg", "z", "erf_inv(z)"); f = boost::math::erfc_inv; plot.clear(); - plot.add(f, find_end_point(f, 0.1, 3, false), 2 + find_end_point(f, -0.1, -3, false, 2), ""); + plot.add(f, find_end_point(f, 0.1, 3, false), find_end_point(f, -0.1, -3, false, 2), ""); plot.plot("erfc_inv", "erfc_inv.svg", "z", "erfc_inv(z)"); f = boost::math::log1p; plot.clear(); - plot.add(f, -1 + find_end_point(f, 0.1, -10, true, -1), 10, ""); + plot.add(f, find_end_point(f, 0.1, -10, true, -1), 10, ""); plot.plot("log1p", "log1p.svg", "z", "log1p(z)"); f = boost::math::expm1; @@ -324,7 +325,7 @@ int main() f = sqrt1pm1; plot.clear(); - plot.add(f, -1 + find_end_point(f, 0.1, -10, true, -1), 5, ""); + plot.add(f, find_end_point(f, 0.1, -10, true, -1), 5, ""); plot.plot("sqrt1pm1", "sqrt1pm1.svg", "z", "sqrt1pm1(z)"); f2 = boost::math::powm1; @@ -359,7 +360,7 @@ int main() f = boost::math::atanh; plot.clear(); - plot.add(f, -1 + find_end_point(f, 0.1, -5, true, -1), 1 + find_end_point(f, -0.1, 5, true, 1), ""); + plot.add(f, find_end_point(f, 0.1, -5, true, -1), find_end_point(f, -0.1, 5, true, 1), ""); plot.plot("atanh", "atanh.svg", "z", "atanh(z)"); f2 = boost::math::tgamma_delta_ratio; @@ -450,15 +451,15 @@ int main() "n = 2"); plot.add(boost::bind(f2u, 3, _1), find_end_point(boost::bind(f2u, 3, _1), -2, 20, false), - 8 + find_end_point(boost::bind(f2u, 3, _1), 1, 20, false, 8), + find_end_point(boost::bind(f2u, 3, _1), 1, 20, false, 8), "n = 3"); plot.add(boost::bind(f2u, 4, _1), find_end_point(boost::bind(f2u, 4, _1), -2, 20, false), - 8 + find_end_point(boost::bind(f2u, 4, _1), 1, 20, true, 8), + find_end_point(boost::bind(f2u, 4, _1), 1, 20, true, 8), "n = 4"); plot.add(boost::bind(f2u, 5, _1), find_end_point(boost::bind(f2u, 5, _1), -2, 20, false), - 8 + find_end_point(boost::bind(f2u, 5, _1), 1, 20, true, 8), + find_end_point(boost::bind(f2u, 5, _1), 1, 20, true, 8), "n = 5"); plot.plot("Laguerre Polynomials", "laguerre.svg", "x", "laguerre(n, x)"); @@ -555,17 +556,17 @@ int main() plot.add(boost::bind(f3, _1, 0.25, boost::math::constants::pi() / 2), find_end_point( boost::bind(f3, _1, 0.25, boost::math::constants::pi() / 2), - 0.5, 4, false, -1) - 1, + 0.5, 4, false, -1), find_end_point( boost::bind(f3, _1, 0.25, boost::math::constants::pi() / 2), - -0.5, 4, true, 1) + 1, "n=0.25 φ=π/2"); + -0.5, 4, true, 1), "n=0.25 φ=π/2"); plot.add(boost::bind(f3, _1, 0.75, boost::math::constants::pi() / 2), find_end_point( boost::bind(f3, _1, 0.75, boost::math::constants::pi() / 2), - 0.5, 4, false, -1) - 1, + 0.5, 4, false, -1), find_end_point( boost::bind(f3, _1, 0.75, boost::math::constants::pi() / 2), - -0.5, 4, true, 1) + 1, "n=0.75 φ=π/2"); + -0.5, 4, true, 1), "n=0.75 φ=π/2"); plot.plot("Elliptic Of the Third Kind", "ellint_3.svg", "k", "ellint_3(k, n, phi)"); f2 = boost::math::jacobi_sn; @@ -696,6 +697,39 @@ int main() plot.add(f, -20, 3, ""); plot.plot("Bi'", "airy_bip.svg", "z", "airy_bi_prime(z)"); + f = boost::math::trigamma; + max_val = 30; + plot.clear(); + plot.add(f, find_end_point(f, 0.1, max_val, false), 5, ""); + plot.add(f, find_end_point(f, 0.1, max_val, false, -1), find_end_point(f, -0.1, max_val, true), ""); + plot.add(f, find_end_point(f, 0.1, max_val, false, -2), find_end_point(f, -0.1, max_val, true, -1), ""); + plot.add(f, find_end_point(f, 0.1, max_val, false, -3), find_end_point(f, -0.1, max_val, true, -2), ""); + plot.add(f, find_end_point(f, 0.1, max_val, false, -4), find_end_point(f, -0.1, max_val, true, -3), ""); + plot.add(f, find_end_point(f, 0.1, max_val, false, -5), find_end_point(f, -0.1, max_val, true, -4), ""); + plot.plot("Trigamma", "trigamma.svg", "x", "trigamma(x)"); + + f2i = boost::math::polygamma; + max_val = -50; + plot.clear(); + plot.add(boost::bind(f2i, 2, _1), find_end_point(boost::bind(f2i, 2, _1), 0.1, max_val, true), 5, ""); + plot.add(boost::bind(f2i, 2, _1), find_end_point(boost::bind(f2i, 2, _1), 0.1, max_val, true, -1), find_end_point(boost::bind(f2i, 2, _1), -0.1, -max_val, true), ""); + plot.add(boost::bind(f2i, 2, _1), find_end_point(boost::bind(f2i, 2, _1), 0.1, max_val, true, -2), find_end_point(boost::bind(f2i, 2, _1), -0.1, -max_val, true, -1), ""); + plot.add(boost::bind(f2i, 2, _1), find_end_point(boost::bind(f2i, 2, _1), 0.1, max_val, true, -3), find_end_point(boost::bind(f2i, 2, _1), -0.1, -max_val, true, -2), ""); + plot.add(boost::bind(f2i, 2, _1), find_end_point(boost::bind(f2i, 2, _1), 0.1, max_val, true, -4), find_end_point(boost::bind(f2i, 2, _1), -0.1, -max_val, true, -3), ""); + plot.add(boost::bind(f2i, 2, _1), find_end_point(boost::bind(f2i, 2, _1), 0.1, max_val, true, -5), find_end_point(boost::bind(f2i, 2, _1), -0.1, -max_val, true, -4), ""); + plot.add(boost::bind(f2i, 2, _1), find_end_point(boost::bind(f2i, 2, _1), 0.1, max_val, true, -6), find_end_point(boost::bind(f2i, 2, _1), -0.1, -max_val, true, -5), ""); + plot.plot("Polygamma", "polygamma2.svg", "x", "polygamma(2, x)"); + + max_val = 800; + plot.clear(); + plot.add(boost::bind(f2i, 3, _1), find_end_point(boost::bind(f2i, 3, _1), 0.1, max_val, false), 5, ""); + plot.add(boost::bind(f2i, 3, _1), find_end_point(boost::bind(f2i, 3, _1), 0.1, max_val, false, -1), find_end_point(boost::bind(f2i, 3, _1), -0.1, max_val, true), ""); + plot.add(boost::bind(f2i, 3, _1), find_end_point(boost::bind(f2i, 3, _1), 0.1, max_val, false, -2), find_end_point(boost::bind(f2i, 3, _1), -0.1, max_val, true, -1), ""); + plot.add(boost::bind(f2i, 3, _1), find_end_point(boost::bind(f2i, 3, _1), 0.1, max_val, false, -3), find_end_point(boost::bind(f2i, 3, _1), -0.1, max_val, true, -2), ""); + plot.add(boost::bind(f2i, 3, _1), find_end_point(boost::bind(f2i, 3, _1), 0.1, max_val, false, -4), find_end_point(boost::bind(f2i, 3, _1), -0.1, max_val, true, -3), ""); + plot.add(boost::bind(f2i, 3, _1), find_end_point(boost::bind(f2i, 3, _1), 0.1, max_val, false, -5), find_end_point(boost::bind(f2i, 3, _1), -0.1, max_val, true, -4), ""); + plot.add(boost::bind(f2i, 3, _1), find_end_point(boost::bind(f2i, 3, _1), 0.1, max_val, false, -6), find_end_point(boost::bind(f2i, 3, _1), -0.1, max_val, true, -5), ""); + plot.plot("Polygamma", "polygamma3.svg", "x", "polygamma(3, x)"); return 0; } diff --git a/doc/graphs/trigamma.png b/doc/graphs/trigamma.png new file mode 100644 index 000000000..3c00ff602 Binary files /dev/null and b/doc/graphs/trigamma.png differ diff --git a/doc/graphs/trigamma.svg b/doc/graphs/trigamma.svg new file mode 100644 index 000000000..faee953d0 --- /dev/null +++ b/doc/graphs/trigamma.svg @@ -0,0 +1,70 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +0 +1 +2 +3 +4 +5 +0 +-1 +-2 +-3 +-4 + +5 +10 +15 +20 +25 +30 + +trigamma(x) + +x + + + + + + + + + +Trigamma + + + diff --git a/doc/html/index.html b/doc/html/index.html index 42ed0f6be..be501b636 100644 --- a/doc/html/index.html +++ b/doc/html/index.html @@ -111,7 +111,7 @@ This manual is also available in -

Last revised: October 30, 2014 at 11:29:37 GMT

+

Last revised: November 23, 2014 at 13:16:35 GMT

diff --git a/doc/html/indexes/s01.html b/doc/html/indexes/s01.html index 66a6067a9..db296919b 100644 --- a/doc/html/indexes/s01.html +++ b/doc/html/indexes/s01.html @@ -24,7 +24,7 @@

-Function Index

+Function Index

A B C D E F G H I J K L M N O P Q R S T U V Z

@@ -1863,6 +1863,7 @@

  • Non-Member Properties

    • +

  • polygamma

  • powm1

  • prime

    @@ -2360,6 +2361,7 @@

  • Ratios of Gamma Functions

    • +

  • trigamma

  • trunc

      diff --git a/doc/html/indexes/s02.html b/doc/html/indexes/s02.html index a4b032dac..3cc591917 100644 --- a/doc/html/indexes/s02.html +++ b/doc/html/indexes/s02.html @@ -24,7 +24,7 @@

    -Class Index

    +Class Index

    B C D E F G H I L M N O P Q R S T U W

    diff --git a/doc/html/indexes/s03.html b/doc/html/indexes/s03.html index ef3f5ee86..73eff3226 100644 --- a/doc/html/indexes/s03.html +++ b/doc/html/indexes/s03.html @@ -24,7 +24,7 @@

    -Typedef Index

    +Typedef Index

    A B C D E F G H I L N O P R S T U V W

    diff --git a/doc/html/indexes/s04.html b/doc/html/indexes/s04.html index 5f971ea77..53a13e325 100644 --- a/doc/html/indexes/s04.html +++ b/doc/html/indexes/s04.html @@ -24,7 +24,7 @@

    -Macro Index

    +Macro Index

    B F

    diff --git a/doc/html/indexes/s05.html b/doc/html/indexes/s05.html index bea2fdf7c..dae02fd4e 100644 --- a/doc/html/indexes/s05.html +++ b/doc/html/indexes/s05.html @@ -23,7 +23,7 @@

    -Index

    +Index

    A B C D E F G H I J K L M N O P Q R S T U V W Z

    @@ -4256,6 +4256,8 @@

  • Weibull Distribution

    • +

  • Polygamma

    • +

  • polygamma

  • Polynomial and Rational Function Evaluation

    • +

  • Trigamma

    • +

  • trigamma

  • trunc

      diff --git a/doc/html/math_toolkit/conventions.html b/doc/html/math_toolkit/conventions.html index e6a10ae56..a2dcab193 100644 --- a/doc/html/math_toolkit/conventions.html +++ b/doc/html/math_toolkit/conventions.html @@ -27,7 +27,7 @@ Document Conventions

    - +

    This documentation aims to use of the following naming and formatting conventions. diff --git a/doc/html/math_toolkit/navigation.html b/doc/html/math_toolkit/navigation.html index d65ceabdb..840a9ffea 100644 --- a/doc/html/math_toolkit/navigation.html +++ b/doc/html/math_toolkit/navigation.html @@ -27,7 +27,7 @@ Navigation

    - +

    Boost.Math documentation is provided in both HTML and PDF formats. diff --git a/doc/html/math_toolkit/sf_gamma.html b/doc/html/math_toolkit/sf_gamma.html index 395d73769..1d4bb80a0 100644 --- a/doc/html/math_toolkit/sf_gamma.html +++ b/doc/html/math_toolkit/sf_gamma.html @@ -30,6 +30,8 @@

    Gamma
    Log Gamma
    Digamma
    +
    Trigamma
    +
    Polygamma
    Ratios of Gamma Functions
    Incomplete Gamma Functions
    Incomplete Gamma Function diff --git a/doc/html/math_toolkit/sf_gamma/digamma.html b/doc/html/math_toolkit/sf_gamma/digamma.html index 58fe9c37f..864f35de2 100644 --- a/doc/html/math_toolkit/sf_gamma/digamma.html +++ b/doc/html/math_toolkit/sf_gamma/digamma.html @@ -7,7 +7,7 @@ - + @@ -20,7 +20,7 @@
    -PrevUpHomeNext +PrevUpHomeNext

    @@ -62,11 +62,6 @@ what level of precision to use etc. Refer to the policy documentation for more details.

    -

    - There is no fully generic version of this function: all the implementations - are tuned to specific accuracy levels, the most precise of which delivers - 34-digits of precision. -

    The return type of this function is computed using the result type calculation rules: the result is of type double when T is an integer type, and type @@ -319,6 +314,17 @@ and BIG=20 for 128-bit reals allows the series to truncated after a suitably small number of terms and evaluated as a polynomial in 1/(x*x).

    +

    + The arbitrary precision version of this function uses recurrence relations + until x > BIG, and then evaluation via the asymptotic expansion above. + As special cases integer and half integer arguments are handled via: +

    +

    + +

    +

    + +

    The rational approximation devised by JM in the range [1,2] is derived as follows. @@ -379,7 +385,7 @@

    -PrevUpHomeNext +PrevUpHomeNext
    diff --git a/doc/html/math_toolkit/sf_gamma/gamma_ratios.html b/doc/html/math_toolkit/sf_gamma/gamma_ratios.html index b2a44d905..19163291f 100644 --- a/doc/html/math_toolkit/sf_gamma/gamma_ratios.html +++ b/doc/html/math_toolkit/sf_gamma/gamma_ratios.html @@ -6,7 +6,7 @@ - + @@ -20,7 +20,7 @@
    -PrevUpHomeNext +PrevUpHomeNext

    @@ -349,7 +349,7 @@
    -PrevUpHomeNext +PrevUpHomeNext
    diff --git a/doc/html/math_toolkit/sf_gamma/polygamma.html b/doc/html/math_toolkit/sf_gamma/polygamma.html new file mode 100644 index 000000000..dd086ac8e --- /dev/null +++ b/doc/html/math_toolkit/sf_gamma/polygamma.html @@ -0,0 +1,288 @@ + + + +Polygamma + + + + + + + + + + + + + + + +
    Boost C++ LibrariesHomeLibrariesPeopleFAQMore
    +
    +
    +PrevUpHomeNext +
    +
    +
    +
    + + Synopsis +
    +
    #include <boost/math/special_functions/polygamma.hpp>
+
    +
    namespace boost{ namespace math{
+
+template <class T>
+calculated-result-type polygamma(int n, T z);
+
+template <class T, class Policy>
+calculated-result-type polygamma(int n, T z, const Policy&);
+
+}} // namespaces
+
    +
    + + Description +
    +

    + Returns the polygamma function of x. Polygamma is defined + as the n'th derivative of the digamma function: +

    +

    + +

    +

    + The following graphs illustrate the behaviour of the function for odd and + even order: +

    +

    + + +

    +

    + The final Policy argument is optional and can + be used to control the behaviour of the function: how it handles errors, + what level of precision to use etc. Refer to the policy + documentation for more details. +

    +

    + The return type of this function is computed using the result + type calculation rules: the result is of type double when T is an integer type, and type + T otherwise. +

    +
    + + Accuracy +
    +

    + The following table shows the peak errors (in units of epsilon) found on + various platforms with various floating point types. Unless otherwise specified + any floating point type that is narrower than the one shown will have effectively zero error. +

    +
    ++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + +
    +

    + Significand Size +

    +     		+

    + Platform and Compiler +

    +     		+

    + Small-medium positive arguments +

    +     		+

    + Small-medium negative x +

    +
    +

    + 53 +

    +     		+

    + Win32 Visual C++ 12 +

    +     		+

    + Peak=5.0 Mean=1 +

    +     		+

    + Peak=1200 Mean=65 +

    +
    +

    + 64 +

    +     		+

    + Win64 Mingw GCC +

    +     		+

    + Peak=16 Mean=3 +

    +     		+

    + Peak=33 Mean=3 +

    +
    +

    + 113 +

    +     		+

    + Win64 Mingw GCC __float128 +

    +     		+

    + Peak=6.5 Mean=1 +

    +     		+

    + Peak=30 Mean=4 +

    +
    +

    + As shown above, error rates are generally very acceptable for moderately + sized arguments. Error rates should stay low for exact inputs, however, please + note that the function becomes exceptionally sensitive to small changes in + input for large n and negative x, indeed for cases where n! + would overflow, the function changes directly from -∞ to +∞ somewhere between + each negative integer - these cases are not handled correctly. +

    +

    + For these reasons results should be treated with extreme + caution when n is large and x negative. +

    +
    + + Testing +
    +

    + Testing is against Mathematica generated spot values to 35 digit precision. +

    +
    + + Implementation +
    +

    + For x < 0 the following reflection formula is used: +

    +

    + +

    +

    + The n'th derivative of cot(x) is tabulated for small + n, and for larger n has the general form: +

    +

    + +

    +

    + The coefficients of the cosine terms can be calculated iteratively starting + from C1,0 = -1 and then using +

    +

    + +

    +

    + to generate coefficients for n+1. +

    +

    + Note that every other coefficient is zero, and therefore what we have are + even or odd polynomials depending on whether n is even or odd. +

    +

    + Once x is positive then we have two methods available to us, for small x + we use the series expansion: +

    +

    + +

    +

    + Note that as this is an alternating series we can accelerate the series using + Algorithm 1 from "Convergence Acceleration of Alternating + Series", Henri Cohen, Fernando Rodriguez Villegas, and Don Zagier, Experimental + Mathematics, 1999. Also note that the evaluation of zeta functions + at integer values is essentially a table lookup as zeta + is optimized for those cases. +

    +

    + For large x we use the asymptotic expansion: +

    +

    + +

    +

    + For x in-between the two extremes we use the relation: +

    +

    + +

    +

    + to make x large enough for the asymptotic expansion to be used. +

    +

    + There are also two special cases: +

    +

    + +

    +

    + +

    +
    + + + +
    Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, + Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert + Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta, + Thijs van den Berg, Daryle Walker and Xiaogang Zhang

    + Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

    +
    +
    +
    +PrevUpHomeNext +
    + + diff --git a/doc/html/math_toolkit/sf_gamma/trigamma.html b/doc/html/math_toolkit/sf_gamma/trigamma.html new file mode 100644 index 000000000..e7ad9a6ba --- /dev/null +++ b/doc/html/math_toolkit/sf_gamma/trigamma.html @@ -0,0 +1,217 @@ + + + +Trigamma + + + + + + + + + + + + + + + +
    Boost C++ LibrariesHomeLibrariesPeopleFAQMore
    +
    +
    +PrevUpHomeNext +
    +
    +
    +
    + + Synopsis +
    +
    #include <boost/math/special_functions/trigamma.hpp>
+
    +
    namespace boost{ namespace math{
+
+template <class T>
+calculated-result-type trigamma(T z);
+
+template <class T, class Policy>
+calculated-result-type trigamma(T z, const Policy&);
+
+}} // namespaces
+
    +
    + + Description +
    +

    + Returns the trigamma function of x. Trigamma is defined + as the derivative of the digamma function: +

    +

    + +

    +

    + +

    +

    + The final Policy argument is optional and can + be used to control the behaviour of the function: how it handles errors, + what level of precision to use etc. Refer to the policy + documentation for more details. +

    +

    + The return type of this function is computed using the result + type calculation rules: the result is of type double when T is an integer type, and type + T otherwise. +

    +
    + + Accuracy +
    +

    + The following table shows the peak errors (in units of epsilon) found on + various platforms with various floating point types. Unless otherwise specified + any floating point type that is narrower than the one shown will have effectively zero error. +

    +
    +++++ + + + + + + + + + + + + + + + + + + + + + + +
    +

    + Significand Size +

    +     		+

    + Platform and Compiler +

    +     		+

    + Random Values +

    +
    +

    + 53 +

    +     		+

    + Win32 Visual C++ 12 +

    +     		+

    + Peak=1.0 Mean=0.4 +

    +
    +

    + 64 +

    +     		+

    + Win64 Mingw GCC +

    +     		+

    + Peak=1.4 Mean=0.4 +

    +
    +

    + 113 +

    +     		+

    + Win64 Mingw GCC __float128 +

    +     		+

    + Peak=1.0 Mean=0.5 +

    +
    +

    + As shown above, error rates are generally very low for built in types. For + multiprecision types, error rates are typically in the order of a few epsilon. +

    +
    + + Testing +
    +

    + Testing is against Mathematica generated spot values to 35 digit precision. +

    +
    + + Implementation +
    +

    + The arbitrary precision version of this function simply calls polygamma. +

    +

    + For built in fixed precision types, negative arguments are first made positive + via: +

    +

    + +

    +

    + Then arguments in the range [0, 1) are shifted to >= 1 via: +

    +

    + +

    +

    + Then evaluation is via one of a number of rational approximations, for small + x these are of the form: +

    +

    + +

    +

    + and for large x of the form: +

    +

    + +

    +
    + + + +
    Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, + Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert + Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta, + Thijs van den Berg, Daryle Walker and Xiaogang Zhang

    + Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +

    +
    +
    +
    +PrevUpHomeNext +
    + + diff --git a/doc/html/math_toolkit/zetas/zeta.html b/doc/html/math_toolkit/zetas/zeta.html index 7f3919801..784056d85 100644 --- a/doc/html/math_toolkit/zetas/zeta.html +++ b/doc/html/math_toolkit/zetas/zeta.html @@ -292,6 +292,23 @@ of: ε/R(0). This saves us quite a few digits when dealing with large z, especially when ε is small.

    +

    + Finally, there are some special cases for integer arguments, there are closed + forms for negative or even integers: +

    +

    + +

    +

    + +

    +

    + +

    +

    + and for positive odd integers we simply cache pre-computed values as these + are of great benefit to some infinite series calculations. +

    diff --git a/doc/html/special.html b/doc/html/special.html index baf8670d2..4efb58f32 100644 --- a/doc/html/special.html +++ b/doc/html/special.html @@ -40,6 +40,8 @@
    Gamma
    Log Gamma
    Digamma
    +
    Trigamma
    +
    Polygamma
    Ratios of Gamma Functions
    Incomplete Gamma Functions
    Incomplete Gamma Function diff --git a/doc/math.qbk b/doc/math.qbk index 58fc7604b..b05dec4ce 100644 --- a/doc/math.qbk +++ b/doc/math.qbk @@ -99,6 +99,8 @@ and use the function's name as the link text.] [/gammas] [def __lgamma [link math_toolkit.sf_gamma.lgamma lgamma]] [def __digamma [link math_toolkit.sf_gamma.digamma digamma]] +[def __trigamma [link math_toolkit.sf_gamma.trigamma trigamma]] +[def __polygamma [link math_toolkit.sf_gamma.polygamma polygamma]] [def __tgamma_ratio [link math_toolkit.sf_gamma.gamma_ratios tgamma_ratio]] [def __tgamma_delta_ratio [link math_toolkit.sf_gamma.gamma_ratios tgamma_delta_ratio]] [def __tgamma [link math_toolkit.sf_gamma.tgamma tgamma]] @@ -464,6 +466,8 @@ and as a CD ISBN 0-9504833-2-X 978-0-9504833-2-0, Classification 519.2-dc22. [include sf/tgamma.qbk] [include sf/lgamma.qbk] [include sf/digamma.qbk] +[include sf/trigamma.qbk] +[include sf/polygamma.qbk] [include sf/gamma_ratios.qbk] [include sf/igamma.qbk] [include sf/igamma_inv.qbk] diff --git a/doc/sf/bernoulli_numbers.qbk b/doc/sf/bernoulli_numbers.qbk new file mode 100644 index 000000000..8551216e3 --- /dev/null +++ b/doc/sf/bernoulli_numbers.qbk @@ -0,0 +1,223 @@ +[section:bernoulli_numbers Bernoulli Numbers] + +[@https://en.wikipedia.org/wiki/Bernoulli_number Bernoulli numbers] +are a sequence of rational numbers useful for the Taylor series expansion, +Euler-Maclaurin formula, and the Riemann zeta function. + +Bernoulli numbers are used in evaluation of some Boost.Math functions, +including the __tgamma, __lgamma and polygamma functions. + +[h4 Single Bernoulli number] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost { namespace math { + + template + T bernoulli_b2n(const int n); // Single Bernoulli number (default policy). + + template + T bernoulli_b2n(const int n, const Policy &pol); // User policy for errors etc. + + }} // namespaces + +[h4 Description] + +Both return the (2 * n)[super th] Bernoulli number B[sub 2n]. + +Note that since all odd numbered Bernoulli numbers are zero (apart from B[sub 1] which is [plusminus][frac12]) +the interface will only return the even numbered Bernoulli numbers. + +This function uses fast table lookup for low-indexed Bernoulli numbers, while larger values are calculated +as needed and then cached. The caching mechanism requires a certain amount of thread safety code, so +`unchecked_bernoulli_b2n` may provide a better interface for performance critical code. + +The final __Policy argument is optional and can be used to control the behaviour of the function: +how it handles errors, what level of precision to use, etc. + +Refer to __policy_section for more details. + +[h4 Examples] + +[import ../../example/bernoulli_example.cpp] +[bernoulli_example_1] + +[bernoulli_output_1] + +[h4 Single (unchecked) Bernoulli number] + +[h4 Synopsis] +`` +#include + +`` + + template <> + struct max_bernoulli_b2n; + + template + inline T unchecked_bernoulli_b2n(unsigned n); + +`unchecked_bernoulli_b2n` provides access to Bernoulli numbers [*without any checks for overflow or invalid parameters]. +It is implemented as a direct (and very fast) table lookup, and while not recomended for general use it can be useful +inside inner loops where the ultimate performance is required, and error checking is moved outside the loop. + +The largest value you can pass to `unchecked_bernoulli_b2n<>` is `max_bernoulli_b2n<>::value`: passing values greater than +that will result in a buffer overrun error, so it's clearly important to place the error handling in your own code +when using this direct interface. + +The value of `boost::math::max_bernoulli_b2n::value` varies by the type T, for types `float`/`double`/`long double` +it's the largest value which doesn't overflow the target type: for example, `boost::math::max_bernoulli_b2n::value` is 129. +However, for multiprecision types, it's the largest value for which the result can be represented as the ratio of two 64-bit +integers, for example `boost::math::max_bernoulli_b2n::value` is just 17. Of course +larger indexes can be passed to `bernoulli_b2n(n)`, but then then you loose fast table lookup (i.e. values may need to be calculated). + +[bernoulli_example_4] +[bernoulli_output_4] + +[h4 Multiple Bernoulli Numbers] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost { namespace math { + + // Multiple Bernoulli numbers (default policy). + template + OutputIterator bernoulli_b2n( + int start_index, + unsigned number_of_bernoullis_b2n, + OutputIterator out_it); + + // Multiple Bernoulli numbers (user policy). + template + OutputIterator bernoulli_b2n( + int start_index, + unsigned number_of_bernoullis_b2n, + OutputIterator out_it, + const Policy& pol); + }} // namespaces + +[h4 Description] + +Two versions of the Bernoulli number function are provided to compute multiple Bernoulli numbers +with one call (one with default policy and the other allowing a user-defined policy). + +These return a series of Bernoulli numbers: + +[:B[sub 2*start_index],B[sub 2*(start_index+1)],...,B[sub 2*(start_index+number_of_bernoullis_b2n-1)]] + +[h4 Examples] +[bernoulli_example_2] +[bernoulli_output_2] +[bernoulli_example_3] +[bernoulli_output_3] + +The source of this example is at [@../../example/bernoulli_example.cpp bernoulli_example.cpp] + +[h4 Accuracy] + +All the functions usually return values within one ULP (unit in the last place) for the floating-point type. + +[h4 Implementation] + +The implementation details are in [@../../include/boost/math/special_functions/detail/bernoulli_details.hpp bernoulli_details.hpp] +and [@../../include/boost/math/special_functions/detail/unchecked_bernoulli.hpp unchecked_bernoulli.hpp]. + +For `i <= max_bernoulli_index::value` this is implemented by simple table lookup from a statically initialized table; +for larger values of `i`, this is implemented by the Tangent Numbers algorithm as described in the paper: +Fast Computation of Bernoulli, Tangent and Secant Numbers, Richard P. Brent and David Harvey, +[@http://arxiv.org/pdf/1108.0286v3.pdf] (2011). + +[@http://mathworld.wolfram.com/TangentNumber.html Tangent (or Zag) numbers] +(an even alternating permutation number) are defined +and their generating function is also given therein. + +The relation of Tangent numbers with Bernoulli numbers ['B[sub i]] +is given by Brent and Harvey's equation 14: + +__spaces[equation tangent_numbers] + +Their relation with Bernoulli numbers ['B[sub i]] are defined by + +if i > 0 and i is even then +__spaces[equation bernoulli_numbers] [br] +elseif i == 0 then ['B[sub i]] = 1 [br] +elseif i == 1 then ['B[sub i]] = -1/2 [br] +elseif i < 0 or i is odd then ['B[sub i]] = 0 + +Note that computed values are stored in a fixed-size table, access is thread safe via atomic operations (i.e. lock +free programming), this imparts a much lower overhead on access to cached values than might overwise be expected - +typically for multiprecision types the cost of thread synchronisation is negligable, while for built in types +this code is not normally executed anyway. For very large arguments which cannot be reasonably computed or +stored in our cache, an asymptotic expansion [@http://www.luschny.de/math/primes/bernincl.html due to Luschny] is used: + +[equation bernoulli_numbers2] + +[endsect] [/section:bernoulli_numbers Bernoulli Numbers] + + +[section:tangent_numbers Tangent Numbers] + +[@http://en.wikipedia.org/wiki/Tangent_numbers Tangent numbers], +also called a zag function. See also +[@http://mathworld.wolfram.com/TangentNumber.html Tangent number]. + +From the number, An, of alternating permutations of the set {1, ..., n}, +the numbers A2n+1 with odd indices are called tangent numbers or zag numbers. +The first few values are 1, 2, 16, 272, 7936, 353792, 22368256, 1903757312 ... +(sequence [@http://oeis.org/A000182 A000182 in OEIS]). +They are called tangent numbers because they appear as +numerators in the Maclaurin series of tan x. + +Tangent numbers are used in the computation of Bernoulli numbers, +but are also made available here. + +[h4 Synopsis] +`` +#include +`` + + template + T tangent_t2n(const int i); // Single tangent number (default policy). + + template + T tangent_t2n(const int i, const Policy &pol); // Single tangent number (user policy). + + // Multiple tangent numbers (default policy). + template + OutputIterator tangent_t2n(const int start_index, + const unsigned number_of_tangent_t2n, + OutputIterator out_it); + + // Multiple tangent numbers (user policy). + template + OutputIterator tangent_t2n(const int start_index, + const unsigned number_of_tangent_t2n, + OutputIterator out_it, + const Policy& pol); + +[h4 Examples] + +[tangent_example_1] + +The output is: +[tangent_output_1] + +The source of this example is at [../../example/bernoulli_example.cpp bernoulli_example.cpp] + +[endsect] [/section:tangent_numbers Tangent Numbers] + +[/ + Copyright 2013, 2014 Nikhar Agrawal, Christopher Kormanyos, John Maddock, Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] diff --git a/doc/sf/digamma.qbk b/doc/sf/digamma.qbk index cf3b9c232..d62466ac3 100644 --- a/doc/sf/digamma.qbk +++ b/doc/sf/digamma.qbk @@ -27,10 +27,6 @@ derivative of the gamma function: [optional_policy] -There is no fully generic version of this function: all the implementations -are tuned to specific accuracy levels, the most precise of which delivers -34-digits of precision. - The return type of this function is computed using the __arg_pomotion_rules: the result is of type `double` when T is an integer type, and type T otherwise. @@ -97,6 +93,14 @@ Choosing BIG=10 for up to 80-bit reals, and BIG=20 for 128-bit reals allows the series to truncated after a suitably small number of terms and evaluated as a polynomial in `1/(x*x)`. +The arbitrary precision version of this function uses recurrence relations until +x > BIG, and then evaluation via the asymptotic expansion above. As special cases +integer and half integer arguments are handled via: + +[equation digamma4] + +[equation digamma5] + The rational approximation [jm_rationals] in the range [1,2] is derived as follows. First a high precision approximation to digamma was constructed using a 60-term diff --git a/doc/sf/polygamma.qbk b/doc/sf/polygamma.qbk new file mode 100644 index 000000000..873403216 --- /dev/null +++ b/doc/sf/polygamma.qbk @@ -0,0 +1,118 @@ +[section:polygamma Polygamma] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + ``__sf_result`` polygamma(int n, T z); + + template + ``__sf_result`` polygamma(int n, T z, const ``__Policy``&); + + }} // namespaces + +[h4 Description] + +Returns the polygamma function of /x/. Polygamma is defined as the n'th +derivative of the digamma function: + +[equation polygamma1] + +The following graphs illustrate the behaviour of the function for odd and even order: + +[graph polygamma2] +[graph polygamma3] + +[optional_policy] + +The return type of this function is computed using the __arg_pomotion_rules: +the result is of type `double` when T is an integer type, and type T otherwise. + +[h4 Accuracy] + +The following table shows the peak errors (in units of epsilon) +found on various platforms with various floating point types. +Unless otherwise specified any floating point type that is narrower +than the one shown will have __zero_error. + +[table +[[Significand Size] [Platform and Compiler] [Small-medium positive arguments] [Small-medium negative x] ] +[[53] [Win32 Visual C++ 12] [Peak=5.0 Mean=1] [Peak=1200 Mean=65]] +[[64] [Win64 Mingw GCC] [Peak=16 Mean=3] [Peak=33 Mean=3] ] +[[113] [Win64 Mingw GCC __float128] [Peak=6.5 Mean=1][Peak=30 Mean=4] ] +] + +As shown above, error rates are generally very acceptable for moderately sized +arguments. Error rates should stay low for exact inputs, however, please note that the +function becomes exceptionally sensitive to small changes in input for large n and negative x, +indeed for cases where ['n!] would overflow, the function changes directly from -[infin] to ++[infin] somewhere between each negative integer - ['these cases are not handled correctly]. + +[*For these reasons results should be treated with extreme caution when /n/ is large and x negative]. + +[h4 Testing] + +Testing is against Mathematica generated spot values to 35 digit precision. + +[h4 Implementation] + +For x < 0 the following reflection formula is used: + +[equation polygamma2] + +The n'th derivative of ['cot(x)] is tabulated for small /n/, and for larger n +has the general form: + +[equation polygamma3] + +The coefficients of the cosine terms can be calculated iteratively starting +from ['C[sub 1,0] = -1] and then using + +[equation polygamma7] + +to generate coefficients for n+1. + +Note that every other coefficient is zero, and therefore what we have are +even or odd polynomials depending on whether n is even or odd. + +Once x is positive then we have two methods available to us, for small x +we use the series expansion: + +[equation polygamma4] + +Note that as this is an alternating series we can accelerate the series using +Algorithm 1 from ['"Convergence Acceleration of Alternating Series", +Henri Cohen, Fernando Rodriguez Villegas, and Don Zagier, Experimental Mathematics, 1999.] +Also note that the evaluation of zeta functions at integer values is essentially a table lookup +as __zeta is optimized for those cases. + +For large x we use the asymptotic expansion: + +[equation polygamma5] + +For x in-between the two extremes we use the relation: + +[equation polygamma6] + +to make x large enough for the asymptotic expansion to be used. + +There are also two special cases: + +[equation polygamma8] + +[equation polygamma9] + +[endsect][/section:polygamma The Polygamma Function] + +[/ + Copyright 2014 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf/trigamma.qbk b/doc/sf/trigamma.qbk new file mode 100644 index 000000000..827c8fb52 --- /dev/null +++ b/doc/sf/trigamma.qbk @@ -0,0 +1,84 @@ +[section:trigamma Trigamma] + +[h4 Synopsis] + +`` +#include +`` + + namespace boost{ namespace math{ + + template + ``__sf_result`` trigamma(T z); + + template + ``__sf_result`` trigamma(T z, const ``__Policy``&); + + }} // namespaces + +[h4 Description] + +Returns the trigamma function of /x/. Trigamma is defined as the +derivative of the digamma function: + +[equation trigamma1] + +[graph trigamma] + +[optional_policy] + +The return type of this function is computed using the __arg_pomotion_rules: +the result is of type `double` when T is an integer type, and type T otherwise. + +[h4 Accuracy] + +The following table shows the peak errors (in units of epsilon) +found on various platforms with various floating point types. +Unless otherwise specified any floating point type that is narrower +than the one shown will have __zero_error. + +[table +[[Significand Size] [Platform and Compiler] [Random Values] ] +[[53] [Win32 Visual C++ 12] [Peak=1.0 Mean=0.4] ] +[[64] [Win64 Mingw GCC] [Peak=1.4 Mean=0.4] ] +[[113] [Win64 Mingw GCC __float128] [Peak=1.0 Mean=0.5] ] +] + +As shown above, error rates are generally very low for built in types. +For multiprecision types, error rates are typically in the order of a +few epsilon. + +[h4 Testing] + +Testing is against Mathematica generated spot values to 35 digit precision. + +[h4 Implementation] + +The arbitrary precision version of this function simply calls __polygamma. + +For built in fixed precision types, negative arguments are first made positive via: + +[equation trigamma2] + +Then arguments in the range \[0, 1) are shifted to >= 1 via: + +[equation trigamma3] + +Then evaluation is via one of a number of rational approximations, for small x these are +of the form: + +[equation trigamma4] + +and for large x of the form: + +[equation trigamma5] + +[endsect][/section:digamma The Trigamma Function] + +[/ + Copyright 2014 John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + diff --git a/doc/sf/zeta.qbk b/doc/sf/zeta.qbk index 08186c6a4..b8520685c 100644 --- a/doc/sf/zeta.qbk +++ b/doc/sf/zeta.qbk @@ -112,6 +112,18 @@ required for R(z-n) is not full machine precision, but an absolute error of: [epsilon]/R(0). This saves us quite a few digits when dealing with large z, especially when [epsilon] is small. +Finally, there are some special cases for integer arguments, there are +closed forms for negative or even integers: + +[equation zeta7] + +[equation zeta8] + +[equation zeta9] + +and for positive odd integers we simply cache pre-computed values as these are of great +benefit to some infinite series calculations. + [endsect] [/ :error_function The Error Functions] diff --git a/include/boost/math/policies/policy.hpp b/include/boost/math/policies/policy.hpp index 49068a6ed..71309fa11 100644 --- a/include/boost/math/policies/policy.hpp +++ b/include/boost/math/policies/policy.hpp @@ -850,6 +850,11 @@ inline int digits(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE(T)) typedef mpl::bool_< std::numeric_limits::is_specialized > tag_type; return detail::digits_imp(tag_type()); } +template +inline int digits_base10(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE(T)) +{ + return boost::math::policies::digits() * 301 / 1000L; +} template inline unsigned long get_max_series_iterations() diff --git a/include/boost/math/special_functions.hpp b/include/boost/math/special_functions.hpp index aa7019a1e..151aca2bf 100644 --- a/include/boost/math/special_functions.hpp +++ b/include/boost/math/special_functions.hpp @@ -1,4 +1,4 @@ -// Copyright John Maddock 2006, 2007, 2012. +// Copyright John Maddock 2006, 2007, 2012, 2014. // Copyright Paul A. Bristow 2006, 2007, 2012 // Use, modification and distribution are subject to the @@ -47,6 +47,7 @@ #include #include #include +#include #include #include #include diff --git a/include/boost/math/special_functions/cos_pi.hpp b/include/boost/math/special_functions/cos_pi.hpp index 18d06c00d..669a2c87a 100644 --- a/include/boost/math/special_functions/cos_pi.hpp +++ b/include/boost/math/special_functions/cos_pi.hpp @@ -25,10 +25,10 @@ T cos_pi_imp(T x, const Policy& pol) BOOST_MATH_STD_USING // ADL of std names // cos of pi*x: bool invert = false; - if(fabs(x) < 0.5) + if(fabs(x) < 0.25) return cos(constants::pi() * x); - if(x < 1) + if(x < 0) { x = -x; } @@ -44,17 +44,30 @@ T cos_pi_imp(T x, const Policy& pol) if(rem == 0.5f) return 0; - rem = cos(constants::pi() * rem); + if(rem > 0.25f) + { + rem = 0.5f - rem; + rem = sin(constants::pi() * rem); + } + else + rem = cos(constants::pi() * rem); return invert ? T(-rem) : rem; } } // namespace detail template -inline typename tools::promote_args::type cos_pi(T x, const Policy& pol) +inline typename tools::promote_args::type cos_pi(T x, const Policy&) { typedef typename tools::promote_args::type result_type; - return boost::math::detail::cos_pi_imp(x, pol); + typedef typename policies::evaluation::type value_type; + typedef typename policies::normalise< + Policy, + policies::promote_float, + policies::promote_double, + policies::discrete_quantile<>, + policies::assert_undefined<> >::type forwarding_policy; + return policies::checked_narrowing_cast(boost::math::detail::cos_pi_imp(x, forwarding_policy()), "cos_pi"); } template diff --git a/include/boost/math/special_functions/detail/polygamma.hpp b/include/boost/math/special_functions/detail/polygamma.hpp new file mode 100644 index 000000000..f9ce7509b --- /dev/null +++ b/include/boost/math/special_functions/detail/polygamma.hpp @@ -0,0 +1,544 @@ + +/////////////////////////////////////////////////////////////////////////////// +// Copyright 2013 Nikhar Agrawal +// Copyright 2013 Christopher Kormanyos +// Copyright 2014 John Maddock +// Copyright 2013 Paul Bristow +// Distributed under the Boost +// Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifndef _BOOST_POLYGAMMA_DETAIL_2013_07_30_HPP_ + #define _BOOST_POLYGAMMA_DETAIL_2013_07_30_HPP_ + + #include + #include + #include + #include + #include + #include + #include + #include + #include + #include + #include + #include + #include + #include + #include + + namespace boost { namespace math { namespace detail{ + + template + T polygamma_atinfinityplus(const int n, const T& x, const Policy& pol, const char* function) // for large values of x such as for x> 400 + { + // See http://functions.wolfram.com/GammaBetaErf/PolyGamma2/06/02/0001/ + BOOST_MATH_STD_USING + // + // sum == current value of accumulated sum. + // term == value of current term to be added to sum. + // part_term == value of current term excluding the Bernoulli number part + // + if(n + x == x) + { + // x is crazy large, just concentrate on the first part of the expression and use logs: + if(n == 1) return 1 / x; + T nlx = n * log(x); + if((nlx < tools::log_max_value()) && (n < max_factorial::value)) + return ((n & 1) ? 1 : -1) * boost::math::factorial(n - 1) * pow(x, -n); + else + return ((n & 1) ? 1 : -1) * exp(boost::math::lgamma(T(n), pol) - n * log(x)); + } + T term, sum, part_term; + T x_squared = x * x; + // + // Start by setting part_term to: + // + // (n-1)! / x^(n+1) + // + // which is common to both the first term of the series (with k = 1) + // and to the leading part. + // We can then get to the leading term by: + // + // part_term * (n + 2 * x) / 2 + // + // and to the first term in the series + // (excluding the Bernoulli number) by: + // + // part_term n * (n + 1) / (2x) + // + // If either the factorial would overflow, + // or the power term underflows, this just gets set to 0 and then we + // know that we have to use logs for the initial terms: + // + part_term = ((n > boost::math::max_factorial::value) && (n * n > tools::log_max_value())) + ? T(0) : static_cast(boost::math::factorial(n - 1, pol) * pow(x, -n - 1)); + if(part_term == 0) + { + // Either n is very large, or the power term underflows, + // set the initial values of part_term, term and sum via logs: + part_term = boost::math::lgamma(n, pol) - (n + 1) * log(x); + sum = exp(part_term + log(n + 2 * x) - boost::math::constants::ln_two()); + part_term += log(T(n) * (n + 1)) - boost::math::constants::ln_two() - log(x); + part_term = exp(part_term); + } + else + { + sum = part_term * (n + 2 * x) / 2; + part_term *= n * (n + 1) / 2; + part_term /= x; + } + // + // If the leading term is 0, so is the result: + // + if(sum == 0) + return sum; + + for(unsigned k = 1;;) + { + term = part_term * boost::math::bernoulli_b2n(k, pol); + sum += term; + // + // Normal termination condition: + // + if(fabs(term / sum) < tools::epsilon()) + break; + // + // Increment our counter, and move part_term on to the next value: + // + ++k; + part_term *= (n + 2 * k - 2) * (n - 1 + 2 * k); + part_term /= (2 * k - 1) * 2 * k; + part_term /= x_squared; + // + // Emergency get out termination condition: + // + if(k > policies::get_max_series_iterations()) + { + return policies::raise_evaluation_error(function, "Series did not converge, closest value was %1%", sum, pol); + } + } + + if((n - 1) & 1) + sum = -sum; + + return sum; + } + + template + T polygamma_attransitionplus(const int n, const T& x, const Policy& pol, const char* function) + { + // See: http://functions.wolfram.com/GammaBetaErf/PolyGamma2/16/01/01/0017/ + + // Use N = (0.4 * digits) + (4 * n) for target value for x: + BOOST_MATH_STD_USING + const int d4d = static_cast(0.4F * policies::digits_base10()); + const int N = d4d + (4 * n); + const int m = n; + const int iter = N - itrunc(x); + + const int minus_m_minus_one = -m - 1; + + T z(x); + T sum0(0); + T z_plus_k_pow_minus_m_minus_one(0); + + // Forward recursion to larger x, need to check for overflow first though: + if(log(z + iter) * minus_m_minus_one > -tools::log_max_value()) + { + for(int k = 1; k <= iter; ++k) + { + z_plus_k_pow_minus_m_minus_one = pow(z, minus_m_minus_one); + sum0 += z_plus_k_pow_minus_m_minus_one; + z += 1; + } + sum0 *= boost::math::factorial(n); + } + else + { + for(int k = 1; k <= iter; ++k) + { + T log_term = log(z) * minus_m_minus_one + boost::math::lgamma(T(n + 1), pol); + sum0 += exp(log_term); + z += 1; + } + } + if((n - 1) & 1) + sum0 = -sum0; + + return sum0 + polygamma_atinfinityplus(n, z, pol, function); + } + + template + T polygamma_nearzero(int n, T x, const Policy& pol, const char* function) + { + BOOST_MATH_STD_USING + // + // If we take this expansion for polygamma: http://functions.wolfram.com/06.15.06.0003.02 + // and substitute in this expression for polygamma(n, 1): http://functions.wolfram.com/06.15.03.0009.01 + // we get an alternating series for polygamma when x is small in terms of zeta functions of + // integer arguments (which are easy to evaluate, at least when the integer is even). + // + // In order to avoid spurious overflow, save the n! term for later, and rescale at the end: + // + T scale = boost::math::factorial(n, pol); + // + // "factorial_part" contains everything except the zeta function + // evaluations in each term: + // + T factorial_part = 1; + // + // "prefix" is what we'll be adding the accumulated sum to, it will + // be n! / z^(n+1), but since we're scaling by n! it's just + // 1 / z^(n+1) for now: + // + T prefix = pow(x, n + 1); + if(prefix == 0) + return boost::math::policies::raise_overflow_error(function, 0, pol); + prefix = 1 / prefix; + // + // First term in the series is necessarily < zeta(2) < 2, so + // ignore the sum if it will have no effect on the result anyway: + // + if(prefix > 2 / policies::get_epsilon()) + return ((n & 1) ? 1 : -1) * + (tools::max_value() / prefix < scale ? policies::raise_overflow_error(function, 0, pol) : prefix * scale); + // + // Since this is an alternating sum we can accelerate convergence using + // Algorithm 1 from "Convergence Acceleration of Alternating Series", + // Henri Cohen, Fernando Rodriguez Villegas, and Don Zagier, + // Experimental Mathematics, 1999. + // While in principle we can determine up front how many terms we will need, + // note that often the prefix term is so large that we need no terms at all, + // or else the series is divergent and we need rather more terms than expected. + // The latter case we try to weed out before this function gets called, but + // just in case set the number of terms to an arbitrary high value, and then + // use the usual heurists to determine when to stop, these next variables + // correspond directly to those in Algorithm 1 of the above paper: + // + int nd = (int)std::min((boost::intmax_t)(boost::math::policies::digits_base10() * 10), (boost::intmax_t)boost::math::policies::get_max_series_iterations()); + T d = pow(3 + sqrt(T(8)), nd); + d = (d + 1 / d) / 2; + T b = -1; + T c = -d; + T sum = 0; + for(int k = 0; k < nd;) + { + // Get the k'th term: + T term = factorial_part * boost::math::zeta(T(k + n + 1), pol); + // Series acceleration: + c = b - c; + sum = sum + c * term; + b = (k + nd) * (k - nd) * b / ((k + 0.5f) * (k + 1)); + // Termination condition: + if(fabs(c * term) < (sum + prefix * d) * boost::math::policies::get_epsilon()) + break; + // + // Move on k and factorial_part: + // + ++k; + factorial_part *= x * (n + k) / k; + } + // + // We need to add the sum to the prefix term and then + // multiply by the scale, at each stage checking for oveflow: + // + sum /= d; + if(boost::math::tools::max_value() - sum < prefix) + return boost::math::policies::raise_overflow_error(function, 0, pol); + sum += prefix; + if(boost::math::tools::max_value() / scale < sum) + return boost::math::policies::raise_overflow_error(function, 0, pol); + sum *= scale; + return n & 1 ? sum : -sum; + } + + // + // Helper function which figures out which slot our coefficient is in + // given an angle multiplier for the cosine term of power: + // + template + typename Table::value_type::reference dereference_table(Table& table, unsigned row, unsigned power) + { + return table[row][power / 2]; + } + + + + template + T poly_cot_pi(int n, T x, T xc, const Policy& pol, const char* function) + { + BOOST_MATH_STD_USING + // Return n'th derivative of cot(pi*x) at x, these are simply + // tabulated for up to n = 9, beyond that it is possible to + // calculate coefficients as follows: + // + // The general form of each derivative is: + // + // pi^n * SUM{k=0, n} C[k,n] * cos^k(pi * x) * csc^(n+1)(pi * x) + // + // With constant C[0,1] = -1 and all other C[k,n] = 0; + // Then for each k < n+1: + // C[k-1, n+1] -= k * C[k, n]; + // C[k+1, n+1] += (k-n-1) * C[k, n]; + // + // Note that there are many different ways of representing this derivative thanks to + // the many trigomonetric identies available. In particular, the sum of powers of + // cosines could be replaced by a sum of cosine multiple angles, and indeed if you + // plug the derivative into Mathematica this is the form it will give. The two + // forms are related via the Chebeshev polynomials of the first kind and + // T_n(cos(x)) = cos(n x). The polynomial form has the great advantage that + // all the cosine terms are zero at half integer arguments - right where this + // function has it's minumum - thus avoiding cancellation error in this region. + // + // And finally, since every other term in the polynomials is zero, we can save + // space by only storing the non-zero terms. This greatly complexifies + // subscripting the tables in the calculation, but halves the storage space + // (and complexity for that matter). + // + T s = fabs(x) < fabs(xc) ? boost::math::sin_pi(x, pol) : boost::math::sin_pi(xc, pol); + T c = boost::math::cos_pi(x, pol); + switch(n) + { + case 1: + return -constants::pi() / (s * s); + case 2: + { + return 2 * constants::pi() * constants::pi() * c / boost::math::pow<3>(s, pol); + } + case 3: + { + int P[] = { -2, -4 }; + return boost::math::pow<3>(constants::pi(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<4>(s, pol); + } + case 4: + { + int P[] = { 16, 8 }; + return boost::math::pow<4>(constants::pi(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<5>(s, pol); + } + case 5: + { + int P[] = { -16, -88, -16 }; + return boost::math::pow<5>(constants::pi(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<6>(s, pol); + } + case 6: + { + int P[] = { 272, 416, 32 }; + return boost::math::pow<6>(constants::pi(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<7>(s, pol); + } + case 7: + { + int P[] = { -272, -2880, -1824, -64 }; + return boost::math::pow<7>(constants::pi(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<8>(s, pol); + } + case 8: + { + int P[] = { 7936, 24576, 7680, 128 }; + return boost::math::pow<8>(constants::pi(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<9>(s, pol); + } + case 9: + { + int P[] = { -7936, -137216, -185856, -31616, -256 }; + return boost::math::pow<9>(constants::pi(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<10>(s, pol); + } + case 10: + { + int P[] = { 353792, 1841152, 1304832, 128512, 512 }; + return boost::math::pow<10>(constants::pi(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<11>(s, pol); + } + case 11: + { + int P[] = { -353792, -9061376, -21253376, -8728576, -518656, -1024}; + return boost::math::pow<11>(constants::pi(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<12>(s, pol); + } + case 12: + { + int P[] = { 22368256, 175627264, 222398464, 56520704, 2084864, 2048 }; + return boost::math::pow<12>(constants::pi(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<13>(s, pol); + } +#ifndef BOOST_NO_LONG_LONG + case 13: + { + long long P[] = { -22368256LL, -795300864LL, -2868264960LL, -2174832640LL, -357888000LL, -8361984LL, -4096 }; + return boost::math::pow<13>(constants::pi(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<14>(s, pol); + } + case 14: + { + long long P[] = { 1903757312LL, 21016670208LL, 41731645440LL, 20261765120LL, 2230947840LL, 33497088LL, 8192 }; + return boost::math::pow<14>(constants::pi(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<15>(s, pol); + } + case 15: + { + long long P[] = { -1903757312LL, -89702612992LL, -460858269696LL, -559148810240LL, -182172651520LL, -13754155008LL, -134094848LL, -16384 }; + return boost::math::pow<15>(constants::pi(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<16>(s, pol); + } + case 16: + { + long long P[] = { 209865342976LL, 3099269660672LL, 8885192097792LL, 7048869314560LL, 1594922762240LL, 84134068224LL, 536608768LL, 32768 }; + return boost::math::pow<16>(constants::pi(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<17>(s, pol); + } + case 17: + { + long long P[] = { -209865342976LL, -12655654469632LL, -87815735738368LL, -155964390375424LL, -84842998005760LL, -13684856848384LL, -511780323328LL, -2146926592LL, -65536 }; + return boost::math::pow<17>(constants::pi(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<18>(s, pol); + } + case 18: + { + long long P[] = { 29088885112832LL, 553753414467584LL, 2165206642589696LL, 2550316668551168LL, 985278548541440LL, 115620218667008LL, 3100738912256LL, 8588754944LL, 131072 }; + return boost::math::pow<18>(constants::pi(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<19>(s, pol); + } + case 19: + { + long long P[] = { -29088885112832LL, -2184860175433728LL, -19686087844429824LL, -48165109676113920LL, -39471306959486976LL, -11124607890751488LL, -965271355195392LL, -18733264797696LL, -34357248000LL, -262144 }; + return boost::math::pow<19>(constants::pi(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<20>(s, pol); + } + case 20: + { + long long P[] = { 4951498053124096LL, 118071834535526400LL, 603968063567560704LL, 990081991141490688LL, 584901762421358592LL, 122829335169859584LL, 7984436548730880LL, 112949304754176LL, 137433710592LL, 524288 }; + return boost::math::pow<20>(constants::pi(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<21>(s, pol); + } +#endif + } + + // + // We'll have to compute the coefficients up to n: + // +#ifdef BOOST_HAS_THREADS + static boost::detail::lightweight_mutex m; + boost::detail::lightweight_mutex::scoped_lock l(m); +#endif + static std::vector > table(1, std::vector(1, T(-1))); + + int index = n - 1; + + if(index >= (int)table.size()) + { + for(int i = (int)table.size() - 1; i < index; ++i) + { + int offset = i & 1; // 1 if the first cos power is 0, otherwise 0. + int sin_order = i + 2; // order of the sin term + int max_cos_order = sin_order - 1; // largest order of the polynomial of cos terms + int max_columns = (max_cos_order - offset) / 2; // How many entries there are in the current row. + int next_offset = offset ? 0 : 1; + int next_max_columns = (max_cos_order + 1 - next_offset) / 2; // How many entries there will be in the next row + table.push_back(std::vector(next_max_columns + 1, T(0))); + + for(int column = 0; column <= max_columns; ++column) + { + int cos_order = 2 * column + offset; // order of the cosine term in entry "column" + BOOST_ASSERT(column < (int)table[i].size()); + BOOST_ASSERT((cos_order + 1) / 2 < (int)table[i + 1].size()); + table[i + 1][(cos_order + 1) / 2] += ((cos_order - sin_order) * table[i][column]) / (sin_order - 1); + if(cos_order) + table[i + 1][(cos_order - 1) / 2] += (-cos_order * table[i][column]) / (sin_order - 1); + } + } + + } + T sum = boost::math::tools::evaluate_even_polynomial(&table[index][0], c, table[index].size()); + if(index & 1) + sum *= c; // First coeffient is order 1, and really an odd polynomial. + if(sum == 0) + return sum; + // + // The remaining terms are computed using logs since the powers and factorials + // get real large real quick: + // + T power_terms = n * log(boost::math::constants::pi()); + if(s == 0) + return sum * boost::math::policies::raise_overflow_error(function, 0, pol); + power_terms -= log(fabs(s)) * (n + 1); + power_terms += boost::math::lgamma(T(n)); + power_terms += log(fabs(sum)); + + if(power_terms > boost::math::tools::log_max_value()) + return sum * boost::math::policies::raise_overflow_error(function, 0, pol); + + return exp(power_terms) * ((s < 0) && ((n + 1) & 1) ? -1 : 1) * boost::math::sign(sum); + } + + template + struct polygamma_initializer + { + struct init + { + init() + { + // Forces initialization of our table of coefficients and mutex: + boost::math::polygamma(30, T(-2.5f), Policy()); + } + void force_instantiate()const{} + }; + static const init initializer; + static void force_instantiate() + { + initializer.force_instantiate(); + } + }; + + template + const typename polygamma_initializer::init polygamma_initializer::initializer; + + template + inline T polygamma_imp(const int n, T x, const Policy &pol) + { + BOOST_MATH_STD_USING + static const char* function = "boost::math::polygamma<%1%>(int, %1%)"; + polygamma_initializer::initializer.force_instantiate(); + if(n < 0) + return policies::raise_domain_error(function, "Order must be >= 0, but got %1%", n, pol); + if(x < 0) + { + if(floor(x) == x) + { + // + // Result is infinity if x is odd, and a pole error if x is even. + // + if(lltrunc(x) & 1) + return policies::raise_overflow_error(function, 0, pol); + else + return policies::raise_pole_error(function, "Evaluation at negative integer %1%", x, pol); + } + T z = 1 - x; + T result = polygamma_imp(n, z, pol) + constants::pi() * poly_cot_pi(n, z, x, pol, function); + return n & 1 ? T(-result) : result; + } + // + // Limit for use of small-x-series is chosen + // so that the series doesn't go too divergent + // in the first few terms. Ordinarily this + // would mean setting the limit to ~ 1 / n, + // but since the series is accelerated, we can + // tolerate a small amount of divergence: + // + T small_x_limit = std::min(T(T(5) / n), T(0.25f)); + if(x < small_x_limit) + { + return polygamma_nearzero(n, x, pol, function); + } + else if(x > 0.4F * policies::digits_base10() + 4 * n) + { + return polygamma_atinfinityplus(n, x, pol, function); + } + else if(x == 1) + { + return (n & 1 ? 1 : -1) * boost::math::factorial(n, pol) * boost::math::zeta(T(n + 1), pol); + } + else if(x == 0.5f) + { + T result = (n & 1 ? 1 : -1) * boost::math::factorial(n, pol) * boost::math::zeta(T(n + 1), pol); + if(fabs(result) >= ldexp(tools::max_value(), -n - 1)) + return boost::math::sign(result) * policies::raise_overflow_error(function, 0, pol); + result *= ldexp(T(1), n + 1) - 1; + return result; + } + else + { + return polygamma_attransitionplus(n, x, pol, function); + } + } + +} } } // namespace boost::math::detail + +#endif // _BOOST_POLYGAMMA_DETAIL_2013_07_30_HPP_ + diff --git a/include/boost/math/special_functions/digamma.hpp b/include/boost/math/special_functions/digamma.hpp index 785cd75c5..d9e576e5e 100644 --- a/include/boost/math/special_functions/digamma.hpp +++ b/include/boost/math/special_functions/digamma.hpp @@ -12,6 +12,7 @@ #include #include +#include #include #include #include @@ -27,7 +28,8 @@ namespace detail{ // inline unsigned digamma_large_lim(const mpl::int_<0>*) { return 20; } - +inline unsigned digamma_large_lim(const mpl::int_<113>*) +{ return 20; } inline unsigned digamma_large_lim(const void*) { return 10; } // @@ -41,7 +43,7 @@ inline unsigned digamma_large_lim(const void*) // This first one gives 34-digit precision for x >= 20: // template -inline T digamma_imp_large(T x, const mpl::int_<0>*) +inline T digamma_imp_large(T x, const mpl::int_<113>*) { BOOST_MATH_STD_USING // ADL of std functions. static const T P[] = { @@ -141,12 +143,47 @@ inline T digamma_imp_large(T x, const mpl::int_<24>*) return result; } // +// Fully generic asymptotic expansion in terms of Bernoulli numbers, see: +// http://functions.wolfram.com/06.14.06.0012.01 +// +template +struct digamma_series_func +{ +private: + int k; + T xx; + T term; +public: + digamma_series_func(T x) : k(1), xx(x * x), term(1 / (x * x)) {} + T operator()() + { + T result = term * boost::math::bernoulli_b2n(k) / (2 * k); + term /= xx; + ++k; + return result; + } + typedef T result_type; +}; + +template +inline T digamma_imp_large(T x, const Policy& pol, const mpl::int_<0>*) +{ + BOOST_MATH_STD_USING + digamma_series_func s(x); + T result = log(x) - 1 / (2 * x); + boost::uintmax_t max_iter = policies::get_max_series_iterations(); + result = boost::math::tools::sum_series(s, boost::math::policies::get_epsilon(), max_iter, -result); + result = -result; + policies::check_series_iterations("boost::math::digamma<%1%>(%1%)", max_iter, pol); + return result; +} +// // Now follow rational approximations over the range [1,2]. // // 35-digit precision: // template -T digamma_imp_1_2(T x, const mpl::int_<0>*) +T digamma_imp_1_2(T x, const mpl::int_<113>*) { // // Now the approximation, we use the form: @@ -410,6 +447,98 @@ T digamma_imp(T x, const Tag* t, const Policy& pol) return result; } +template +T digamma_imp(T x, const mpl::int_<0>* t, const Policy& pol) +{ + // + // This handles reflection of negative arguments, and all our + // error handling, then forwards to the T-specific approximation. + // + BOOST_MATH_STD_USING // ADL of std functions. + + T result = 0; + // + // Check for negative arguments and use reflection: + // + if(x <= -1) + { + // Reflect: + x = 1 - x; + // Argument reduction for tan: + T remainder = x - floor(x); + // Shift to negative if > 0.5: + if(remainder > 0.5) + { + remainder -= 1; + } + // + // check for evaluation at a negative pole: + // + if(remainder == 0) + { + return policies::raise_pole_error("boost::math::digamma<%1%>(%1%)", 0, (1 - x), pol); + } + result = constants::pi() / tan(constants::pi() * remainder); + } + if(x == 0) + return policies::raise_pole_error("boost::math::digamma<%1%>(%1%)", 0, x, pol); + // + // If we're above the lower-limit for the + // asymptotic expansion then use it, the + // limit is a linear interpolation with + // limit = 10 at 50 bit precision and + // limit = 250 at 1000 bit precision. + // + T lim = 10 + (tools::digits() - 50) * 240 / 950; + T two_x = ldexp(x, 1); + if(x >= lim) + { + result += digamma_imp_large(x, pol, t); + } + else if(floor(x) == x) + { + // + // Special case for integer arguments, see + // http://functions.wolfram.com/06.14.03.0001.01 + // + result = -constants::euler(); + T val = 1; + while(val < x) + { + result += 1 / val; + val += 1; + } + } + else if(floor(two_x) == two_x) + { + // + // Special case for half integer arguments, see: + // http://functions.wolfram.com/06.14.03.0007.01 + // + result = -2 * constants::ln_two() - constants::euler(); + int n = itrunc(x); + if(n) + { + for(int k = 1; k < n; ++k) + result += 1 / T(k); + for(int k = n; k <= 2 * n - 1; ++k) + result += 2 / T(k); + } + } + else + { + // + // Rescale so we can use the asymptotic expansion: + // + while(x < lim) + { + result -= 1 / x; + x += 1; + } + result += digamma_imp_large(x, pol, t); + } + return result; +} // // Initializer: ensure all our constants are initialized prior to the first call of main: // @@ -419,10 +548,16 @@ struct digamma_initializer struct init { init() + { + typedef typename policies::precision::type precision_type; + do_init(mpl::bool_()); + } + void do_init(const mpl::true_&) { boost::math::digamma(T(1.5), Policy()); boost::math::digamma(T(500), Policy()); } + void do_init(const mpl::false_&){} void force_instantiate()const{} }; static const init initializer; @@ -439,7 +574,7 @@ const typename digamma_initializer::init digamma_initializer inline typename tools::promote_args::type - digamma(T x, const Policy& pol) + digamma(T x, const Policy&) { typedef typename tools::promote_args::type result_type; typedef typename policies::evaluation::type value_type; @@ -447,7 +582,7 @@ inline typename tools::promote_args::type typedef typename mpl::if_< mpl::or_< mpl::less_equal >, - mpl::greater > + mpl::greater > >, mpl::int_<0>, typename mpl::if_< @@ -456,17 +591,28 @@ inline typename tools::promote_args::type typename mpl::if_< mpl::less >, mpl::int_<53>, - mpl::int_<64> + typename mpl::if_< + mpl::less >, + mpl::int_<64>, + mpl::int_<113> + >::type >::type >::type >::type tag_type; + typedef typename policies::normalise< + Policy, + policies::promote_float, + policies::promote_double, + policies::discrete_quantile<>, + policies::assert_undefined<> >::type forwarding_policy; + // Force initialization of constants: - detail::digamma_initializer::force_instantiate(); + detail::digamma_initializer::force_instantiate(); return policies::checked_narrowing_cast(detail::digamma_imp( static_cast(x), - static_cast(0), pol), "boost::math::digamma<%1%>(%1%)"); + static_cast(0), forwarding_policy()), "boost::math::digamma<%1%>(%1%)"); } template diff --git a/include/boost/math/special_functions/factorials.hpp b/include/boost/math/special_functions/factorials.hpp index de24642ac..e36a098bb 100644 --- a/include/boost/math/special_functions/factorials.hpp +++ b/include/boost/math/special_functions/factorials.hpp @@ -151,7 +151,7 @@ T rising_factorial_imp(T x, int n, const Policy& pol) if((x < 1) && (x + n < 0)) { T val = boost::math::tgamma_delta_ratio(1 - x, static_cast(-n), pol); - return (n & 1) ? -val : val; + return (n & 1) ? T(-val) : val; } // // We don't optimise this for small n, because diff --git a/include/boost/math/special_functions/math_fwd.hpp b/include/boost/math/special_functions/math_fwd.hpp index e952dcdb5..f91d21eda 100644 --- a/include/boost/math/special_functions/math_fwd.hpp +++ b/include/boost/math/special_functions/math_fwd.hpp @@ -463,6 +463,20 @@ namespace boost template typename tools::promote_args::type digamma(T x, const Policy&); + // trigamma: + template + typename tools::promote_args::type trigamma(T x); + + template + typename tools::promote_args::type trigamma(T x, const Policy&); + + // polygamma: + template + typename tools::promote_args::type polygamma(int n, T x); + + template + typename tools::promote_args::type polygamma(int n, T x, const Policy&); + // Hypotenuse function sqrt(x ^ 2 + y ^ 2). template typename tools::promote_args::type @@ -1205,6 +1219,12 @@ namespace boost template \ inline typename boost::math::tools::promote_args::type digamma(T x){ return boost::math::digamma(x, Policy()); }\ \ + template \ + inline typename boost::math::tools::promote_args::type trigamma(T x){ return boost::math::trigamma(x, Policy()); }\ +\ + template \ + inline typename boost::math::tools::promote_args::type polygamma(int n, T x){ return boost::math::polygamma(n, x, Policy()); }\ + \ template \ inline typename boost::math::tools::promote_args::type \ hypot(T1 x, T2 y){ return boost::math::hypot(x, y, Policy()); }\ diff --git a/include/boost/math/special_functions/polygamma.hpp b/include/boost/math/special_functions/polygamma.hpp new file mode 100644 index 000000000..6b7815d5e --- /dev/null +++ b/include/boost/math/special_functions/polygamma.hpp @@ -0,0 +1,83 @@ + +/////////////////////////////////////////////////////////////////////////////// +// Copyright 2013 Nikhar Agrawal +// Copyright 2013 Christopher Kormanyos +// Copyright 2014 John Maddock +// Copyright 2013 Paul Bristow +// Distributed under the Boost +// Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifndef _BOOST_POLYGAMMA_2013_07_30_HPP_ + #define _BOOST_POLYGAMMA_2013_07_30_HPP_ + +#include +#include +#include + +namespace boost { namespace math { + + + template + inline typename tools::promote_args::type polygamma(const int n, T x, const Policy& pol) + { + // + // Filter off special cases right at the start: + // + if(n == 0) + return boost::math::digamma(x, pol); + if(n == 1) + return boost::math::trigamma(x, pol); + // + // We've found some standard library functions to misbehave if any FPU exception flags + // are set prior to their call, this code will clear those flags, then reset them + // on exit: + // + BOOST_FPU_EXCEPTION_GUARD + // + // The type of the result - the common type of T and U after + // any integer types have been promoted to double: + // + typedef typename tools::promote_args::type result_type; + // + // The type used for the calculation. This may be a wider type than + // the result in order to ensure full precision: + // + typedef typename policies::evaluation::type value_type; + // + // The type of the policy to forward to the actual implementation. + // We disable promotion of float and double as that's [possibly] + // happened already in the line above. Also reset to the default + // any policies we don't use (reduces code bloat if we're called + // multiple times with differing policies we don't actually use). + // Also normalise the type, again to reduce code bloat in case we're + // called multiple times with functionally identical policies that happen + // to be different types. + // + typedef typename policies::normalise< + Policy, + policies::promote_float, + policies::promote_double, + policies::discrete_quantile<>, + policies::assert_undefined<> >::type forwarding_policy; + // + // Whew. Now we can make the actual call to the implementation. + // Arguments are explicitly cast to the evaluation type, and the result + // passed through checked_narrowing_cast which handles things like overflow + // according to the policy passed: + // + return policies::checked_narrowing_cast( + detail::polygamma_imp(n, static_cast(x), forwarding_policy()), + "boost::math::polygamma<%1%>(int, %1%)"); + } + + template + inline typename tools::promote_args::type polygamma(const int n, T x) + { + return boost::math::polygamma(n, x, policies::policy<>()); + } + +} } // namespace boost::math + +#endif // _BOOST_BERNOULLI_2013_05_30_HPP_ + diff --git a/include/boost/math/special_functions/sin_pi.hpp b/include/boost/math/special_functions/sin_pi.hpp index 16aed51d2..ae6b3e744 100644 --- a/include/boost/math/special_functions/sin_pi.hpp +++ b/include/boost/math/special_functions/sin_pi.hpp @@ -53,10 +53,17 @@ T sin_pi_imp(T x, const Policy& pol) } // namespace detail template -inline typename tools::promote_args::type sin_pi(T x, const Policy& pol) +inline typename tools::promote_args::type sin_pi(T x, const Policy&) { typedef typename tools::promote_args::type result_type; - return boost::math::detail::sin_pi_imp(x, pol); + typedef typename policies::evaluation::type value_type; + typedef typename policies::normalise< + Policy, + policies::promote_float, + policies::promote_double, + policies::discrete_quantile<>, + policies::assert_undefined<> >::type forwarding_policy; + return policies::checked_narrowing_cast(boost::math::detail::sin_pi_imp(x, forwarding_policy()), "cos_pi"); } template diff --git a/include/boost/math/special_functions/trigamma.hpp b/include/boost/math/special_functions/trigamma.hpp new file mode 100644 index 000000000..6fccb36a3 --- /dev/null +++ b/include/boost/math/special_functions/trigamma.hpp @@ -0,0 +1,469 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifndef BOOST_MATH_SF_TRIGAMMA_HPP +#define BOOST_MATH_SF_TRIGAMMA_HPP + +#ifdef _MSC_VER +#pragma once +#endif + +#include +#include +#include +#include +#include +#include +#include +#include +#include + +namespace boost{ +namespace math{ +namespace detail{ + +template +T polygamma_imp(const int n, T x, const Policy &pol); + +template +T trigamma_prec(T x, const mpl::int_<53>*, const Policy&) +{ + // Max error in interpolated form: 3.736e-017 + static const T offset = BOOST_MATH_BIG_CONSTANT(T, 53, 2.1093254089355469); + static const T P_1_2[] = { + BOOST_MATH_BIG_CONSTANT(T, 53, -1.1093280605946045), + BOOST_MATH_BIG_CONSTANT(T, 53, -3.8310674472619321), + BOOST_MATH_BIG_CONSTANT(T, 53, -3.3703848401898283), + BOOST_MATH_BIG_CONSTANT(T, 53, 0.28080574467981213), + BOOST_MATH_BIG_CONSTANT(T, 53, 1.6638069578676164), + BOOST_MATH_BIG_CONSTANT(T, 53, 0.64468386819102836), + }; + static const T Q_1_2[] = { + BOOST_MATH_BIG_CONSTANT(T, 53, 1.0), + BOOST_MATH_BIG_CONSTANT(T, 53, 3.4535389668541151), + BOOST_MATH_BIG_CONSTANT(T, 53, 4.5208926987851437), + BOOST_MATH_BIG_CONSTANT(T, 53, 2.7012734178351534), + BOOST_MATH_BIG_CONSTANT(T, 53, 0.64468798399785611), + BOOST_MATH_BIG_CONSTANT(T, 53, -0.20314516859987728e-6), + }; + // Max error in interpolated form: 1.159e-017 + static const T P_2_4[] = { + BOOST_MATH_BIG_CONSTANT(T, 53, -0.13803835004508849e-7), + BOOST_MATH_BIG_CONSTANT(T, 53, 0.50000049158540261), + BOOST_MATH_BIG_CONSTANT(T, 53, 1.6077979838469348), + BOOST_MATH_BIG_CONSTANT(T, 53, 2.5645435828098254), + BOOST_MATH_BIG_CONSTANT(T, 53, 2.0534873203680393), + BOOST_MATH_BIG_CONSTANT(T, 53, 0.74566981111565923), + }; + static const T Q_2_4[] = { + BOOST_MATH_BIG_CONSTANT(T, 53, 1.0), + BOOST_MATH_BIG_CONSTANT(T, 53, 2.8822787662376169), + BOOST_MATH_BIG_CONSTANT(T, 53, 4.1681660554090917), + BOOST_MATH_BIG_CONSTANT(T, 53, 2.7853527819234466), + BOOST_MATH_BIG_CONSTANT(T, 53, 0.74967671848044792), + BOOST_MATH_BIG_CONSTANT(T, 53, -0.00057069112416246805), + }; + // Maximum Deviation Found: 6.896e-018 + // Expected Error Term : -6.895e-018 + // Maximum Relative Change in Control Points : 8.497e-004 + static const T P_4_inf[] = { + 0.68947581948701249e-17L, + 0.49999999999998975L, + 1.0177274392923795L, + 2.498208511343429L, + 2.1921221359427595L, + 1.5897035272532764L, + 0.40154388356961734L, + }; + static const T Q_4_inf[] = { + 1.0L, + 1.7021215452463932L, + 4.4290431747556469L, + 2.9745631894384922L, + 2.3013614809773616L, + 0.28360399799075752L, + 0.022892987908906897L, + }; + + if(x <= 2) + { + return (offset + boost::math::tools::evaluate_polynomial(P_1_2, x) / tools::evaluate_polynomial(Q_1_2, x)) / (x * x); + } + else if(x <= 4) + { + T y = 1 / x; + return (1 + tools::evaluate_polynomial(P_2_4, y) / tools::evaluate_polynomial(Q_2_4, y)) / x; + } + T y = 1 / x; + return (1 + tools::evaluate_polynomial(P_4_inf, y) / tools::evaluate_polynomial(Q_4_inf, y)) / x; +} + +template +T trigamma_prec(T x, const mpl::int_<64>*, const Policy&) +{ + // Max error in interpolated form: 1.178e-020 + static const T offset_1_2 = BOOST_MATH_BIG_CONSTANT(T, 64, 2.109325408935546875); + static const T P_1_2[] = { + BOOST_MATH_BIG_CONSTANT(T, 64, -1.10932535608960258341), + BOOST_MATH_BIG_CONSTANT(T, 64, -4.18793841543017129052), + BOOST_MATH_BIG_CONSTANT(T, 64, -4.63865531898487734531), + BOOST_MATH_BIG_CONSTANT(T, 64, -0.919832884430500908047), + BOOST_MATH_BIG_CONSTANT(T, 64, 1.68074038333180423012), + BOOST_MATH_BIG_CONSTANT(T, 64, 1.21172611429185622377), + BOOST_MATH_BIG_CONSTANT(T, 64, 0.259635673503366427284), + }; + static const T Q_1_2[] = { + BOOST_MATH_BIG_CONSTANT(T, 64, 1.0), + BOOST_MATH_BIG_CONSTANT(T, 64, 3.77521119359546982995), + BOOST_MATH_BIG_CONSTANT(T, 64, 5.664338024578956321), + BOOST_MATH_BIG_CONSTANT(T, 64, 4.25995134879278028361), + BOOST_MATH_BIG_CONSTANT(T, 64, 1.62956638448940402182), + BOOST_MATH_BIG_CONSTANT(T, 64, 0.259635512844691089868), + BOOST_MATH_BIG_CONSTANT(T, 64, 0.629642219810618032207e-8), + }; + // Max error in interpolated form: 3.912e-020 + static const T P_2_8[] = { + BOOST_MATH_BIG_CONSTANT(T, 64, -0.387540035162952880976e-11), + BOOST_MATH_BIG_CONSTANT(T, 64, 0.500000000276430504), + BOOST_MATH_BIG_CONSTANT(T, 64, 3.21926880986360957306), + BOOST_MATH_BIG_CONSTANT(T, 64, 10.2550347708483445775), + BOOST_MATH_BIG_CONSTANT(T, 64, 18.9002075150709144043), + BOOST_MATH_BIG_CONSTANT(T, 64, 21.0357215832399705625), + BOOST_MATH_BIG_CONSTANT(T, 64, 13.4346512182925923978), + BOOST_MATH_BIG_CONSTANT(T, 64, 3.98656291026448279118), + }; + static const T Q_2_8[] = { + BOOST_MATH_BIG_CONSTANT(T, 64, 1.0), + BOOST_MATH_BIG_CONSTANT(T, 64, 6.10520430478613667724), + BOOST_MATH_BIG_CONSTANT(T, 64, 18.475001060603645512), + BOOST_MATH_BIG_CONSTANT(T, 64, 31.7087534567758405638), + BOOST_MATH_BIG_CONSTANT(T, 64, 31.908814523890465398), + BOOST_MATH_BIG_CONSTANT(T, 64, 17.4175479039227084798), + BOOST_MATH_BIG_CONSTANT(T, 64, 3.98749106958394941276), + BOOST_MATH_BIG_CONSTANT(T, 64, -0.000115917322224411128566), + }; + // Maximum Deviation Found: 2.635e-020 + // Expected Error Term : 2.635e-020 + // Maximum Relative Change in Control Points : 1.791e-003 + static const T P_8_inf[] = { + BOOST_MATH_BIG_CONSTANT(T, 64, -0.263527875092466899848e-19), + BOOST_MATH_BIG_CONSTANT(T, 64, 0.500000000000000058145), + BOOST_MATH_BIG_CONSTANT(T, 64, 0.0730121433777364138677), + BOOST_MATH_BIG_CONSTANT(T, 64, 1.94505878379957149534), + BOOST_MATH_BIG_CONSTANT(T, 64, 0.0517092358874932620529), + BOOST_MATH_BIG_CONSTANT(T, 64, 1.07995383547483921121), + }; + static const T Q_8_inf[] = { + BOOST_MATH_BIG_CONSTANT(T, 64, 1.0), + BOOST_MATH_BIG_CONSTANT(T, 64, -0.187309046577818095504), + BOOST_MATH_BIG_CONSTANT(T, 64, 3.95255391645238842975), + BOOST_MATH_BIG_CONSTANT(T, 64, -1.14743283327078949087), + BOOST_MATH_BIG_CONSTANT(T, 64, 2.52989799376344914499), + BOOST_MATH_BIG_CONSTANT(T, 64, -0.627414303172402506396), + BOOST_MATH_BIG_CONSTANT(T, 64, 0.141554248216425512536), + }; + + if(x <= 2) + { + return (offset_1_2 + boost::math::tools::evaluate_polynomial(P_1_2, x) / tools::evaluate_polynomial(Q_1_2, x)) / (x * x); + } + else if(x <= 8) + { + T y = 1 / x; + return (1 + tools::evaluate_polynomial(P_2_8, y) / tools::evaluate_polynomial(Q_2_8, y)) / x; + } + T y = 1 / x; + return (1 + tools::evaluate_polynomial(P_8_inf, y) / tools::evaluate_polynomial(Q_8_inf, y)) / x; +} + +template +T trigamma_prec(T x, const mpl::int_<113>*, const Policy&) +{ + // Max error in interpolated form: 1.916e-035 + + static const T P_1_2[] = { + BOOST_MATH_BIG_CONSTANT(T, 113, -0.999999999999999082554457936871832533), + BOOST_MATH_BIG_CONSTANT(T, 113, -4.71237311120865266379041700054847734), + BOOST_MATH_BIG_CONSTANT(T, 113, -7.94125711970499027763789342500817316), + BOOST_MATH_BIG_CONSTANT(T, 113, -5.74657746697664735258222071695644535), + BOOST_MATH_BIG_CONSTANT(T, 113, -0.404213349456398905981223965160595687), + BOOST_MATH_BIG_CONSTANT(T, 113, 2.47877781178642876561595890095758896), + BOOST_MATH_BIG_CONSTANT(T, 113, 2.07714151702455125992166949812126433), + BOOST_MATH_BIG_CONSTANT(T, 113, 0.858877899162360138844032265418028567), + BOOST_MATH_BIG_CONSTANT(T, 113, 0.20499222604410032375789018837922397), + BOOST_MATH_BIG_CONSTANT(T, 113, 0.0272103140348194747360175268778415049), + BOOST_MATH_BIG_CONSTANT(T, 113, 0.0015764849020876949848954081173520686), + }; + static const T Q_1_2[] = { + BOOST_MATH_BIG_CONSTANT(T, 113, 1.0), + BOOST_MATH_BIG_CONSTANT(T, 113, 4.71237311120863419878375031457715223), + BOOST_MATH_BIG_CONSTANT(T, 113, 9.58619118655339853449127952145877467), + BOOST_MATH_BIG_CONSTANT(T, 113, 11.0940067269829372437561421279054968), + BOOST_MATH_BIG_CONSTANT(T, 113, 8.09075424749327792073276309969037885), + BOOST_MATH_BIG_CONSTANT(T, 113, 3.87705890159891405185343806884451286), + BOOST_MATH_BIG_CONSTANT(T, 113, 1.22758678701914477836330837816976782), + BOOST_MATH_BIG_CONSTANT(T, 113, 0.249092040606385004109672077814668716), + BOOST_MATH_BIG_CONSTANT(T, 113, 0.0295750413900655597027079600025569048), + BOOST_MATH_BIG_CONSTANT(T, 113, 0.00157648490200498142247694709728858139), + BOOST_MATH_BIG_CONSTANT(T, 113, 0.161264050344059471721062360645432809e-14), + }; + + // Max error in interpolated form: 8.958e-035 + static const T P_2_4[] = { + BOOST_MATH_BIG_CONSTANT(T, 113, -2.55843734739907925764326773972215085), + BOOST_MATH_BIG_CONSTANT(T, 113, -12.2830208240542011967952466273455887), + BOOST_MATH_BIG_CONSTANT(T, 113, -23.9195022162767993526575786066414403), + BOOST_MATH_BIG_CONSTANT(T, 113, -24.9256431504823483094158828285470862), + BOOST_MATH_BIG_CONSTANT(T, 113, -14.7979122765478779075108064826412285), + BOOST_MATH_BIG_CONSTANT(T, 113, -4.46654453928610666393276765059122272), + BOOST_MATH_BIG_CONSTANT(T, 113, -0.0191439033405649675717082465687845002), + BOOST_MATH_BIG_CONSTANT(T, 113, 0.515412052554351265708917209749037352), + BOOST_MATH_BIG_CONSTANT(T, 113, 0.195378348786064304378247325360320038), + BOOST_MATH_BIG_CONSTANT(T, 113, 0.0334761282624174313035014426794245393), + BOOST_MATH_BIG_CONSTANT(T, 113, 0.002373665205942206348500250056602687), + }; + static const T Q_2_4[] = { + BOOST_MATH_BIG_CONSTANT(T, 113, 1.0), + BOOST_MATH_BIG_CONSTANT(T, 113, 4.80098558454419907830670928248659245), + BOOST_MATH_BIG_CONSTANT(T, 113, 9.99220727843170133895059300223445265), + BOOST_MATH_BIG_CONSTANT(T, 113, 11.8896146167631330735386697123464976), + BOOST_MATH_BIG_CONSTANT(T, 113, 8.96613256683809091593793565879092581), + BOOST_MATH_BIG_CONSTANT(T, 113, 4.47254136149624110878909334574485751), + BOOST_MATH_BIG_CONSTANT(T, 113, 1.48600982028196527372434773913633152), + BOOST_MATH_BIG_CONSTANT(T, 113, 0.319570735766764237068541501137990078), + BOOST_MATH_BIG_CONSTANT(T, 113, 0.0407358345787680953107374215319322066), + BOOST_MATH_BIG_CONSTANT(T, 113, 0.00237366520593271641375755486420859837), + BOOST_MATH_BIG_CONSTANT(T, 113, 0.239554887903526152679337256236302116e-15), + BOOST_MATH_BIG_CONSTANT(T, 113, -0.294749244740618656265237072002026314e-17), + }; + + static const T y_offset_2_4 = BOOST_MATH_BIG_CONSTANT(T, 113, 3.558437347412109375); + + // Max error in interpolated form: 4.319e-035 + static const T P_4_8[] = { + BOOST_MATH_BIG_CONSTANT(T, 113, 0.166626112697021464248967707021688845e-16), + BOOST_MATH_BIG_CONSTANT(T, 113, 0.499999999999997739552090249208808197), + BOOST_MATH_BIG_CONSTANT(T, 113, 6.40270945019053817915772473771553187), + BOOST_MATH_BIG_CONSTANT(T, 113, 41.3833374155000608013677627389343329), + BOOST_MATH_BIG_CONSTANT(T, 113, 166.803341854562809335667241074035245), + BOOST_MATH_BIG_CONSTANT(T, 113, 453.39964786925369319960722793414521), + BOOST_MATH_BIG_CONSTANT(T, 113, 851.153712317697055375935433362983944), + BOOST_MATH_BIG_CONSTANT(T, 113, 1097.70657567285059133109286478004458), + BOOST_MATH_BIG_CONSTANT(T, 113, 938.431232478455316020076349367632922), + BOOST_MATH_BIG_CONSTANT(T, 113, 487.268001604651932322080970189930074), + BOOST_MATH_BIG_CONSTANT(T, 113, 119.953445242335730062471193124820659), + }; + static const T Q_4_8[] = { + BOOST_MATH_BIG_CONSTANT(T, 113, 1.0), + BOOST_MATH_BIG_CONSTANT(T, 113, 12.4720855670474488978638945855932398), + BOOST_MATH_BIG_CONSTANT(T, 113, 78.6093129753298570701376952709727391), + BOOST_MATH_BIG_CONSTANT(T, 113, 307.470246050318322489781182863190127), + BOOST_MATH_BIG_CONSTANT(T, 113, 805.140686101151538537565264188630079), + BOOST_MATH_BIG_CONSTANT(T, 113, 1439.12019760292146454787601409644413), + BOOST_MATH_BIG_CONSTANT(T, 113, 1735.6105285756048831268586001383127), + BOOST_MATH_BIG_CONSTANT(T, 113, 1348.32500712856328019355198611280536), + BOOST_MATH_BIG_CONSTANT(T, 113, 607.225985860570846699704222144650563), + BOOST_MATH_BIG_CONSTANT(T, 113, 119.952317857277045332558673164517227), + BOOST_MATH_BIG_CONSTANT(T, 113, 0.000140165918355036060868680809129436084), + }; + + // Maximum Deviation Found: 2.867e-035 + // Expected Error Term : 2.866e-035 + // Maximum Relative Change in Control Points : 2.662e-004 + static const T P_8_16[] = { + BOOST_MATH_BIG_CONSTANT(T, 113, -0.184828315274146610610872315609837439e-19), + BOOST_MATH_BIG_CONSTANT(T, 113, 0.500000000000000004122475157735807738), + BOOST_MATH_BIG_CONSTANT(T, 113, 3.02533865247313349284875558880415875), + BOOST_MATH_BIG_CONSTANT(T, 113, 13.5995927517457371243039532492642734), + BOOST_MATH_BIG_CONSTANT(T, 113, 35.3132224283087906757037999452941588), + BOOST_MATH_BIG_CONSTANT(T, 113, 67.1639424550714159157603179911505619), + BOOST_MATH_BIG_CONSTANT(T, 113, 83.5767733658513967581959839367419891), + BOOST_MATH_BIG_CONSTANT(T, 113, 71.073491212235705900866411319363501), + BOOST_MATH_BIG_CONSTANT(T, 113, 35.8621515614725564575893663483998663), + BOOST_MATH_BIG_CONSTANT(T, 113, 8.72152231639983491987779743154333318), + }; + static const T Q_8_16[] = { + BOOST_MATH_BIG_CONSTANT(T, 113, 1.0), + BOOST_MATH_BIG_CONSTANT(T, 113, 5.71734397161293452310624822415866372), + BOOST_MATH_BIG_CONSTANT(T, 113, 25.293404179620438179337103263274815), + BOOST_MATH_BIG_CONSTANT(T, 113, 62.2619767967468199111077640625328469), + BOOST_MATH_BIG_CONSTANT(T, 113, 113.955048909238993473389714972250235), + BOOST_MATH_BIG_CONSTANT(T, 113, 130.807138328938966981862203944329408), + BOOST_MATH_BIG_CONSTANT(T, 113, 102.423146902337654110717764213057753), + BOOST_MATH_BIG_CONSTANT(T, 113, 44.0424772805245202514468199602123565), + BOOST_MATH_BIG_CONSTANT(T, 113, 8.89898032477904072082994913461386099), + BOOST_MATH_BIG_CONSTANT(T, 113, -0.0296627336872039988632793863671456398), + }; + // Maximum Deviation Found: 1.079e-035 + // Expected Error Term : -1.079e-035 + // Maximum Relative Change in Control Points : 7.884e-003 + static const T P_16_inf[] = { + BOOST_MATH_BIG_CONSTANT(T, 113, 0.0), + BOOST_MATH_BIG_CONSTANT(T, 113, 0.500000000000000000000000000000087317), + BOOST_MATH_BIG_CONSTANT(T, 113, 0.345625669885456215194494735902663968), + BOOST_MATH_BIG_CONSTANT(T, 113, 9.62895499360842232127552650044647769), + BOOST_MATH_BIG_CONSTANT(T, 113, 3.5936085382439026269301003761320812), + BOOST_MATH_BIG_CONSTANT(T, 113, 49.459599118438883265036646019410669), + BOOST_MATH_BIG_CONSTANT(T, 113, 7.77519237321893917784735690560496607), + BOOST_MATH_BIG_CONSTANT(T, 113, 74.4536074488178075948642351179304121), + BOOST_MATH_BIG_CONSTANT(T, 113, 2.75209340397069050436806159297952699), + BOOST_MATH_BIG_CONSTANT(T, 113, 23.9292359711471667884504840186561598), + }; + static const T Q_16_inf[] = { + BOOST_MATH_BIG_CONSTANT(T, 113, 1.0), + BOOST_MATH_BIG_CONSTANT(T, 113, 0.357918006437579097055656138920742037), + BOOST_MATH_BIG_CONSTANT(T, 113, 19.1386039850709849435325005484512944), + BOOST_MATH_BIG_CONSTANT(T, 113, 0.874349081464143606016221431763364517), + BOOST_MATH_BIG_CONSTANT(T, 113, 98.6516097434855572678195488061432509), + BOOST_MATH_BIG_CONSTANT(T, 113, -16.1051972833382893468655223662534306), + BOOST_MATH_BIG_CONSTANT(T, 113, 154.316860216253720989145047141653727), + BOOST_MATH_BIG_CONSTANT(T, 113, -40.2026880424378986053105969312264534), + BOOST_MATH_BIG_CONSTANT(T, 113, 60.1679136674264778074736441126810223), + BOOST_MATH_BIG_CONSTANT(T, 113, -13.3414844622256422644504472438320114), + BOOST_MATH_BIG_CONSTANT(T, 113, 2.53795636200649908779512969030363442), + }; + + if(x <= 2) + { + return (2 + boost::math::tools::evaluate_polynomial(P_1_2, x) / tools::evaluate_polynomial(Q_1_2, x)) / (x * x); + } + else if(x <= 4) + { + return (y_offset_2_4 + boost::math::tools::evaluate_polynomial(P_2_4, x) / tools::evaluate_polynomial(Q_2_4, x)) / (x * x); + } + else if(x <= 8) + { + T y = 1 / x; + return (1 + tools::evaluate_polynomial(P_4_8, y) / tools::evaluate_polynomial(Q_4_8, y)) / x; + } + else if(x <= 16) + { + T y = 1 / x; + return (1 + tools::evaluate_polynomial(P_8_16, y) / tools::evaluate_polynomial(Q_8_16, y)) / x; + } + T y = 1 / x; + return (1 + tools::evaluate_polynomial(P_16_inf, y) / tools::evaluate_polynomial(Q_16_inf, y)) / x; +} + +template +T trigamma_imp(T x, const Tag* t, const Policy& pol) +{ + // + // This handles reflection of negative arguments, and all our + // error handling, then forwards to the T-specific approximation. + // + BOOST_MATH_STD_USING // ADL of std functions. + + T result = 0; + // + // Check for negative arguments and use reflection: + // + if(x <= 0) + { + // Reflect: + T z = 1 - x; + // Argument reduction for tan: + if(floor(x) == x) + { + return policies::raise_pole_error("boost::math::trigamma<%1%>(%1%)", 0, (1-x), pol); + } + T s = fabs(x) < fabs(z) ? boost::math::sin_pi(x, pol) : boost::math::sin_pi(z, pol); + return -trigamma_imp(z, t, pol) + boost::math::pow<2>(constants::pi()) / (s * s); + } + if(x < 1) + { + result = 1 / (x * x); + x += 1; + } + return result + trigamma_prec(x, t, pol); +} + +template +T trigamma_imp(T x, const mpl::int_<0>*, const Policy& pol) +{ + return polygamma_imp(1, x, pol); +} +// +// Initializer: ensure all our constants are initialized prior to the first call of main: +// +template +struct trigamma_initializer +{ + struct init + { + init() + { + typedef typename policies::precision::type precision_type; + do_init(mpl::bool_()); + } + void do_init(const mpl::true_&) + { + boost::math::trigamma(T(2.5), Policy()); + } + void do_init(const mpl::false_&){} + void force_instantiate()const{} + }; + static const init initializer; + static void force_instantiate() + { + initializer.force_instantiate(); + } +}; + +template +const typename trigamma_initializer::init trigamma_initializer::initializer; + +} // namespace detail + +template +inline typename tools::promote_args::type + trigamma(T x, const Policy&) +{ + typedef typename tools::promote_args::type result_type; + typedef typename policies::evaluation::type value_type; + typedef typename policies::precision::type precision_type; + typedef typename mpl::if_< + mpl::or_< + mpl::less_equal >, + mpl::greater > + >, + mpl::int_<0>, + typename mpl::if_< + mpl::less >, + mpl::int_<53>, + typename mpl::if_< + mpl::less >, + mpl::int_<64>, + mpl::int_<113> + >::type + >::type + >::type tag_type; + + typedef typename policies::normalise< + Policy, + policies::promote_float, + policies::promote_double, + policies::discrete_quantile<>, + policies::assert_undefined<> >::type forwarding_policy; + + // Force initialization of constants: + detail::trigamma_initializer::force_instantiate(); + + return policies::checked_narrowing_cast(detail::trigamma_imp( + static_cast(x), + static_cast(0), forwarding_policy()), "boost::math::trigamma<%1%>(%1%)"); +} + +template +inline typename tools::promote_args::type + trigamma(T x) +{ + return trigamma(x, policies::policy<>()); +} + +} // namespace math +} // namespace boost +#endif + diff --git a/include/boost/math/special_functions/zeta.hpp b/include/boost/math/special_functions/zeta.hpp index b176f2017..1ba282f4d 100644 --- a/include/boost/math/special_functions/zeta.hpp +++ b/include/boost/math/special_functions/zeta.hpp @@ -1,4 +1,4 @@ -// Copyright John Maddock 2007. +// Copyright John Maddock 2007, 2014. // Use, modification and distribution are subject to the // Boost Software License, Version 1.0. (See accompanying file // LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) @@ -16,6 +16,7 @@ #include #include #include +#include #include namespace boost{ namespace math{ namespace detail{ @@ -167,6 +168,8 @@ T zeta_imp_prec(T s, T sc, const Policy& pol, const mpl::int_<0>&) { BOOST_MATH_STD_USING T result; + if(s >= policies::digits()) + return 1; result = zeta_polynomial_series(s, sc, pol); #if 0 // Old code archived for future reference: @@ -863,6 +866,34 @@ T zeta_imp_prec(T s, T sc, const Policy&, const mpl::int_<113>&) return result; } +template +T zeta_imp_odd_integer(int s, const T&, const Policy&, const mpl::true_&) +{ + static const T results[] = { + BOOST_MATH_BIG_CONSTANT(T, 113, 1.2020569031595942853997381615114500), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0369277551433699263313654864570342), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0083492773819228268397975498497968), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0020083928260822144178527692324121), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0004941886041194645587022825264699), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0001227133475784891467518365263574), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000305882363070204935517285106451), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000076371976378997622736002935630), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000019082127165539389256569577951), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000004769329867878064631167196044), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000001192199259653110730677887189), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000298035035146522801860637051), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000074507117898354294919810042), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000018626597235130490064039099), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000004656629065033784072989233), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000001164155017270051977592974), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000291038504449709968692943), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000072759598350574810145209), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000018189896503070659475848), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000004547473783042154026799), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000001136868407680227849349), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000284217097688930185546), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000071054273952108527129), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000017763568435791203275), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000004440892103143813364), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000001110223025141066134), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000277555756213612417), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000069388939045441537), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000017347234760475766), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000004336808690020650), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000001084202172494241), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000271050543122347), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000067762635780452), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000016940658945098), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000004235164736273), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000001058791184068), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000264697796017), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000066174449004), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000016543612251), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000004135903063), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000001033975766), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000258493941), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000064623485), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000016155871), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000004038968), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000001009742), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000252435), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000063109), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000015777), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000003944), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000000986), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000000247), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000000062), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000000015), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000000004), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000000001), + }; + return s > 113 ? 1 : results[(s - 3) / 2]; +} + +template +T zeta_imp_odd_integer(int s, const T& sc, const Policy& pol, const mpl::false_&) +{ + static bool is_init = false; + static T results[50] = {}; + if(!is_init) + { + is_init = true; + for(int k = 0; k < sizeof(results) / sizeof(results[0]); ++k) + { + T arg = k * 2 + 3; + T c_arg = 1 - arg; + results[k] = zeta_polynomial_series(arg, c_arg, pol); + } + } + int index = (s - 3) / 2; + return index >= sizeof(results) / sizeof(results[0]) ? zeta_polynomial_series(T(s), sc, pol): results[index]; +} + template T zeta_imp(T s, T sc, const Policy& pol, const Tag& tag) { @@ -874,6 +905,45 @@ T zeta_imp(T s, T sc, const Policy& pol, const Tag& tag) "Evaluation of zeta function at pole %1%", s, pol); T result; + // + // Trivial case: + // + if(s > policies::digits()) + return 1; + // + // Start by seeing if we have a simple closed form: + // + if(floor(s) == s) + { + try + { + int v = itrunc(s); + if(v == s) + { + if(v < 0) + { + if(((-v) & 1) == 0) + return 0; + int n = (-v + 1) / 2; + if(n <= boost::math::max_bernoulli_b2n::value) + return T((-v & 1) ? -1 : 1) * boost::math::unchecked_bernoulli_b2n(n) / (1 - v); + } + else if((v & 1) == 0) + { + if(((v / 2) <= boost::math::max_bernoulli_b2n::value) && (v <= boost::math::max_factorial::value)) + return T(((v / 2 - 1) & 1) ? -1 : 1) * ldexp(T(1), v - 1) * pow(constants::pi(), v) * + boost::math::unchecked_bernoulli_b2n(v / 2) / boost::math::unchecked_factorial(v); + return T(((v / 2 - 1) & 1) ? -1 : 1) * ldexp(T(1), v - 1) * pow(constants::pi(), v) * + boost::math::bernoulli_b2n(v / 2) / boost::math::factorial(v); + } + else + return zeta_imp_odd_integer(v, sc, pol, mpl::bool_<(Tag::value <= 113) && Tag::value>()); + } + } + catch(const boost::math::rounding_error&){} // Just fall through, s is too large to round + catch(const std::overflow_error&){} + } + if(fabs(s) < tools::root_epsilon()) { result = -0.5f - constants::log_root_two_pi() * s; @@ -922,8 +992,8 @@ struct zeta_initializer { do_init(tag()); } - static void do_init(const mpl::int_<0>&){} - static void do_init(const mpl::int_<53>&){} + static void do_init(const mpl::int_<0>&){ boost::math::zeta(static_cast(5), Policy()); } + static void do_init(const mpl::int_<53>&){ boost::math::zeta(static_cast(5), Policy()); } static void do_init(const mpl::int_<64>&) { boost::math::zeta(static_cast(0.5), Policy()); @@ -932,6 +1002,8 @@ struct zeta_initializer boost::math::zeta(static_cast(6.5), Policy()); boost::math::zeta(static_cast(14.5), Policy()); boost::math::zeta(static_cast(40.5), Policy()); + + boost::math::zeta(static_cast(5), Policy()); } static void do_init(const mpl::int_<113>&) { @@ -941,8 +1013,10 @@ struct zeta_initializer boost::math::zeta(static_cast(5.5), Policy()); boost::math::zeta(static_cast(9.5), Policy()); boost::math::zeta(static_cast(16.5), Policy()); - boost::math::zeta(static_cast(25), Policy()); - boost::math::zeta(static_cast(70), Policy()); + boost::math::zeta(static_cast(25.5), Policy()); + boost::math::zeta(static_cast(70.5), Policy()); + + boost::math::zeta(static_cast(5), Policy()); } void force_instantiate()const{} }; diff --git a/minimax/f.cpp b/minimax/f.cpp index 3435f0175..74ff97638 100644 --- a/minimax/f.cpp +++ b/minimax/f.cpp @@ -323,6 +323,20 @@ boost::math::ntl::RR f(const boost::math::ntl::RR& x, int variant) // cbrt over [0.5, 1] return boost::math::cbrt(x); } + case 30: + { + // trigamma over [x,y] + boost::math::ntl::RR y = x; + y = sqrt(y); + return boost::math::trigamma(x) * (x * x); + } + case 31: + { + // trigamma over [x, INF] + if(x == 0) return 1; + boost::math::ntl::RR y = (x == 0) ? (std::numeric_limits::max)() / 2 : 1/x; + return boost::math::trigamma(y) * y; + } } return 0; } diff --git a/minimax/main.cpp b/minimax/main.cpp index a12657ed8..bc430ec4f 100644 --- a/minimax/main.cpp +++ b/minimax/main.cpp @@ -3,6 +3,7 @@ // Boost Software License, Version 1.0. (See accompanying file // LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +#define BOOST_TEST_MODULE foobar #define BOOST_UBLAS_TYPE_CHECK_EPSILON (type_traits::type_sqrt (boost::math::tools::epsilon ())) #define BOOST_UBLAS_TYPE_CHECK_MIN (type_traits::type_sqrt ( boost::math::tools::min_value())) #define BOOST_UBLAS_NDEBUG @@ -21,6 +22,7 @@ using boost::math::ntl::pow; #include #include #include // for test_main +#include extern boost::math::ntl::RR f(const boost::math::ntl::RR& x, int variant); extern void show_extra( @@ -210,6 +212,41 @@ void graph(const char*, const char*) do_graph(3); } +template +boost::math::ntl::RR convert_to_rr(const T& val) +{ + return val; +} +template +boost::math::ntl::RR convert_to_rr(const boost::multiprecision::number& val) +{ + return boost::lexical_cast(val.str()); +} + +namespace boost{ namespace math{ namespace tools{ + +template <> +boost::multiprecision::cpp_bin_float_double_extended real_cast(boost::math::ntl::RR val) +{ + unsigned p = NTL::RR::OutputPrecision(); + NTL::RR::SetOutputPrecision(20); + boost::multiprecision::cpp_bin_float_double_extended r = boost::lexical_cast(val); + NTL::RR::SetOutputPrecision(p); + return r; +} +template <> +boost::multiprecision::cpp_bin_float_quad real_cast(boost::math::ntl::RR val) +{ + unsigned p = NTL::RR::OutputPrecision(); + NTL::RR::SetOutputPrecision(35); + boost::multiprecision::cpp_bin_float_quad r = boost::lexical_cast(val); + NTL::RR::SetOutputPrecision(p); + return r; +} + +}}} + + template void do_test(T, const char* name) { @@ -260,8 +297,8 @@ void do_test(T, const char* name) { boost::math::ntl::RR true_result = the_function(zeros[i]); T absissa = boost::math::tools::real_cast(zeros[i]); - boost::math::ntl::RR test_result = n.evaluate(absissa) / d.evaluate(absissa); - boost::math::ntl::RR cheb_result = boost::math::tools::evaluate_chebyshev(cn, absissa) / boost::math::tools::evaluate_chebyshev(cd, absissa); + boost::math::ntl::RR test_result = convert_to_rr(n.evaluate(absissa) / d.evaluate(absissa)); + boost::math::ntl::RR cheb_result = convert_to_rr(boost::math::tools::evaluate_chebyshev(cn, absissa) / boost::math::tools::evaluate_chebyshev(cd, absissa)); boost::math::ntl::RR err, cheb_err; if(rel_error) { @@ -287,8 +324,8 @@ void do_test(T, const char* name) { boost::math::ntl::RR true_result = the_function(cheb[i]); T absissa = boost::math::tools::real_cast(cheb[i]); - boost::math::ntl::RR test_result = n.evaluate(absissa) / d.evaluate(absissa); - boost::math::ntl::RR cheb_result = boost::math::tools::evaluate_chebyshev(cn, absissa) / boost::math::tools::evaluate_chebyshev(cd, absissa); + boost::math::ntl::RR test_result = convert_to_rr(n.evaluate(absissa) / d.evaluate(absissa)); + boost::math::ntl::RR cheb_result = convert_to_rr(boost::math::tools::evaluate_chebyshev(cn, absissa) / boost::math::tools::evaluate_chebyshev(cd, absissa)); boost::math::ntl::RR err, cheb_err; if(rel_error) { @@ -334,6 +371,16 @@ void test_long(const char*, const char*) do_test((long double)(0), "long double"); } +void test_float80(const char*, const char*) +{ + do_test((boost::multiprecision::cpp_bin_float_double_extended)(0), "float80"); +} + +void test_float128(const char*, const char*) +{ + do_test((boost::multiprecision::cpp_bin_float_quad)(0), "float128"); +} + void test_all(const char*, const char*) { do_test(float(0), "float"); @@ -383,9 +430,12 @@ void do_test_n(T, const char* name, unsigned count) for(boost::math::ntl::RR x = a; x <= b; x += step) { boost::math::ntl::RR true_result = the_function(x); + //std::cout << true_result << std::endl; T absissa = boost::math::tools::real_cast(x); - boost::math::ntl::RR test_result = n.evaluate(absissa) / d.evaluate(absissa); - boost::math::ntl::RR cheb_result = boost::math::tools::evaluate_chebyshev(cn, absissa) / boost::math::tools::evaluate_chebyshev(cd, absissa); + boost::math::ntl::RR test_result = convert_to_rr(n.evaluate(absissa) / d.evaluate(absissa)); + //std::cout << test_result << std::endl; + boost::math::ntl::RR cheb_result = convert_to_rr(boost::math::tools::evaluate_chebyshev(cn, absissa) / boost::math::tools::evaluate_chebyshev(cd, absissa)); + //std::cout << cheb_result << std::endl; boost::math::ntl::RR err, cheb_err; if(rel_error) { @@ -441,6 +491,16 @@ void test_long_n(unsigned n) do_test_n((long double)(0), "long double", n); } +void test_float80_n(unsigned n) +{ + do_test_n((boost::multiprecision::cpp_bin_float_double_extended)(0), "float80", n); +} + +void test_float128_n(unsigned n) +{ + do_test_n((boost::multiprecision::cpp_bin_float_quad)(0), "float128", n); +} + void rotate(const char*, const char*) { if(p_remez) @@ -504,7 +564,7 @@ BOOST_AUTO_TEST_CASE( test_main ) while(std::getline(std::cin, line)) { if(parse(line.c_str(), str_p("quit"), space_p).full) - return 0; + return; if(false == parse(line.c_str(), ( @@ -569,6 +629,14 @@ BOOST_AUTO_TEST_CASE( test_main ) str_p("test") && str_p("long") && uint_p[&test_long_n] || str_p("test") && str_p("long")[&test_long] + || + str_p("test") && str_p("float80") && uint_p[&test_float80_n] + || + str_p("test") && str_p("float80")[&test_float80] + || + str_p("test") && str_p("float128") && uint_p[&test_float128_n] + || + str_p("test") && str_p("float128")[&test_float128] || str_p("test") && str_p("all")[&test_all] || diff --git a/test/Jamfile.v2 b/test/Jamfile.v2 index 131943151..d57581907 100644 --- a/test/Jamfile.v2 +++ b/test/Jamfile.v2 @@ -603,6 +603,8 @@ run test_nc_t.cpp pch ../../test/build//boost_unit_test_framework run test_normal.cpp pch ../../test/build//boost_unit_test_framework ; run test_owens_t.cpp ../../test/build//boost_unit_test_framework ; run test_pareto.cpp ../../test/build//boost_unit_test_framework ; +run test_polygamma.cpp test_instances//test_instances pch_light ../../test/build//boost_unit_test_framework ; +run test_trigamma.cpp test_instances//test_instances ../../test/build//boost_unit_test_framework ; run test_poisson.cpp ../../test/build//boost_unit_test_framework : # command line : # input files @@ -666,6 +668,7 @@ run test_skew_normal.cpp ../../test/build//boost_unit_test_framework ; run test_tgamma_ratio.cpp test_instances//test_instances pch_light ../../test/build//boost_unit_test_framework ; run test_toms748_solve.cpp pch ../../test/build//boost_unit_test_framework ; run test_triangular.cpp pch ../../test/build//boost_unit_test_framework ; +run test_trig.cpp test_instances//test_instances pch_light ../../test/build//boost_unit_test_framework ; run test_uniform.cpp pch ../../test/build//boost_unit_test_framework ; run test_weibull.cpp ../../test/build//boost_unit_test_framework ; run test_zeta.cpp ../../test/build//boost_unit_test_framework test_instances//test_instances pch_light ; @@ -772,6 +775,7 @@ run compile_test/sf_binomial_incl_test.cpp compile_test_main ; run compile_test/sf_cbrt_incl_test.cpp compile_test_main ; run compile_test/sf_cos_pi_incl_test.cpp compile_test_main ; run compile_test/sf_digamma_incl_test.cpp compile_test_main ; +run compile_test/sf_polygamma_incl_test.cpp compile_test_main ; run compile_test/sf_ellint_1_incl_test.cpp compile_test_main ; run compile_test/sf_ellint_2_incl_test.cpp compile_test_main ; run compile_test/sf_ellint_3_incl_test.cpp compile_test_main ; diff --git a/test/compile_test/instantiate.hpp b/test/compile_test/instantiate.hpp index c6a9d5a7e..776829a3a 100644 --- a/test/compile_test/instantiate.hpp +++ b/test/compile_test/instantiate.hpp @@ -166,6 +166,8 @@ void instantiate(RealType) boost::math::lgamma(v1); boost::math::lgamma(v1, &i); boost::math::digamma(v1); + boost::math::trigamma(v1); + boost::math::polygamma(i, v1); boost::math::tgamma_ratio(v1, v2); boost::math::tgamma_delta_ratio(v1, v2); boost::math::factorial(i); @@ -351,6 +353,8 @@ void instantiate(RealType) boost::math::lgamma(v1 * 1); boost::math::lgamma(v1 * 1, &i); boost::math::digamma(v1 * 1); + boost::math::trigamma(v1 * 1); + boost::math::polygamma(i, v1 * 1); boost::math::tgamma_ratio(v1 * 1, v2 + 0); boost::math::tgamma_delta_ratio(v1 * 1, v2 + 0); boost::math::factorial(i); @@ -525,6 +529,8 @@ void instantiate(RealType) boost::math::lgamma(v1, pol); boost::math::lgamma(v1, &i, pol); boost::math::digamma(v1, pol); + boost::math::trigamma(v1, pol); + boost::math::polygamma(i, v1, pol); boost::math::tgamma_ratio(v1, v2, pol); boost::math::tgamma_delta_ratio(v1, v2, pol); boost::math::factorial(i, pol); @@ -702,6 +708,8 @@ void instantiate(RealType) test::lgamma(v1); test::lgamma(v1, &i); test::digamma(v1); + test::trigamma(v1); + test::polygamma(i, v1); test::tgamma_ratio(v1, v2); test::tgamma_delta_ratio(v1, v2); test::factorial(i); @@ -893,6 +901,8 @@ void instantiate_mixed(RealType) boost::math::lgamma(i); boost::math::lgamma(i, &i); boost::math::digamma(i); + boost::math::trigamma(i); + boost::math::polygamma(i, i); boost::math::tgamma_ratio(i, l); boost::math::tgamma_ratio(fr, lr); boost::math::tgamma_delta_ratio(i, s); @@ -1035,6 +1045,8 @@ void instantiate_mixed(RealType) boost::math::lgamma(i, pol); boost::math::lgamma(i, &i, pol); boost::math::digamma(i, pol); + boost::math::trigamma(i, pol); + boost::math::polygamma(i, i, pol); boost::math::tgamma_ratio(i, l, pol); boost::math::tgamma_ratio(fr, lr, pol); boost::math::tgamma_delta_ratio(i, s, pol); @@ -1170,6 +1182,8 @@ void instantiate_mixed(RealType) test::lgamma(i); test::lgamma(i, &i); test::digamma(i); + test::trigamma(i); + test::polygamma(i, i); test::tgamma_ratio(i, l); test::tgamma_ratio(fr, lr); test::tgamma_delta_ratio(i, s); diff --git a/test/compile_test/sf_polygamma_incl_test.cpp b/test/compile_test/sf_polygamma_incl_test.cpp new file mode 100644 index 000000000..883644993 --- /dev/null +++ b/test/compile_test/sf_polygamma_incl_test.cpp @@ -0,0 +1,28 @@ +// Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void compile_and_link_test() +{ + check_result(boost::math::trigamma(f)); + check_result(boost::math::trigamma(d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::trigamma(l)); +#endif + check_result(boost::math::polygamma(1, f)); + check_result(boost::math::polygamma(1, d)); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + check_result(boost::math::polygamma(1, l)); +#endif +} diff --git a/test/float128/test_digamma.cpp b/test/float128/test_digamma.cpp index 7981c6367..9856223be 100644 --- a/test/float128/test_digamma.cpp +++ b/test/float128/test_digamma.cpp @@ -15,13 +15,20 @@ void expected_results() // Define the max and mean errors expected for // various compilers and platforms. // + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + ".*", // test type(s) + ".*negative.*", // test data group + ".*", 400, 200); // test function add_expected_result( ".*", // compiler ".*", // stdlib ".*", // platform ".*", // test type(s) ".*", // test data group - ".*", 350, 100); // test function + ".*", 2, 2); // test function // // Finish off by printing out the compiler/stdlib/platform names, // we do this to make it easier to mark up expected error rates. diff --git a/test/float128/test_polygamma.cpp b/test/float128/test_polygamma.cpp new file mode 100644 index 000000000..4debc41a9 --- /dev/null +++ b/test/float128/test_polygamma.cpp @@ -0,0 +1,51 @@ +/////////////////////////////////////////////////////////////// +// Copyright 2011 John Maddock. Distributed under the Boost +// Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_ + +#include "setup.hpp" +#include "table_type.hpp" + +#include +#include "libs/math/test/test_polygamma.hpp" + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + ".*", // test type(s) + ".*bug cases.*", // test data group + ".*", 100000, 40000); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + ".*", // test type(s) + ".*", // test data group + ".*", 350, 100); // test function + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} + +template +void test(T t, const char* p) +{ + test_polygamma(t, p); +} + +BOOST_AUTO_TEST_CASE( test_main ) +{ + expected_results(); + ALL_TESTS +} + diff --git a/test/float128/test_trigamma.cpp b/test/float128/test_trigamma.cpp new file mode 100644 index 000000000..8862b53d0 --- /dev/null +++ b/test/float128/test_trigamma.cpp @@ -0,0 +1,44 @@ +/////////////////////////////////////////////////////////////// +// Copyright 2011 John Maddock. Distributed under the Boost +// Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_ + +#include "setup.hpp" +#include "table_type.hpp" + +#include +#include "libs/math/test/test_trigamma.hpp" + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + ".*", // test type(s) + ".*", // test data group + ".*", 2, 2); // test function + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} + +template +void test(T t, const char* p) +{ + test_trigamma(t, p); +} + +BOOST_AUTO_TEST_CASE( test_main ) +{ + expected_results(); + ALL_TESTS +} + diff --git a/test/test_digamma.cpp b/test/test_digamma.cpp index 7926e7031..23ffc236a 100644 --- a/test/test_digamma.cpp +++ b/test/test_digamma.cpp @@ -42,6 +42,20 @@ void expected_results() ".*", // test type(s) ".*Negative.*", // test data group ".*", 300, 40); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + ".*Near the Positive Root.*", // test data group + ".*", 25000, 3000); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "real_concept", // test type(s) + ".*Half.*", // test data group + ".*", 15, 10); // test function add_expected_result( ".*", // compiler ".*", // stdlib @@ -72,8 +86,8 @@ void test_spots(T, const char* t) BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast(0.125)), static_cast(-8.3884926632958548678027429230863430000514460424495L), tolerance); BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast(0.5)), static_cast(-1.9635100260214234794409763329987555671931596046604L), tolerance); BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast(1)), static_cast(-0.57721566490153286060651209008240243104215933593992L), tolerance); - BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast(1.5)), static_cast(0.036489973978576520559023667001244432806840395339566L), tolerance); - BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast(1.5) - static_cast(1)/32), static_cast(0.00686541147073577672813890866512415766586241385896200579891429L), tolerance); + BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast(1.5)), static_cast(0.036489973978576520559023667001244432806840395339566L), tolerance * 40); + BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast(1.5) - static_cast(1)/32), static_cast(0.00686541147073577672813890866512415766586241385896200579891429L), tolerance * 100); BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast(2)), static_cast(0.42278433509846713939348790991759756895784066406008L), tolerance); BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast(8)), static_cast(2.0156414779556099965363450527747404261006978069172L), tolerance); BOOST_CHECK_CLOSE(::boost::math::digamma(static_cast(12)), static_cast(2.4426616799758120167383652547949424463027180089374L), tolerance); diff --git a/test/test_digamma.hpp b/test/test_digamma.hpp index 3653d1699..5d1de7cd7 100644 --- a/test/test_digamma.hpp +++ b/test/test_digamma.hpp @@ -82,6 +82,16 @@ void test_digamma(T, const char* name) }}; do_test_digamma(digamma_bugs, name, "Digamma Function: Values near 0"); + static const boost::array, 40> digamma_integers = { { + { 1, SC_(-0.57721566490153286060651209008240243) }, { 2, SC_(0.42278433509846713939348790991759757) }, { 3, SC_(0.92278433509846713939348790991759757) }, { 4, SC_(1.2561176684318004727268212432509309) }, { 5, SC_(1.5061176684318004727268212432509309) }, { 6, SC_(1.7061176684318004727268212432509309) }, { 7, SC_(1.8727843350984671393934879099175976) }, { 8, SC_(2.0156414779556099965363450527747404) }, { 9, SC_(2.1406414779556099965363450527747404) }, { SC_(10.0), SC_(2.2517525890667211076474561638858515) }, { SC_(11.0), SC_(2.3517525890667211076474561638858515) }, { SC_(12.0), SC_(2.4426616799758120167383652547949424) }, { SC_(13.0), SC_(2.5259950133091453500716985881282758) }, { SC_(14.0), SC_(2.6029180902322222731486216650513527) }, { SC_(15.0), SC_(2.6743466616607937017200502364799241) }, { SC_(16.0), SC_(2.7410133283274603683867169031465908) }, { SC_(17.0), SC_(2.8035133283274603683867169031465908) }, { SC_(18.0), SC_(2.8623368577392250742690698443230614) }, { SC_(19.0), SC_(2.9178924132947806298246253998786169) }, { SC_(20.0), SC_(2.9705239922421490508772569788259854) }, { SC_(21.0), SC_(3.0205239922421490508772569788259854) }, { SC_(22.0), SC_(3.0681430398611966699248760264450330) }, { SC_(23.0), SC_(3.1135975853157421244703305718995784) }, { SC_(24.0), SC_(3.1570758461853073418616349197256654) }, { SC_(25.0), SC_(3.1987425128519740085283015863923321) }, { SC_(26.0), SC_(3.2387425128519740085283015863923321) }, { SC_(27.0), SC_(3.2772040513135124700667631248538705) }, { SC_(28.0), SC_(3.3142410883505495071038001618909076) }, { SC_(29.0), SC_(3.3499553740648352213895144476051933) }, { SC_(30.0), SC_(3.3844381326855248765619282407086415) }, { SC_(31.0), SC_(3.4177714660188582098952615740419749) }, { SC_(32.0), SC_(3.4500295305349872421533260901710071) }, { SC_(33.0), SC_(3.4812795305349872421533260901710071) }, { SC_(34.0), SC_(3.5115825608380175451836291204740374) }, { SC_(35.0), SC_(3.5409943255438998981248055910622727) }, { SC_(36.0), SC_(3.5695657541153284695533770196337013) }, { SC_(37.0), SC_(3.5973435318931062473311547974114791) }, { SC_(38.0), SC_(3.6243705589201332743581818244385061) }, { SC_(39.0), SC_(3.6506863483938174848844976139121903) }, { SC_(40.0), SC_(3.6763273740348431259101386395532160) } + } }; + do_test_digamma(digamma_integers, name, "Digamma Function: Integer arguments"); + + static const boost::array, 41> digamma_half_integers = { { + { SC_(0.5), SC_(-1.9635100260214234794409763329987556) }, { SC_(1.5), SC_(0.036489973978576520559023667001244433) }, { SC_(2.5), SC_(0.70315664064524318722569033366791110) }, { SC_(3.5), SC_(1.1031566406452431872256903336679111) }, { SC_(4.5), SC_(1.3888709263595289015114046193821968) }, { SC_(5.5), SC_(1.6110931485817511237336268416044190) }, { SC_(6.5), SC_(1.7929113303999329419154450234226009) }, { SC_(7.5), SC_(1.9467574842460867880692911772687547) }, { SC_(8.5), SC_(2.0800908175794201214026245106020880) }, { SC_(9.5), SC_(2.1977378764029495331673303929550292) }, { SC_(10.5), SC_(2.3030010342976863752725935508497661) }, { SC_(11.5), SC_(2.3982391295357816133678316460878613) }, { SC_(12.5), SC_(2.4851956512749120481504403417400352) }, { SC_(13.5), SC_(2.5651956512749120481504403417400352) }, { SC_(14.5), SC_(2.6392697253489861222245144158141093) }, { SC_(15.5), SC_(2.7082352425903654325693420020210058) }, { SC_(16.5), SC_(2.7727513716226234970854710342790703) }, { SC_(17.5), SC_(2.8333574322286841031460770948851310) }, { SC_(18.5), SC_(2.8905002893715412460032199520279881) }, { SC_(19.5), SC_(2.9445543434255953000572740060820421) }, { SC_(20.5), SC_(2.9958363947076465821085560573640934) }, { SC_(21.5), SC_(3.0446168825125246308890438622421422) }, { SC_(22.5), SC_(3.0911285104195013750750903738700492) }, { SC_(23.5), SC_(3.1355729548639458195195348183144936) }, { SC_(24.5), SC_(3.1781261463533075216471943927825787) }, { SC_(25.5), SC_(3.2189424728839197665451535764560481) }, { SC_(26.5), SC_(3.2581581591584295704667222039070285) }, { SC_(27.5), SC_(3.2958940082150333440516278642843870) }, { SC_(28.5), SC_(3.3322576445786697076879915006480234) }, { SC_(29.5), SC_(3.3673453638769153217230792199462690) }, { SC_(30.5), SC_(3.4012436689616610844349436267259300) }, { SC_(31.5), SC_(3.4340305542075627237792059218078972) }, { SC_(32.5), SC_(3.4657765859535944698109519535539290) }, { SC_(33.5), SC_(3.4965458167228252390417211843231597) }, { SC_(34.5), SC_(3.5263965629914819554596316320843538) }, { SC_(35.5), SC_(3.5553820702378587670538345306350784) }, { SC_(36.5), SC_(3.5835510843223658093073556573956418) }, { SC_(37.5), SC_(3.6109483445963384120470816847929021) }, { SC_(38.5), SC_(3.6376150112630050787137483514595687) }, { SC_(39.5), SC_(3.6635890372370310527397223774335947) }, { SC_(40.5), SC_(3.6889054929332335843852919976867593) } + } }; + do_test_digamma(digamma_half_integers, name, "Digamma Function: Half integer arguments"); + BOOST_CHECK_THROW(boost::math::digamma(T(0)), std::domain_error); BOOST_CHECK_THROW(boost::math::digamma(T(-1)), std::domain_error); BOOST_CHECK_THROW(boost::math::digamma(T(-2)), std::domain_error); diff --git a/test/test_instances/double_test_instances_5.cpp b/test/test_instances/double_test_instances_5.cpp index 3c4d0a336..62dbbe2f1 100644 --- a/test/test_instances/double_test_instances_5.cpp +++ b/test/test_instances/double_test_instances_5.cpp @@ -9,6 +9,7 @@ #define BOOST_MATH_OVERFLOW_ERROR_POLICY ignore_error #include #include +#include #endif #define BOOST_MATH_TEST_TYPE double diff --git a/test/test_instances/double_test_instances_6.cpp b/test/test_instances/double_test_instances_6.cpp index 2edbc5d37..5fc890867 100644 --- a/test/test_instances/double_test_instances_6.cpp +++ b/test/test_instances/double_test_instances_6.cpp @@ -13,6 +13,8 @@ #include #include #include +#include +#include #endif #define BOOST_MATH_TEST_TYPE double diff --git a/test/test_instances/float_test_instances_5.cpp b/test/test_instances/float_test_instances_5.cpp index 13e9e34f3..4e2315c06 100644 --- a/test/test_instances/float_test_instances_5.cpp +++ b/test/test_instances/float_test_instances_5.cpp @@ -9,6 +9,7 @@ #define BOOST_MATH_OVERFLOW_ERROR_POLICY ignore_error #include #include +#include #endif #define BOOST_MATH_TEST_TYPE float diff --git a/test/test_instances/float_test_instances_6.cpp b/test/test_instances/float_test_instances_6.cpp index 580a9adac..8d8400131 100644 --- a/test/test_instances/float_test_instances_6.cpp +++ b/test/test_instances/float_test_instances_6.cpp @@ -13,6 +13,8 @@ #include #include #include +#include +#include #endif #define BOOST_MATH_TEST_TYPE float diff --git a/test/test_instances/ldouble_test_instances_5.cpp b/test/test_instances/ldouble_test_instances_5.cpp index 5e0e04c23..6e4599b64 100644 --- a/test/test_instances/ldouble_test_instances_5.cpp +++ b/test/test_instances/ldouble_test_instances_5.cpp @@ -9,6 +9,7 @@ #define BOOST_MATH_OVERFLOW_ERROR_POLICY ignore_error #include #include +#include #endif #ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS diff --git a/test/test_instances/ldouble_test_instances_6.cpp b/test/test_instances/ldouble_test_instances_6.cpp index 5a03396b2..c56695fec 100644 --- a/test/test_instances/ldouble_test_instances_6.cpp +++ b/test/test_instances/ldouble_test_instances_6.cpp @@ -13,6 +13,8 @@ #include #include #include +#include +#include #endif #ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS diff --git a/test/test_instances/real_concept_test_instances_5.cpp b/test/test_instances/real_concept_test_instances_5.cpp index 14fc0909b..6dd7d0433 100644 --- a/test/test_instances/real_concept_test_instances_5.cpp +++ b/test/test_instances/real_concept_test_instances_5.cpp @@ -9,6 +9,7 @@ #define BOOST_MATH_OVERFLOW_ERROR_POLICY ignore_error #include #include +#include #endif #include diff --git a/test/test_instances/real_concept_test_instances_6.cpp b/test/test_instances/real_concept_test_instances_6.cpp index 05ad5324b..18137510d 100644 --- a/test/test_instances/real_concept_test_instances_6.cpp +++ b/test/test_instances/real_concept_test_instances_6.cpp @@ -13,6 +13,8 @@ #include #include #include +#include +#include #endif #include diff --git a/test/test_instances/test_instances.hpp b/test/test_instances/test_instances.hpp index 889e9b2f1..949618fab 100644 --- a/test/test_instances/test_instances.hpp +++ b/test/test_instances/test_instances.hpp @@ -287,8 +287,13 @@ namespace boost{ namespace math{ // digamma: template tools::promote_args::type digamma(BOOST_MATH_TEST_TYPE x); - template tools::promote_args::type digamma(BOOST_MATH_TEST_TYPE x, const policies::policy<>&); + // trigamma: + template tools::promote_args::type trigamma(BOOST_MATH_TEST_TYPE x); + template tools::promote_args::type trigamma(BOOST_MATH_TEST_TYPE x, const policies::policy<>&); + // polygamma: + template tools::promote_args::type polygamma(int, BOOST_MATH_TEST_TYPE x); + template tools::promote_args::type polygamma(int, BOOST_MATH_TEST_TYPE x, const policies::policy<>&); #endif #ifdef TEST_GROUP_6 // Hypotenuse function sqrt(x ^ 2 + y ^ 2). @@ -332,6 +337,12 @@ namespace boost{ namespace math{ template tools::promote_args::type sqrt1pm1(const BOOST_MATH_TEST_TYPE& val); template tools::promote_args::type sqrt1pm1(const BOOST_MATH_TEST_TYPE& val, const policies::policy<>&); + + // sin_pi, cos_pi + template tools::promote_args::type sin_pi(const BOOST_MATH_TEST_TYPE val); + template tools::promote_args::type sin_pi(const BOOST_MATH_TEST_TYPE val, const policies::policy<>&); + template tools::promote_args::type cos_pi(const BOOST_MATH_TEST_TYPE val); + template tools::promote_args::type cos_pi(const BOOST_MATH_TEST_TYPE val, const policies::policy<>&); #endif #ifdef TEST_GROUP_7 // Bessel functions: diff --git a/test/test_polygamma.cpp b/test/test_polygamma.cpp new file mode 100644 index 000000000..f87ae95c6 --- /dev/null +++ b/test/test_polygamma.cpp @@ -0,0 +1,94 @@ +// (C) Copyright John Maddock 2014. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include "test_polygamma.hpp" +//#include + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + const char* largest_type; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + if(boost::math::policies::digits >() == boost::math::policies::digits >()) + { + largest_type = "(long\\s+)?double|real_concept"; + } + else + { + largest_type = "long double|real_concept"; + } +#else + largest_type = "(long\\s+)?double|real_concept"; +#endif + + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*large arguments", // test data group + ".*", 400, 200); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*negative.*", // test data group + ".*", 800, 400); // test function + if((std::numeric_limits::digits > std::numeric_limits::digits) + && (std::numeric_limits::digits - std::numeric_limits::digits < 20)) + { + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + "double", // test type(s) + ".*bug cases.*", // test data group + ".*", 100, 30); // test function + } + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*bug cases.*", // test data group + ".*", 100000, 50000); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*", // test data group + ".*", 30, 10); // test function + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} + +BOOST_AUTO_TEST_CASE( test_main ) +{ + expected_results(); + BOOST_MATH_CONTROL_FP; + + test_polygamma(0.0F, "float"); + test_polygamma(0.0, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_polygamma(0.0L, "long double"); + test_polygamma(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif +#ifdef BOOST_FLOAT128_C + //test_polygamma(BOOST_FLOAT128_C(0.0), "float128_t"); +#endif +} + + + diff --git a/test/test_polygamma.hpp b/test/test_polygamma.hpp new file mode 100644 index 000000000..e7559bf18 --- /dev/null +++ b/test/test_polygamma.hpp @@ -0,0 +1,203 @@ +// Copyright John Maddock 20 +// Copyright Paul A. Bristow 2007, 2009 +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define BOOST_MATH_OVERFLOW_ERROR_POLICY ignore_error + +#include +#include +#define BOOST_TEST_MAIN +#include +#include +#include +#include +#include +#include +#include +#include "functor.hpp" + +#include "handle_test_result.hpp" +#include "table_type.hpp" + +#ifndef SC_ +#ifndef BOOST_FLOAT128_C +#define SC_(x) static_cast::type>(BOOST_JOIN(x, L)) +#else +#define SC_(x) static_cast::type>(BOOST_FLOAT128_C(x)) +#endif +#endif + +template +void do_test_polygamma(const T& data, const char* type_name, const char* test_name) +{ + typedef Real value_type; + + typedef value_type(*pg)(int, value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::polygamma; +#else + pg funcp = boost::math::polygamma; +#endif + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test polygamma against data: + // + result = boost::math::tools::test_hetero( + data, + bind_func_int1(funcp, 0, 1), + extract_result(2)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::polygamma", test_name); + std::cout << std::endl; +} + +template +void test_polygamma(T, const char* name) +{ + typedef typename table_type::type value_type; + + boost::array, 484> data = + { { + { 1, SC_(0.1250000), SC_(65.388133444988034473142999334395961) }, { 1, SC_(2.250000), SC_(0.55732915450711073927131911933522402) }, { 1, SC_(4.375000), SC_(0.25666408805722660683906428275458774) }, { 1, SC_(6.500000), SC_(0.16628453574995823763989666631218566) }, { 1, SC_(8.625000), SC_(0.12292237374423990274075995315923687) }, { 1, SC_(10.75000), SC_(0.097483848201852104395946001854344927) }, { 1, SC_(12.87500), SC_(0.080764208092843621858393487209278638) }, { 1, SC_(15.00000), SC_(0.068938227847683806226155216756371670) }, { 1, SC_(17.12500), SC_(0.060132263162293455894576107399989891) }, { 1, SC_(19.25000), SC_(0.053320703915617277211139745531295189) }, { 1, SC_(21.37500), SC_(0.047895038036916716105500109226810942) }, { 1, SC_(23.50000), SC_(0.043471416266946770249685779030294199) }, { 1, SC_(25.62500), SC_(0.039795743807625238080963836217545550) }, { 1, SC_(27.75000), SC_(0.036693131333593477569076090983653779) }, { 1, SC_(29.87500), SC_(0.034039266877179098641898178094001935) }, { 1, SC_(32.00000), SC_(0.031743366520302090126581680438741427) }, { 1, SC_(34.12500), SC_(0.029737585673522726661363528635488348) }, { 1, SC_(36.25000), SC_(0.027970204614894933106878169214392067) }, { 1, SC_(38.37500), SC_(0.026401106865951764123858232364900665) }, { 1, SC_(40.50000), SC_(0.024998698201356741322280011143883922) }, { 1, SC_(42.62500), SC_(0.023737757818075642720991864115881164) }, { 1, SC_(44.75000), SC_(0.022597908441287364284087900916682571) }, { 1, SC_(46.87500), SC_(0.021562506914486557632388530071308920) }, { 1, SC_(49.00000), SC_(0.020617826354560516060031453062401102) }, { 1, SC_(51.12500), SC_(0.019752444228552790805040230288386135) }, { 1, SC_(53.25000), SC_(0.018956778300513446216011889734099110) }, { 1, SC_(55.37500), SC_(0.018222730375562878627773770314126276) }, { 1, SC_(57.50000), SC_(0.017543409716574620734228673575882677) }, { 1, SC_(59.62500), SC_(0.016912916093398919581541485278641593) }, { 1, SC_(61.75000), SC_(0.016326167985389235938281994221076159) }, { 1, SC_(63.87500), SC_(0.015778765341125640054231784371915358) }, { 1, SC_(66.00000), SC_(0.015266879048806385777045778219279459) }, { 1, SC_(68.12500), SC_(0.014787161242916062152535888615850260) }, { 1, SC_(70.25000), SC_(0.014336672004276912093324664070489886) }, { 1, SC_(72.37500), SC_(0.013912819061256593888508241399678441) }, { 1, SC_(74.50000), SC_(0.013513307879079644573772830117155790) }, { 1, SC_(76.62500), SC_(0.013136100107643257539226305043251166) }, { 1, SC_(78.75000), SC_(0.012779378799112389298746113446648982) }, { 1, SC_(80.87500), SC_(0.012441519142554280549428650598112729) }, { 1, SC_(83.00000), SC_(0.012121063720980953787190412456820037) }, { 1, SC_(85.12500), SC_(0.011816701495952671887865236708275449) }, { 1, SC_(87.25000), SC_(0.011527249880640584213306489206202069) }, { 1, SC_(89.37500), SC_(0.011251639384481213426240562514542477) }, { 1, SC_(91.50000), SC_(0.010988900409103388104670465071922664) }, { 1, SC_(93.62500), SC_(0.010738151851930343087942933225290955) }, { 1, SC_(95.75000), SC_(0.010498591235178571927709342934117247) }, { 1, SC_(97.87500), SC_(0.010269486127251686167359605922136260) }, { 1, SC_(100.0000), SC_(0.010050166663333571395245668465701423) }, + { 2, SC_(0.1250000), -SC_(1025.7533381181356825956689300565174) }, { 2, SC_(2.250000), -SC_(0.30373993753692033333796717884398989) }, { 2, SC_(4.375000), -SC_(0.065528725397877855792664680804766330) }, { 2, SC_(6.500000), -SC_(0.027587910706876798794117450572831562) }, { 2, SC_(8.625000), -SC_(0.015091062676061564388822078971884395) }, { 2, SC_(10.75000), -SC_(0.0094956196449265900776488965631791775) }, { 2, SC_(12.87500), -SC_(0.0065193261909178260169885198194705291) }, { 2, SC_(15.00000), -SC_(0.0047506027165515547467791223768191188) }, { 2, SC_(17.12500), -SC_(0.0036148020016626802195565448384834283) }, { 2, SC_(19.25000), -SC_(0.0028424250740909855631736845850335535) }, { 2, SC_(21.37500), -SC_(0.0022934967923297751145806065882712900) }, { 2, SC_(23.50000), -SC_(0.0018894667868625909895476094900477678) }, { 2, SC_(25.62500), -SC_(0.0015834924252652953218803111387922996) }, { 2, SC_(27.75000), -SC_(0.0013462349527320363170378913859580860) }, { 2, SC_(29.87500), -SC_(0.0011585598948326545653381467670779893) }, { 2, SC_(32.00000), -SC_(0.0010075567602140907392185110593117265) }, { 2, SC_(34.12500), -SC_(0.00088425886906787461365402045268994574) }, { 2, SC_(36.25000), -SC_(0.00078228136778540401396894643668418462) }, { 2, SC_(38.37500), -SC_(0.00069697797537723210697105880606724159) }, { 2, SC_(40.50000), -SC_(0.00062490237932923289658683588812998732) }, { 2, SC_(42.62500), -SC_(0.00056345469641484389456121935536309197) }, { 2, SC_(44.75000), -SC_(0.00051064374134295093656385520125286421) }, { 2, SC_(46.87500), -SC_(0.00046492369550612086723038302919171885) }, { 2, SC_(49.00000), -SC_(0.00042507970884222510504308867471824948) }, { 2, SC_(51.12500), -SC_(0.00039014637079445100439645401142194535) }, { 2, SC_(53.25000), -SC_(0.00035934868438211062777790080207505881) }, { 2, SC_(55.37500), -SC_(0.00033205871518117505559928737076624192) }, { 2, SC_(57.50000), -SC_(0.00030776333242771756953580263456220628) }, { 2, SC_(59.62500), -SC_(0.00028603991345613245802207114123372414) }, { 2, SC_(61.75000), -SC_(0.00026653784162151616772904552992231739) }, { 2, SC_(63.87500), -SC_(0.00024896427102287300629668349085350311) }, { 2, SC_(66.00000), -SC_(0.00023307306946180321476975587412226332) }, { 2, SC_(68.12500), -SC_(0.00021865615382096049855997233359992101) }, { 2, SC_(70.25000), -SC_(0.00020553664405312492990454318541834320) }, { 2, SC_(72.37500), -SC_(0.00019356341228034103704385457246984425) }, { 2, SC_(74.50000), -SC_(0.00018260671130395075946075978013398780) }, { 2, SC_(76.62500), -SC_(0.00017255464497899193114313963889049197) }, { 2, SC_(78.75000), -SC_(0.00016331030013937037094502788633900241) }, { 2, SC_(80.87500), -SC_(0.00015478940207215993505449988937898640) }, { 2, SC_(83.00000), -SC_(0.00014691838710034332570083901051760808) }, { 2, SC_(85.12500), -SC_(0.00013963280957351165370765015817038278) }, { 2, SC_(87.25000), -SC_(0.00013287601856558888563037791437997733) }, { 2, SC_(89.37500), -SC_(0.00012659805332837531702241737466058293) }, { 2, SC_(91.50000), -SC_(0.00012075471712784846612240303590084686) }, { 2, SC_(93.62500), -SC_(0.00011530679728326736002584104318350829) }, { 2, SC_(95.75000), -SC_(0.00011021940561565095981861374083495299) }, { 2, SC_(97.87500), -SC_(0.00010546141852085692313553484671110170) }, { 2, SC_(100.0000), -SC_(0.00010100499983334999700083300446059382) }, + { 3, SC_(0.1250000), SC_(24580.143419063566218511004446647010) }, + { 3, SC_(2.250000), SC_(0.32454400918839602279124382903505677) }, + { 3, SC_(4.375000), SC_(0.033289201205556512286673394063908247) }, + { 3, SC_(6.500000), SC_(0.0091336635043781405048050655353658508) }, + { 3, SC_(8.625000), SC_(0.0037008460120616755687668778806054328) }, + { 3, SC_(10.75000), SC_(0.0018484328773891268475922730275004208) }, + { 3, SC_(12.87500), SC_(0.0010519186241203463873018930415985105) }, + { 3, SC_(15.00000), SC_(0.00065447977828273734841733708901750530) }, + { 3, SC_(17.12500), SC_(0.00043447135395979214393136263736764201) }, + { 3, SC_(19.25000), SC_(0.00030297696415718385912539504824566570) }, + { 3, SC_(21.37500), SC_(0.00021961042047214603636739002880115737) }, + { 3, SC_(23.50000), SC_(0.00016422394665239820056068884910278128) }, + { 3, SC_(25.62500), SC_(0.00012599930135391694524067581601102767) }, + { 3, SC_(27.75000), SC_(0.000098773010741268191080883613389009048) }, + { 3, SC_(29.87500), SC_(0.000078857844307768334137453508928722671) }, + { 3, SC_(32.00000), SC_(0.000063955754777976299576673673574642423) }, + { 3, SC_(34.12500), SC_(0.000052583702941352521813139097855885393) }, + { 3, SC_(36.25000), SC_(0.000043755438122384003302367816109980192) }, + { 3, SC_(38.37500), SC_(0.000036797707573246758497786056137200200) }, + { 3, SC_(40.50000), SC_(0.000031240239710655936422248716017582946) }, + { 3, SC_(42.62500), SC_(0.000026747791378195202271687207455056302) }, + { 3, SC_(44.75000), SC_(0.000023076997706632396178157160717335565) }, + { 3, SC_(46.87500), SC_(0.000020048287815935784161580614536631137) }, + { 3, SC_(49.00000), SC_(0.000017527197874026635143654793647828466) }, + { 3, SC_(51.12500), SC_(0.000015411686998050652806770684541876757) }, + { 3, SC_(53.25000), SC_(0.000013623370966529973793708331754693666) }, + { 3, SC_(55.37500), SC_(0.000012101363299593660023105041673385376) }, + { 3, SC_(57.50000), SC_(0.000010797882726896889062768134787291884) }, + { 3, SC_(59.62500), SC_(9.6750769605707758334815857396558355e-6) }, + { 3, SC_(61.75000), SC_(8.7026966259856900409582330070865268e-6) }, + { 3, SC_(63.87500), SC_(7.8563716857343030149244371160376495e-6) }, + { 3, SC_(66.00000), SC_(7.1163203299934132027680033779014197e-6) }, + { 3, SC_(68.12500), SC_(6.4663719906627627544882175944382226e-6) }, + { 3, SC_(70.25000), SC_(5.8932210515208497328937862935232496e-6) }, + { 3, SC_(72.37500), SC_(5.3858517367657111634358444590928906e-6) }, + { 3, SC_(74.50000), SC_(4.9350912436686398240670814427342376e-6) }, + { 3, SC_(76.62500), SC_(4.5332598242018110991734569100539592e-6) }, + { 3, SC_(78.75000), SC_(4.1738947808706992483423293125251133e-6) }, + { 3, SC_(80.87500), SC_(3.8515312659502461861484363442184908e-6) }, + { 3, SC_(83.00000), SC_(3.5615270636518630626786644585266764e-6) }, + { 3, SC_(85.12500), SC_(3.2999216708248315372665461067377996e-6) }, + { 3, SC_(87.25000), SC_(3.0633223042636924720337703682152499e-6) }, + { 3, SC_(89.37500), SC_(2.8488111819903247190925375784544775e-6) }, + { 3, SC_(91.50000), SC_(2.6538697141941088454844912572034392e-6) }, + { 3, SC_(93.62500), SC_(2.4763162120667457234467192228222670e-6) }, + { 3, SC_(95.75000), SC_(2.3142544621402797124759791203351867e-6) }, + { 3, SC_(97.87500), SC_(2.1660310796121506708372890790676070e-6) }, + { 3, SC_(100.0000), SC_(2.0301999900013330334332872946449855e-6) }, + + {4, SC_(0.1250000), SC_(-786445.98543106378579320120709638297)}, { 4, SC_(2.250000), SC_(-0.51106863793373355822715252195899576) }, { 4, SC_(4.375000), SC_(-0.025241882074562753654505456518792508) }, { 4, SC_(6.500000), SC_(-0.0045259302803220607103496010172883494) }, { 4, SC_(8.625000), SC_(-0.0013596929106556359886012273921730071) }, { 4, SC_(10.75000), SC_(-0.00053930828860547053427689740051604949) }, { 4, SC_(12.87500), SC_(-0.00025445997266103206171225355293726865) }, { 4, SC_(15.00000), SC_(-0.00013519619187519276575068431301221388) }, { 4, SC_(17.12500), SC_(-0.000078306716822025065793290599586946118) }, { 4, SC_(19.25000), SC_(-0.000048430513924395561509999792045098081) }, { 4, SC_(21.37500), SC_(-0.000031536705873201397805285969259914559) }, { 4, SC_(23.50000), SC_(-0.000021407049050977062373311093992814121) }, { 4, SC_(25.62500), SC_(-0.000015036764178343754461565939319516893) }, { 4, SC_(27.75000), SC_(-0.000010869219781124567432001391245344089) }, { 4, SC_(29.87500), SC_(-8.0504604926168752698902834477348881e-6) }, { 4, SC_(32.00000), SC_(-6.0889806370027137702207132674152980e-6) }, { 4, SC_(34.12500), SC_(-4.6901011823268163094417546531023856e-6) }, { 4, SC_(36.25000), SC_(-3.6708283086618325558290000911536581e-6) }, { 4, SC_(38.37500), SC_(-2.9139932244650290336213711603586301e-6) }, { 4, SC_(40.50000), SC_(-2.3425302303691304543510535971847968e-6) }, { 4, SC_(42.62500), SC_(-1.9045296486050512642801504306671373e-6) }, { 4, SC_(44.75000), SC_(-1.5642760453399369047824852836708697e-6) }, { 4, SC_(46.87500), SC_(-1.2967233822729326645555539185332769e-6) }, { 4, SC_(49.00000), SC_(-1.0840030171141394983603065427611252e-6) }, { 4, SC_(51.12500), SC_(-9.1316643010960703020517255319750485e-7) }, { 4, SC_(53.25000), SC_(-7.7469609700462921021995856208285815e-7) }, { 4, SC_(55.37500), SC_(-6.6150474476547305805911346012045165e-7) }, { 4, SC_(57.50000), SC_(-5.6825133351570053043820949353677183e-7) }, { 4, SC_(59.62500), SC_(-4.9086620236193161135912086142572580e-7) }, { 4, SC_(61.75000), SC_(-4.2621666772058776224801114961267801e-7) }, { 4, SC_(63.87500), SC_(-3.7186839586672891333608898041067112e-7) }, { 4, SC_(66.00000), SC_(-3.2591301914111222705427951896642626e-7) }, { 4, SC_(68.12500), SC_(-2.8684217920848561693233665614958777e-7) }, { 4, SC_(70.25000), 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SC_(2.4178516392292583494123520000000000e24), SC_(-2.7774698720523025435197111896762965e-701) }, { SC_(30.0), SC_(4.8357032784585166988247040000000000e24), SC_(-2.5867203921566741015014262629362602e-710) }, { SC_(30.0), SC_(9.6714065569170333976494080000000000e24), SC_(-2.4090710954337139627909495039162813e-719) }, { SC_(30.0), SC_(1.9342813113834066795298816000000000e25), SC_(-2.2436222950310576360588405614024340e-728) }, { SC_(30.0), SC_(3.8685626227668133590597632000000000e25), SC_(-2.0895360922730133273255449640176061e-737) }, { SC_(30.0), SC_(7.7371252455336267181195264000000000e25), SC_(-1.9460321332076688551581879695116458e-746) }, { SC_(30.0), SC_(1.5474250491067253436239052800000000e26), SC_(-1.8123836565834180034307649180969011e-755) }, { SC_(30.0), SC_(3.0948500982134506872478105600000000e26), SC_(-1.6879138132402841033700526042422183e-764) }, { SC_(30.0), SC_(6.1897001964269013744956211200000000e26), SC_(-1.5719922382759713599179429591983936e-773) }, { SC_(30.0), SC_(1.2379400392853802748991242240000000e27), SC_(-1.4640318586267264186571752095136003e-782) }, { SC_(30.0), SC_(2.4758800785707605497982484480000000e27), SC_(-1.3634859198953271085928894585406723e-791) }, { SC_(30.0), SC_(4.9517601571415210995964968960000000e27), SC_(-1.2698452173688701433994709089494920e-800) }, { SC_(30.0), SC_(9.9035203142830421991929937920000000e27), SC_(-1.1826355172031280988822420193126128e-809) }, { SC_(30.0), SC_(1.9807040628566084398385987584000000e28), SC_(-1.1014151547133252945563215933898419e-818) }, { SC_(30.0), SC_(3.9614081257132168796771975168000000e28), SC_(-1.0257727976081197099353387886404875e-827) }, { SC_(30.0), SC_(7.9228162514264337593543950336000000e28), SC_(-9.5532536283891621039746216353614015e-837) }, { SC_(30.0), SC_(1.5845632502852867518708790067200000e29), SC_(-8.8971607651460562869670071028704372e-846) }, { SC_(30.0), SC_(3.1691265005705735037417580134400000e29), SC_(-8.2861266705636459281360796675545348e-855) }, { SC_(30.0), SC_(6.3382530011411470074835160268800000e29), SC_(-7.7170568244193177000955489160105929e-864) }, { SC_(30.0), SC_(1.2676506002282294014967032053760000e30), SC_(-7.1870692301721476950641245729469486e-873) }, + { SC_(31.0), SC_(2.0000000000000000000000000000000000), SC_(1.9145332544935048093264355986676073e24) }, { SC_(31.0), SC_(4.0000000000000000000000000000000000), SC_(4.4611518561919816922776795853710092e14) }, { SC_(31.0), SC_(8.0000000000000000000000000000000000), SC_(106267.95619808419253963903669279065) }, { SC_(31.0), SC_(16.000000000000000000000000000000000), SC_(0.000028318136062027075392587779376335821) }, { SC_(31.0), SC_(32.000000000000000000000000000000000), SC_(9.0814734303237413772295578988209818e-15) } + } }; + do_test_polygamma(big_data, name, "Mathematica Data - large arguments"); + + boost::array, 654> neg_data = + { { + { SC_(1.0), SC_(-12.750), SC_(19.663772856722737612034697464751605) }, { SC_(1.0), SC_(-12.250), SC_(19.660817549236368273654684043826967) }, { SC_(1.0), SC_(-11.750), SC_(19.657621376522814505537196503582823) }, { SC_(1.0), SC_(-11.250), SC_(19.654153659190554029589711115880278) }, { SC_(1.0), SC_(-10.750), SC_(19.650378280099093364749509767503149) }, { SC_(1.0), SC_(-10.250), SC_(19.646252424622652795021809881312377) }, { SC_(1.0), SC_(-9.7500), SC_(19.641724953976865133273035997897957) }, { SC_(1.0), SC_(-9.2500), SC_(19.636734280660725370869519577921538) }, { SC_(1.0), SC_(-8.7500), SC_(19.631205558842085383108670448917024) }, { SC_(1.0), SC_(-8.2500), SC_(19.625046917622010980803778160828770) }, { SC_(1.0), SC_(-7.7500), SC_(19.618144334352289464741323510141514) }, { SC_(1.0), SC_(-7.2500), SC_(19.610354539293269015698176691590937) }, { SC_(1.0), SC_(-6.7500), SC_(19.601495010731061577124257953429755) }, { SC_(1.0), SC_(-6.2500), SC_(19.591329569019785068016844943671793) }, { SC_(1.0), SC_(-5.7500), SC_(19.579547136931335925546754524074474) }, { SC_(1.0), SC_(-5.2500), SC_(19.565729569019785068016844943671793) }, { SC_(1.0), SC_(-4.7500), SC_(19.549301390239464469970194977760674) }, { SC_(1.0), SC_(-4.2500), SC_(19.529448389881463072551992335962043) }, { SC_(1.0), SC_(-3.7500), SC_(19.504980060599575273294294700752364) }, { SC_(1.0), SC_(-3.2500), SC_(19.474085068082155114074483685443011) }, { SC_(1.0), SC_(-2.7500), SC_(19.433868949488464162183183589641253) }, { SC_(1.0), SC_(-2.2500), SC_(19.379410511869137362595193744614609) }, { SC_(1.0), SC_(-1.7500), SC_(19.301637544529786476232770366500757) }, { SC_(1.0), SC_(-1.2500), SC_(19.181879647671606498397662880417078) }, { SC_(1.0), SC_(-0.75000), SC_(18.975106932284888517049096897113002) }, { SC_(1.0), SC_(-0.25000), SC_(18.541879647671606498397662880417078) }, + { SC_(2.0), SC_(-12.750), SC_(-124.03079461415823384604153251543681) }, { SC_(2.0), SC_(-12.250), SC_(124.01896466745858356132308878716344) }, { SC_(2.0), SC_(-11.750), SC_(-124.03175955222881001960976796032603) }, { SC_(2.0), SC_(-11.250), SC_(124.01787668541028735821044014586602) }, { SC_(2.0), SC_(-10.750), SC_(-124.03299241970518808612682102178640) }, { SC_(2.0), SC_(-10.250), SC_(124.01647202148710491650947992638728) }, { SC_(2.0), SC_(-9.7500), SC_(-124.03460234084420729198290916496876) }, { SC_(2.0), SC_(-9.2500), SC_(124.01461482266526541911391108670126) }, { SC_(2.0), SC_(-8.7500), SC_(-124.03676016548723903560636876475972) }, { SC_(2.0), SC_(-8.2500), SC_(124.01208782525148933477537240192445) }, { SC_(2.0), SC_(-7.7500), SC_(-124.03974558822776381694747663647984) }, { SC_(2.0), SC_(-7.2500), SC_(124.00852603656573370687098416695770) }, { SC_(2.0), SC_(-6.7500), SC_(-124.04404218787195165891317097369578) }, { SC_(2.0), SC_(-6.2500), SC_(124.00327776890408296268303058132483) }, { SC_(2.0), SC_(-5.7500), SC_(-124.05054526159038888901020902683808) }, { SC_(2.0), SC_(-5.2500), SC_(123.99508576890408296268303058132483) }, { SC_(2.0), SC_(-4.7500), SC_(-124.06106552130930069964553408642549) }, { SC_(2.0), SC_(-4.2500), SC_(123.98126436732757934536308673076874) }, { SC_(2.0), SC_(-3.7500), SC_(-124.07972713378925404561433420306057) }, { SC_(2.0), SC_(-3.2500), SC_(123.95521103942202265902072971875978) }, { SC_(2.0), SC_(-2.7500), SC_(-124.11765305971517997154026012898649) }, { SC_(2.0), SC_(-2.2500), SC_(123.89694977406016558118732052440384) }, { SC_(2.0), SC_(-1.7500), SC_(-124.21382135423058192495874247308867) }, { SC_(2.0), SC_(-1.2500), SC_(123.72136678366236036856729308956159) }, { SC_(2.0), SC_(-0.75000), SC_(-124.58699919679617959259722643810325) }, { SC_(2.0), SC_(-0.25000), SC_(122.69736678366236036856729308956159) }, + { SC_(3.0), SC_(-12.750), SC_(1558.5445992104061926890981987122713) }, { SC_(3.0), SC_(-12.250), SC_(1558.5444945580353369268010524200916) }, { SC_(3.0), SC_(-11.750), SC_(1558.5443721661542924129644962546503) }, { SC_(3.0), SC_(-11.250), SC_(1558.5442281134520807137938731609983) }, { SC_(3.0), SC_(-10.750), SC_(1558.5440573914794724810878018559796) }, { SC_(3.0), SC_(-10.250), SC_(1558.5438535364058987293402837691373) }, { SC_(3.0), SC_(-9.7500), SC_(1558.5436081111616066561977307462543) }, { SC_(3.0), SC_(-9.2500), SC_(1558.5433099660190188764440197184975) }, { SC_(3.0), SC_(-8.7500), SC_(1558.5429441651175968889289739463186) }, { SC_(3.0), SC_(-8.2500), SC_(1558.5424903992902266328747639288402) }, { SC_(3.0), SC_(-7.7500), SC_(1558.5419205916065598210405941045860) }, { SC_(3.0), SC_(-7.2500), SC_(1558.5411952034044973136368045706704) }, { SC_(3.0), SC_(-6.7500), SC_(1558.5402573917442935596345188772766) }, { SC_(3.0), SC_(-6.2500), SC_(1558.5390235064410556263866168800637) }, { SC_(3.0), SC_(-5.7500), SC_(1558.5373671367583214573691686314356) }, { SC_(3.0), SC_(-5.2500), SC_(1558.5350913464410556263866168800637) }, { SC_(3.0), SC_(-4.7500), SC_(1558.5318783056006283387768251220856) }, { SC_(3.0), SC_(-4.2500), SC_(1558.5271934026830535593466489654603) }, { SC_(3.0), SC_(-3.7500), SC_(1558.5200920240343420150070566273687) }, { SC_(3.0), SC_(-3.2500), SC_(1558.5088028182791311925167498981598) }, { SC_(3.0), SC_(-2.7500), SC_(1558.4897512832936012742663158866280) }, { SC_(3.0), SC_(-2.2500), SC_(1558.4550231887143400437474491033697) }, { SC_(3.0), SC_(-1.7500), SC_(1558.3848404165495264159916078748802) }, { SC_(3.0), SC_(-1.2500), SC_(1558.2209125348505997602540791902467) }, { SC_(3.0), SC_(-0.75000), SC_(1557.7451069721513589857542067919980) }, { SC_(3.0), SC_(-0.25000), SC_(1555.7633125348505997602540791902467) }, + { SC_(4.0), SC_(-12.750), SC_(-24481.574976569827769932951761311307) }, { SC_(4.0), SC_(-12.250), SC_(24481.574556933476371183897773040987) }, { SC_(4.0), SC_(-11.750), SC_(-24481.575047799396993548993707180364) }, { SC_(4.0), SC_(-11.250), SC_(24481.574469931163471195977061446181) }, { SC_(4.0), SC_(-10.750), SC_(-24481.575154956733102461973007401188) }, { SC_(4.0), SC_(-10.250), SC_(24481.574336748213717601504674106852) }, { SC_(4.0), SC_(-9.7500), SC_(-24481.575322130804866489839080372249) }, { SC_(4.0), SC_(-9.2500), SC_(24481.574124623184691317447595452944) }, { SC_(4.0), SC_(-8.7500), SC_(-24481.575594518925485881539083161966) }, { SC_(4.0), SC_(-8.2500), SC_(24481.573770215950618995904133489849) }, { SC_(4.0), SC_(-7.7500), SC_(-24481.576062438244817112573771089615) }, { SC_(4.0), SC_(-7.2500), SC_(24481.573142242187841144152395619221) }, { SC_(4.0), SC_(-6.7500), SC_(-24481.576920863980180344267229271453) }, { SC_(4.0), SC_(-6.2500), SC_(24481.571944064552838833945395514059) }, { SC_(4.0), SC_(-5.7500), SC_(-24481.578633607675571219683733120840) }, { SC_(4.0), SC_(-5.2500), SC_(24481.569427482152838833945395514059) }, { SC_(4.0), SC_(-4.7500), SC_(-24481.582451925002662084791450344735) }, { SC_(4.0), SC_(-4.2500), SC_(24481.563410001194361068581610436266) }, { SC_(4.0), SC_(-3.7500), SC_(-24481.592377214742692673229150129760) }, { SC_(4.0), SC_(-3.2500), SC_(24481.546101215873022370388764255277) }, { SC_(4.0), SC_(-2.7500), SC_(-24481.624740671532816130019273586550) }, { SC_(4.0), SC_(-2.2500), SC_(24481.479910902562510187288086353997) }, { SC_(4.0), SC_(-1.7500), SC_(-24481.777338295887834105691576149093) }, { SC_(4.0), SC_(-1.2500), SC_(24481.063714184582527461077650952890) }, { SC_(4.0), SC_(-0.75000), SC_(-24483.239586168797931089091350052823) }, { SC_(4.0), SC_(-0.25000), SC_(24473.199394184582527461077650952890) }, + { SC_(5.0), SC_(-12.750), SC_(492231.26705220367447285602722829798) }, { SC_(5.0), SC_(-12.250), SC_(492231.26703986858726773334478356804) }, { SC_(5.0), SC_(-11.750), SC_(492231.26702427051007143797156325129) }, { SC_(5.0), SC_(-11.250), SC_(492231.26700435743914528929551352935) }, { SC_(5.0), SC_(-10.750), SC_(492231.26697867164364211329952060413) }, { SC_(5.0), SC_(-10.250), SC_(492231.26694516501703258064111915631) }, { SC_(5.0), SC_(-9.7500), SC_(492231.26690091626142628638506805946) }, { SC_(5.0), SC_(-9.2500), SC_(492231.26684168939311732012547103246) }, { SC_(5.0), SC_(-8.7500), SC_(492231.26676123004572403423122047499) }, { SC_(5.0), SC_(-8.2500), SC_(492231.26665011791524038956143753889) }, { SC_(5.0), SC_(-7.7500), SC_(492231.26649384757753475935425594490) }, { SC_(5.0), SC_(-7.2500), SC_(492231.26626952775598108546947519306) }, { SC_(5.0), SC_(-6.7500), SC_(492231.26594002452246170664879905339) }, { SC_(5.0), SC_(-6.2500), SC_(492231.26544319835253121636119925846) }, { SC_(5.0), SC_(-5.7500), SC_(492231.26467132548883883596990731311) }, { SC_(5.0), SC_(-5.2500), SC_(492231.26342993243253121636119925846) }, { SC_(5.0), SC_(-4.7500), SC_(492231.26135104955223808370232711841) }, { SC_(5.0), SC_(-4.2500), SC_(492231.25769899818636191601473727961) }, { SC_(5.0), SC_(-3.7500), SC_(492231.25090337614167956955737997628) }, { SC_(5.0), SC_(-3.2500), SC_(492231.23733572133772815343491824315) }, { SC_(5.0), SC_(-2.7500), SC_(492231.20775210042151496050388203390) }, { SC_(5.0), SC_(-2.2500), SC_(492231.13550447009078633328002916426) }, { SC_(5.0), SC_(-1.7500), SC_(492230.93030187432148227746333192018) }, { SC_(5.0), SC_(-1.2500), SC_(492230.21062287457971360836795049513) }, { SC_(5.0), SC_(-0.75000), SC_(492226.75245080886406232489254933809) }, { SC_(5.0), SC_(-0.25000), SC_(492198.75334287457971360836795049513) }, + { SC_(6.0), SC_(-12.750), SC_(-1.1791224761262553923199740571475659e7) }, { SC_(6.0), SC_(-12.250), SC_(1.1791224761212959953338096554813394e7) }, { SC_(6.0), SC_(-11.750), SC_(-1.1791224761275698941741584362376857e7) }, { SC_(6.0), SC_(-11.250), SC_(1.1791224761195566737931185183742355e7) }, { SC_(6.0), SC_(-10.750), SC_(-1.1791224761298983469279962918313528e7) }, { SC_(6.0), SC_(-10.250), SC_(1.1791224761163997446137740568065356e7) }, { SC_(6.0), SC_(-9.7500), SC_(-1.1791224761342381822144610498473088e7) }, { SC_(6.0), SC_(-9.2500), SC_(1.1791224761103426349211734412564015e7) }, { SC_(6.0), SC_(-8.7500), SC_(-1.1791224761428342570269073362379294e7) }, { SC_(6.0), SC_(-8.2500), SC_(1.1791224760979163768967238911569317e7) }, { SC_(6.0), SC_(-7.7500), SC_(-1.1791224761611690548456004706583543e7) }, { SC_(6.0), SC_(-7.2500), 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SC_(35853.714744150436439369297880163123) }, { SC_(11.), SC_(-0.49999976158142089843750000000000000), SC_(3.2699873393997058844655604408250598e11) }, { SC_(12.), SC_(-0.49999976158142089843750000000000000), SC_(4.6184127480583821271680125577145403e7) }, { SC_(13.), SC_(-0.49999976158142089843750000000000000), SC_(2.0404703892667592897735663251491103e14) }, { SC_(14.), SC_(-0.49999976158142089843750000000000000), SC_(4.0665644827064704770358560360332387e10) }, { SC_(15.), SC_(-0.49999976158142089843750000000000000), SC_(1.7139949675920960909557128308150980e17) }, { SC_(16.), SC_(-0.49999976158142089843750000000000000), SC_(4.4439689184846657075559083382657063e13) }, { SC_(17.), SC_(-0.49999976158142089843750000000000000), SC_(1.8648265054954223096603082369745267e20) }, { SC_(18.), SC_(-0.49999976158142089843750000000000000), SC_(6.0819661971925807709274166342293111e16) }, { SC_(19.), SC_(-0.49999976158142089843750000000000000), SC_(2.5510826566134726713883235580466596e23) }, { SC_(20.), SC_(-0.49999976158142089843750000000000000), SC_(1.0218139657405687413065653078640133e20) }, { SC_(21.), SC_(-0.49999976158142089843750000000000000), SC_(4.2858188626062232387116210610555278e26) }, + } }; + do_test_polygamma(neg_data, name, "Mathematica Data - negative arguments"); + + boost::array, 90> small_data = + { { + { SC_(0.0), SC_(0.12500000000000000000), SC_(-8.3884926632958548678027429230863430) }, { SC_(0.0), SC_(0.062500000000000000000), SC_(-16.478853490060104366505723782801995) }, { SC_(0.0), SC_(0.031250000000000000000), SC_(-32.526953288606118111369026129964135) }, { SC_(0.0), SC_(0.015625000000000000000), SC_(-64.551802973167856670965920212624596) }, { SC_(0.0), SC_(0.0078125000000000000000), SC_(-128.56443747297672763722041223143322) }, { SC_(0.0), SC_(0.0039062500000000000000), SC_(-256.57080841886464838984737508407824) }, { SC_(0.0), SC_(0.0019531250000000000000), SC_(-512.57400748048652546824732749592750) }, { SC_(0.0), SC_(0.00097656250000000000000), SC_(-1024.5756104293406219086220979096446) }, { SC_(0.0), SC_(0.00048828125000000000000), SC_(-2048.5764127609059633822989920937770) }, { SC_(0.0), SC_(0.00024414062500000000000), SC_(-4096.5768141413027972625364884707221) }, { SC_(0.0), SC_(0.00012207031250000000000), SC_(-8192.5770148851960259755970875167303) }, { SC_(0.0), SC_(0.000061035156250000000000), SC_(-16384.577115270571506673278270921248) }, { SC_(0.0), SC_(0.000030517578125000000000), SC_(-32768.577165466617109315191527852551) }, { SC_(0.0), SC_(0.000015258789062500000000), SC_(-65536.577190565479456940587995127301) }, { SC_(0.0), SC_(7.6293945312500000000e-6), SC_(-131072.57720311512052742189906385878) }, { SC_(0.0), SC_(3.8146972656250000000e-6), SC_(-262144.57720938999353809133982546559) }, { SC_(0.0), SC_(1.9073486328125000000e-6), SC_(-524288.57721252744316244096483807868) }, { SC_(0.0), SC_(9.5367431640625000000e-7), SC_(-1.0485765772140961712543892173131386e6) }, + { SC_(1.0), SC_(0.1250000000), SC_(65.388133444988034473142999334395961) }, { SC_(1.0), SC_(0.06250000000), SC_(257.50642004291541426394984152786018) }, { SC_(1.0), SC_(0.03125000000), SC_(1025.5728544782377088851896549789956) }, { SC_(1.0), SC_(0.01562500000), SC_(4097.6081469812325471140472931934309) }, { SC_(1.0), SC_(0.007812500000), SC_(16385.626348148031663597978251925972) }, { SC_(1.0), SC_(0.003906250000), SC_(65537.635592296074077546680509110271) }, { SC_(1.0), SC_(0.001953125000), SC_(262145.64025088744769438583827382756) }, { SC_(1.0), SC_(0.0009765625000), SC_(1.0485776425893921526170408061678298e6) }, + { SC_(2.0), SC_(0.1250000000), SC_(-1025.7533381181356825956689300565174) }, { SC_(2.0), SC_(0.06250000000), SC_(-8194.0423055503627202407284588855458) }, { SC_(2.0), SC_(0.03125000000), SC_(-65538.212736402744663973874571262931) }, { SC_(2.0), SC_(0.01562500000), SC_(-524290.30560802491992997062359624105) }, { SC_(2.0), SC_(0.007812500000), SC_(-4.1943063541297826472489756741474152e6) }, { SC_(2.0), SC_(0.003906250000), SC_(-3.3554434378935516909394862712904232e7) }, { SC_(2.0), SC_(0.001953125000), SC_(-2.6843545839147764655287988662280398e8) }, { SC_(2.0), SC_(0.0009765625000), SC_(-2.1474836503977839163960630063909364e9) }, + { SC_(3.0), SC_(0.1250000000), SC_(24580.143419063566218511004446647010) }, { SC_(3.0), SC_(0.06250000000), SC_(393221.15036999967974263906424748910) }, { SC_(3.0), SC_(0.03125000000), SC_(6.2914617723523498519444110540563165e6) }, { SC_(3.0), SC_(0.01562500000), SC_(1.0066330211954465631968224525194028e8) }, { SC_(3.0), SC_(0.007812500000), SC_(1.6106127423031841473495039368910042e9) }, { SC_(3.0), SC_(0.003906250000), SC_(2.5769803782397651667126674423858834e10) }, { SC_(3.0), SC_(0.001953125000), SC_(4.1231686042244556536661660289768233e11) }, { SC_(3.0), SC_(0.0009765625000), SC_(6.5970697666624696945083422683684831e12) }, + { SC_(4.0), SC_(0.1250000000), SC_(-786445.98543106378579320120709638297) }, { SC_(4.0), SC_(0.06250000000), SC_(-2.5165842491343080297812493001330106e7) }, { SC_(4.0), SC_(0.03125000000), SC_(-8.0530638940150780088628643019449463e8) }, { SC_(4.0), SC_(0.01562500000), SC_(-2.5769803799064252508878172105001377e10) }, { SC_(4.0), SC_(0.007812500000), SC_(-8.2463372085595426712260437166652246e11) }, { SC_(4.0), SC_(0.003906250000), SC_(-2.6388279066648414875708308182697262e13) }, { SC_(4.0), SC_(0.001953125000), SC_(-8.4442493013199264920484069629201316e14) }, { SC_(4.0), SC_(0.0009765625000), SC_(-2.7021597764223000767391638642998277e16) }, + { SC_(5.0), SC_(0.1250000000), SC_(3.1457340659602645662942019557433307e7) }, { SC_(5.0), SC_(0.06250000000), SC_(2.0132660051504893647288429933363318e9) }, { SC_(5.0), SC_(0.03125000000), SC_(1.2884901898167237155557768979087387e11) }, { SC_(5.0), SC_(0.01562500000), SC_(8.2463372084313301694493047619778287e12) }, { SC_(5.0), SC_(0.007812500000), SC_(5.2776558133259656048321206289799569e14) }, { SC_(5.0), SC_(0.003906250000), SC_(3.3776997205278839283396175959111085e16) }, { SC_(5.0), SC_(0.001953125000), SC_(2.1617278211378382006727785506307164e18) }, { SC_(5.0), SC_(0.0009765625000), SC_(1.3835058055282163724137457865337740e20) }, + { SC_(8.0), SC_(0.1250000000), SC_(-5.4116588069756277838328695722389595e12) }, { SC_(8.0), SC_(0.06250000000), SC_(-2.7707693020189462516270216110672946e15) }, { SC_(8.0), SC_(0.03125000000), SC_(-1.4186338826217368769684713903764732e18) }, { SC_(8.0), SC_(0.01562500000), SC_(-7.2634054790231363002428528775599797e20) }, { SC_(8.0), SC_(0.007812500000), SC_(-3.7188636052598456061623085548707189e23) }, { SC_(8.0), SC_(0.003906250000), SC_(-1.9040581658930409501626172937578262e26) }, { SC_(8.0), SC_(0.001953125000), SC_(-9.7487778093723696648306072338399010e28) }, { SC_(8.0), SC_(0.0009765625000), SC_(-4.9913742383986532683932688751727976e31) }, + { SC_(15.0), SC_(0.1250000000), SC_(3.6807761227792200230957850246407904e26) }, { SC_(15.0), SC_(0.06250000000), SC_(2.4122334398245883325511911077327284e31) }, { SC_(15.0), SC_(0.03125000000), SC_(1.5808813071234422095882610857505513e36) }, { SC_(15.0), SC_(0.01562500000), SC_(1.0360463734364190864757622613687259e41) }, { SC_(15.0), SC_(0.007812500000), SC_(6.7898335129529161251275555560392101e45) }, { SC_(15.0), SC_(0.003906250000), SC_(4.4497852910488231117635948092058560e50) }, { SC_(15.0), SC_(0.001953125000), SC_(2.9162112883417567145253894941611498e55) }, { SC_(15.0), SC_(0.0009765625000), SC_(1.9111682299276536804313592588934511e60) }, + { SC_(22.0), SC_(0.1250000000), SC_(-6.6349292044725783891012472579673570e41) }, { SC_(22.0), SC_(0.06250000000), SC_(-5.5657820204072306855435555719642654e48) }, { SC_(22.0), SC_(0.03125000000), SC_(-4.6689163582644258586596154617085763e55) }, { SC_(22.0), SC_(0.01562500000), SC_(-3.9165709114267828873358919539012256e62) }, { SC_(22.0), SC_(0.007812500000), SC_(-3.2854578080162002342968961931631453e69) }, { SC_(22.0), SC_(0.003906250000), SC_(-2.7560417651987161415024817781137906e76) }, { SC_(22.0), SC_(0.001953125000), SC_(-2.3119353999880071814336850663739568e83) }, { SC_(22.0), SC_(0.0009765625000), SC_(-1.9393919791822596946232062017265105e90) }, + { SC_(35.0), SC_(0.1250000000), SC_(3.3532982327901451846973629635910627e72) }, { SC_(35.0), SC_(0.06250000000), SC_(2.3043689989709229438987285737704404e83) }, { SC_(35.0), SC_(0.03125000000), SC_(1.5835503181594194718369731136519624e94) }, { SC_(35.0), SC_(0.01562500000), SC_(1.0882074924904162473416628106826351e105) }, { SC_(35.0), SC_(0.007812500000), SC_(7.4781049464136054012151819362261716e115) }, { SC_(35.0), SC_(0.003906250000), SC_(5.1389145889443628300269064119650184e126) }, { SC_(35.0), SC_(0.001953125000), SC_(3.5314352154325314429637711743107931e137) }, { SC_(35.0), SC_(0.0009765625000), SC_(2.4267838013160699267233738410387795e148) } + } }; + do_test_polygamma(small_data, name, "Mathematica Data - small arguments"); + + boost::array, 23> bug_cases = + { { + { SC_(171.0), SC_(2.0), SC_(2.073093314165313149880140394410e257) }, { SC_(171.0), SC_(5.0), SC_(7.42911976071332889749264626321716781e188) }, + { SC_(166.0), SC_(2.0), SC_(-4.8129498903508823293044351695484095e247) }, { SC_(166.0), SC_(3.0), SC_(-1.8843912448604502196243093626013895e218) }, + { SC_(171.0), SC_(23.0), SC_(7.53143916217078889612817829861181739e74) }, { SC_(168.0), SC_(150.0), SC_(-6.5266062780306068333215312257902920e-66) }, + { SC_(169.0), SC_(202.0), SC_(9.2734049986021958613510169328055599e-88) }, + { SC_(20.0), SC_(-9.5), SC_(-0.0010307637762451416598018081796345932)}, { SC_(21.0), SC_(-9.5), SC_(4.2858188624279670465725116159418790e26) }, { SC_(22.0), SC_(-9.5), SC_(-0.0041914420015886426448664611125724338) }, { SC_(23.0), SC_(-9.5), SC_(8.6744973773084910367753904944787155e29) }, { SC_(24.0), SC_(-9.5), SC_(-0.020482527998674199369359987617445420) }, { SC_(25.0), SC_(-9.5), SC_(2.0818793705474855280713094147422278e33) }, { SC_(26.0), SC_(-9.5), SC_(-0.11840299831136879082790399023048278) }, { SC_(27.0), SC_(-9.5), SC_(5.8459172724952950454469355072783445e36) }, { SC_(28.0), SC_(-9.5), SC_(-0.79896867888629450211348696225656872) }, { SC_(29.0), SC_(-9.5), SC_(1.8987539301063980537054871851063348e40) }, { SC_(30.0), SC_(-9.5), SC_(-6.2224465669723272001827279817965087) }, + { SC_(1.0), ldexp(value_type(1), 120), SC_(7.5231638452626400509999138382223723e-37) }, { SC_(2.0), ldexp(value_type(1), 120), SC_(-5.6597994242666952296931995568048699e-73) }, { SC_(3.0), ldexp(value_type(1), 120), SC_(8.5159196800163014398201074324946177e-109) }, { SC_(10.0), ldexp(value_type(1), 120), SC_(-2.1075031678562075551498983356333178e-356) }, { SC_(15.0), ldexp(value_type(1), 120), SC_(1.2201582392961399809842378412624410e-531) }, + } }; + do_test_polygamma(bug_cases, name, "Mathematica Data - Large orders and other bug cases"); +} + diff --git a/test/test_trig.cpp b/test/test_trig.cpp new file mode 100644 index 000000000..142910c81 --- /dev/null +++ b/test/test_trig.cpp @@ -0,0 +1,82 @@ +// Copyright John Maddock 2014. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include "test_trig.hpp" + +// +// DESCRIPTION: +// ~~~~~~~~~~~~ +// +// This file tests the functions sin_pi, cos_pi, with test data +// generated at high precision naively using sin(pi*x) rather than +// our routines. +// + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + const char* largest_type; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + if(boost::math::policies::digits >() == boost::math::policies::digits >()) + { + largest_type = "(long\\s+)?double|real_concept"; + } + else + { + largest_type = "long double|real_concept"; + } +#else + largest_type = "(long\\s+)?double"; +#endif + // + // On MacOS X erfc has much higher error levels than + // expected: given that the implementation is basically + // just a rational function evaluation combined with + // exponentiation, we conclude that exp and pow are less + // accurate on this platform, especially when the result + // is outside the range of a double. + // + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*", // test data group + ".*", 2, 1); // test function + + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} + +BOOST_AUTO_TEST_CASE( test_main ) +{ + BOOST_MATH_CONTROL_FP; + expected_results(); + + test_trig(0.1F, "float"); + test_trig(0.1, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_trig(0.1L, "long double"); +#ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS +#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) + test_trig(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif +#endif +#else + std::cout << "The long double tests have been disabled on this platform " + "either because the long double overloads of the usual math functions are " + "not available at all, or because they are too inaccurate for these tests " + "to pass." << std::cout; +#endif +} + diff --git a/test/test_trig.hpp b/test/test_trig.hpp new file mode 100644 index 000000000..c3b1887ca --- /dev/null +++ b/test/test_trig.hpp @@ -0,0 +1,81 @@ +// Copyright John Maddock 2014. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#define BOOST_TEST_MAIN +#include +#include +#include +#include +#include +#include +#include "functor.hpp" + +#include "handle_test_result.hpp" +#include "table_type.hpp" + +#ifndef SC_ +#define SC_(x) static_cast::type>(BOOST_JOIN(x, L)) +#endif + +template +void do_test_trig(const T& data, const char* type_name, const char* test_name) +{ + typedef Real value_type; + + typedef value_type (*pg)(value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::sin_pi; +#else + pg funcp = boost::math::sin_pi; +#endif + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test sin_pi against data: + // + result = boost::math::tools::test_hetero( + data, + bind_func(funcp, 0), + extract_result(1)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::sin_pi", test_name); + // + // test cos_pi against data: + // +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + funcp = boost::math::cos_pi; +#else + funcp = boost::math::cos_pi; +#endif + result = boost::math::tools::test_hetero( + data, + bind_func(funcp, 0), + extract_result(2)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::cos_pi", test_name); + std::cout << std::endl; +} + +template +void test_trig(T, const char* name) +{ + // + // The actual test data is rather verbose, so it's in a separate file + // + // The contents are as follows, each row of data contains + // three items, input value a, input value b and erf(a, b): + // +# include "trig_data.ipp" + + do_test_trig(trig_data, name, "sin_pi and cos_pi"); + +# include "trig_data2.ipp" + + do_test_trig(trig_data2, name, "sin_pi and cos_pi near integers and half integers"); +} + diff --git a/test/test_trigamma.cpp b/test/test_trigamma.cpp new file mode 100644 index 000000000..9e471dc1c --- /dev/null +++ b/test/test_trigamma.cpp @@ -0,0 +1,56 @@ +// (C) Copyright John Maddock 2014. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include "test_trigamma.hpp" + +void expected_results() +{ + // + // Define the max and mean errors expected for + // various compilers and platforms. + // + const char* largest_type; +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + if(boost::math::policies::digits >() == boost::math::policies::digits >()) + { + largest_type = "(long\\s+)?double|real_concept"; + } + else + { + largest_type = "long double|real_concept"; + } +#else + largest_type = "(long\\s+)?double|real_concept"; +#endif + + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*", // test data group + ".*", 20, 10); // test function + // + // Finish off by printing out the compiler/stdlib/platform names, + // we do this to make it easier to mark up expected error rates. + // + std::cout << "Tests run with " << BOOST_COMPILER << ", " + << BOOST_STDLIB << ", " << BOOST_PLATFORM << std::endl; +} + +BOOST_AUTO_TEST_CASE( test_main ) +{ + expected_results(); + BOOST_MATH_CONTROL_FP; + + test_trigamma(0.0F, "float"); + test_trigamma(0.0, "double"); +#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS + test_trigamma(0.0L, "long double"); + test_trigamma(boost::math::concepts::real_concept(0.1), "real_concept"); +#endif +} + diff --git a/test/test_trigamma.hpp b/test/test_trigamma.hpp new file mode 100644 index 000000000..9535bbba8 --- /dev/null +++ b/test/test_trigamma.hpp @@ -0,0 +1,74 @@ +// Copyright John Maddock 2014 +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define BOOST_MATH_OVERFLOW_ERROR_POLICY ignore_error + +#include +#include +#define BOOST_TEST_MAIN +#include +#include +#include +#include +#include +#include +#include +#include "functor.hpp" + +#include "handle_test_result.hpp" +#include "table_type.hpp" + +#ifndef SC_ +#ifndef BOOST_FLOAT128_C +#define SC_(x) static_cast::type>(BOOST_JOIN(x, L)) +#else +#define SC_(x) static_cast::type>(BOOST_FLOAT128_C(x)) +#endif +#endif + +template +void do_test_trigamma(const T& data, const char* type_name, const char* test_name) +{ + typedef Real value_type; + + typedef value_type(*pg)(value_type); +#if defined(BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS) + pg funcp = boost::math::trigamma; +#else + pg funcp = boost::math::trigamma; +#endif + + boost::math::tools::test_result result; + + std::cout << "Testing " << test_name << " with type " << type_name + << "\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n"; + + // + // test trigamma against data: + // + result = boost::math::tools::test_hetero( + data, + bind_func(funcp, 0), + extract_result(1)); + handle_test_result(result, data[result.worst()], result.worst(), type_name, "boost::math::trigamma", test_name); + std::cout << std::endl; +} + +template +void test_trigamma(T, const char* name) +{ + typedef typename table_type::type value_type; + + boost::array, 659> data = + { { + { SC_(1.0), SC_(1.6449340668482264364724151666460252) }, { SC_(2.0), SC_(0.64493406684822643647241516664602519) }, { SC_(3.0), SC_(0.39493406684822643647241516664602519) }, { SC_(4.0), SC_(0.28382295573711532536130405553491408) }, { SC_(5.0), SC_(0.22132295573711532536130405553491408) }, { SC_(6.0), SC_(0.18132295573711532536130405553491408) }, { SC_(7.0), SC_(0.15354517795933754758352627775713630) }, { SC_(8.0), SC_(0.13313701469403142513454668592040161) }, { SC_(9.0), SC_(0.11751201469403142513454668592040161) }, { SC_(10.0), SC_(0.10516633568168574612220100690805593) }, { SC_(11.0), SC_(0.095166335681685746122201006908055927) }, { SC_(12.0), SC_(0.086901872871768390750300180461774936) }, { SC_(13.0), SC_(0.079957428427323946305855736017330491) }, { SC_(14.0), SC_(0.074040268664010336838400114715555343) }, { SC_(15.0), SC_(0.068938227847683806226155216756371670) }, { SC_(16.0), SC_(0.064493783403239361781710772311927225) }, { SC_(17.0), SC_(0.060587533403239361781710772311927225) }, { SC_(18.0), SC_(0.057127325790782614376866481654487779) }, { SC_(19.0), SC_(0.054040906037696194623780061901401359) }, { SC_(20.0), SC_(0.051270822935203119831536294588381969) }, { SC_(21.0), SC_(0.048770822935203119831536294588381969) }, { SC_(22.0), SC_(0.046503249239057995114983006606522558) }, { SC_(23.0), SC_(0.044437133536578656272007799994952310) }, { SC_(24.0), SC_(0.042546774368336690298472828350339834) }, { SC_(25.0), 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SC_(1.0485792158394087991015231685011609e6) }, { SC_(-14.000976562500000000000000000000000), SC_(1.0485792209407379096480891892744762e6) }, { SC_(-15.000976562500000000000000000000000), SC_(1.0485792253846037068979581619480639e6) }, { SC_(-16.000976562500000000000000000000000), SC_(1.0485792292903769133919482794005111e6) }, { SC_(-17.000976562500000000000000000000000), SC_(1.0485792327501870178663179616292787e6) }, { SC_(-18.000976562500000000000000000000000), SC_(1.0485792358362719002281530732844340e6) }, { SC_(-19.000976562500000000000000000000000), SC_(1.0485792386060702710649859237414384e6) }, { SC_(-20.000976562500000000000000000000000), SC_(1.0485792411058261483202152741904670e6) }, + } }; + do_test_trigamma(data, name, "Mathematica Data"); +} + diff --git a/test/test_zeta.cpp b/test/test_zeta.cpp index bd8713d4f..8683b7c78 100644 --- a/test/test_zeta.cpp +++ b/test/test_zeta.cpp @@ -62,13 +62,20 @@ void expected_results() "real_concept", // test type(s) ".*Random values less than 1", // test data group ".*", 1200, 500); // test function + add_expected_result( + ".*", // compiler + ".*", // stdlib + ".*", // platform + largest_type, // test type(s) + ".*Integer.*", // test data group + ".*", 30, 15); // test function add_expected_result( ".*", // compiler ".*", // stdlib ".*", // platform largest_type, // test type(s) ".*", // test data group - ".*", 3, 1); // test function + ".*", 3, 3); // test function add_expected_result( ".*", // compiler ".*", // stdlib diff --git a/test/test_zeta.hpp b/test/test_zeta.hpp index d70abd442..a93bfd09d 100644 --- a/test/test_zeta.hpp +++ b/test/test_zeta.hpp @@ -79,6 +79,12 @@ void test_zeta(T, const char* name) do_test_zeta(zeta_1_up_data, name, "Zeta: Values close to and greater than 1"); #include "zeta_1_below_data.ipp" do_test_zeta(zeta_1_below_data, name, "Zeta: Values close to and less than 1"); + + boost::array, 90> integer_data = { { + { 2, SC_(1.6449340668482264364724151666460252) }, { 3, SC_(1.2020569031595942853997381615114500) }, { 4, SC_(1.0823232337111381915160036965411679) }, { 5, SC_(1.0369277551433699263313654864570342) }, { 6, SC_(1.0173430619844491397145179297909205) }, { 7, SC_(1.0083492773819228268397975498497968) }, { 8, SC_(1.0040773561979443393786852385086525) }, { 9, SC_(1.0020083928260822144178527692324121) }, { SC_(10.0), SC_(1.0009945751278180853371459589003190) }, { SC_(11.0), SC_(1.0004941886041194645587022825264699) }, { SC_(12.0), SC_(1.0002460865533080482986379980477397) }, { SC_(13.0), SC_(1.0001227133475784891467518365263574) }, { SC_(14.0), SC_(1.0000612481350587048292585451051353) }, { SC_(15.0), SC_(1.0000305882363070204935517285106451) }, { SC_(16.0), SC_(1.0000152822594086518717325714876367) }, { SC_(17.0), SC_(1.0000076371976378997622736002935630) }, { SC_(18.0), SC_(1.0000038172932649998398564616446219) }, { SC_(19.0), SC_(1.0000019082127165539389256569577951) }, { SC_(20.0), SC_(1.0000009539620338727961131520386834) }, { SC_(21.0), SC_(1.0000004769329867878064631167196044) }, { SC_(22.0), SC_(1.0000002384505027277329900036481868) }, { SC_(23.0), SC_(1.0000001192199259653110730677887189) }, { SC_(24.0), SC_(1.0000000596081890512594796124402079) }, { SC_(25.0), SC_(1.0000000298035035146522801860637051) }, { SC_(26.0), SC_(1.0000000149015548283650412346585066) }, { SC_(27.0), SC_(1.0000000074507117898354294919810042) }, { SC_(28.0), SC_(1.0000000037253340247884570548192040) }, { SC_(29.0), SC_(1.0000000018626597235130490064039099) }, { SC_(30.0), SC_(1.0000000009313274324196681828717647) }, { SC_(31.0), SC_(1.0000000004656629065033784072989233) }, { SC_(32.0), SC_(1.0000000002328311833676505492001456) }, { SC_(33.0), SC_(1.0000000001164155017270051977592974) }, { SC_(34.0), SC_(1.0000000000582077208790270088924369) }, { SC_(35.0), SC_(1.0000000000291038504449709968692943) }, { SC_(36.0), SC_(1.0000000000145519218910419842359296) }, { SC_(37.0), SC_(1.0000000000072759598350574810145209) }, { SC_(38.0), SC_(1.0000000000036379795473786511902372) }, { SC_(39.0), SC_(1.0000000000018189896503070659475848) }, { SC_(40.0), SC_(1.0000000000009094947840263889282533) }, { SC_(41.0), SC_(1.0000000000004547473783042154026799) }, { SC_(42.0), SC_(1.0000000000002273736845824652515227) }, { SC_(43.0), SC_(1.0000000000001136868407680227849349) }, { SC_(44.0), SC_(1.0000000000000568434198762758560928) }, { SC_(45.0), SC_(1.0000000000000284217097688930185546) }, { SC_(46.0), SC_(1.0000000000000142108548280316067698) }, { SC_(47.0), SC_(1.0000000000000071054273952108527129) }, { SC_(48.0), SC_(1.0000000000000035527136913371136733) }, { SC_(49.0), SC_(1.0000000000000017763568435791203275) }, { SC_(50.0), SC_(1.0000000000000008881784210930815903) }, { SC_(51.0), SC_(1.0000000000000004440892103143813364) }, { SC_(52.0), SC_(1.0000000000000002220446050798041984) }, { SC_(53.0), SC_(1.0000000000000001110223025141066134) }, { SC_(54.0), SC_(1.0000000000000000555111512484548124) }, { SC_(55.0), SC_(1.0000000000000000277555756213612417) }, { SC_(56.0), SC_(1.0000000000000000138777878097252328) }, { SC_(57.0), SC_(1.0000000000000000069388939045441537) }, { SC_(58.0), SC_(1.0000000000000000034694469521659226) }, { SC_(59.0), SC_(1.0000000000000000017347234760475766) }, { SC_(60.0), SC_(1.0000000000000000008673617380119934) }, + { SC_(-61.0), SC_(-3.3066089876577576725680214670439210e34) }, { SC_(-59.0), SC_(3.5666582095375556109684574608651829e32) }, { SC_(-57.0), SC_(-4.1147288792557978697665486067619336e30) }, { SC_(-55.0), SC_(5.0890659468662289689766332915911925e28) }, { SC_(-53.0), SC_(-6.7645882379292820990945242301798478e26) }, { SC_(-51.0), SC_(9.6899578874635940656497942894654088e24) }, { SC_(-49.0), SC_(-1.5001733492153928733711440151515152e23) }, { SC_(-47.0), SC_(2.5180471921451095697089023320225526e21) }, { SC_(-45.0), SC_(-4.5979888343656503490437943262411348e19) }, { SC_(-43.0), SC_(9.1677436031953307756992753623188406e17) }, { SC_(-41.0), SC_(-2.0040310656516252738108421663238939e16) }, { SC_(-39.0), SC_(4.8241448354850170371581670362158167e14) }, { SC_(-37.0), SC_(-1.2850850499305083333333333333333333e13) }, { SC_(-35.0), SC_(3.8087931125245368811553022079337869e11) }, { SC_(-33.0), SC_(-1.2635724795916666666666666666666667e10) }, { SC_(-31.0), SC_(4.7238486772162990196078431372549020e8) }, { SC_(-29.0), SC_(-2.0052695796688078946143462272494531e7) }, { SC_(-27.0), SC_(974936.82385057471264367816091954023) }, { SC_(-25.0), SC_(-54827.583333333333333333333333333333) }, { SC_(-23.0), SC_(3607.5105463980463980463980463980464) }, { SC_(-21.0), SC_(-281.46014492753623188405797101449275) }, { SC_(-19.0), SC_(26.456212121212121212121212121212121) }, { SC_(-17.0), SC_(-3.0539543302701197438039543302701197) }, { SC_(-15.0), SC_(0.44325980392156862745098039215686275) }, { SC_(-13.0), SC_(-0.083333333333333333333333333333333333) }, { SC_(-11.0), SC_(0.021092796092796092796092796092796093) }, { -9, SC_(-0.0075757575757575757575757575757575758) }, { -7, SC_(0.0041666666666666666666666666666666667) }, { -5, SC_(-0.0039682539682539682539682539682539683) }, { -3, SC_(0.0083333333333333333333333333333333333) }, { -1, SC_(-0.083333333333333333333333333333333333) } + } }; + do_test_zeta(integer_data, name, "Zeta: Integer arguments"); } extern "C" double zetac(double); diff --git a/test/trig_data.ipp b/test/trig_data.ipp new file mode 100644 index 000000000..406fde947 --- /dev/null +++ b/test/trig_data.ipp @@ -0,0 +1,421 @@ +#ifndef SC_ +# define SC_(x) static_cast(BOOST_JOIN(x, L)) +#endif + static const boost::array, 414> trig_data = {{ + { SC_(-3.9774532318115234375000000000000000000000e+00), SC_(7.0773544860026243860279929113278325110375e-02), SC_(9.9749240866682582129171337066426038082437e-01) }, + { SC_(-3.9617328643798828125000000000000000000000e+00), SC_(1.1993037621708977961146677085715171980751e-01), SC_(9.9278230486870952597534166945986069243176e-01) }, + { SC_(-3.9047842025756835937500000000000000000000e+00), SC_(2.9468823519354637105911947101495882286179e-01), SC_(9.5559345123253806128916838776381320091963e-01) }, + { SC_(-3.8916873931884765625000000000000000000000e+00), SC_(3.3374547909601862041893513083013201963857e-01), SC_(9.4266322469001038147288152092736823391568e-01) }, + { SC_(-3.8578090667724609375000000000000000000000e+00), SC_(4.3199709903403737307912472091970588030864e-01), SC_(9.0187499490016691899377344383391047319671e-01) }, + { SC_(-3.7453374862670898437500000000000000000000e+00), SC_(7.1738805664233213419617848590715510878805e-01), SC_(6.9667379467505311174160335392136633824863e-01) }, + { SC_(-3.7244319915771484375000000000000000000000e+00), SC_(7.6156366259470177260070281063647817588426e-01), SC_(6.4809010779006898537856851402643268133964e-01) }, + { SC_(-3.7143068313598632812500000000000000000000e+00), SC_(7.8179011765786965192070724412252980346206e-01), SC_(6.2354166816059239705031197761170482474277e-01) }, + { SC_(-3.7084703445434570312500000000000000000000e+00), SC_(7.9309123554739645631663479916579239803262e-01), SC_(6.0910285838920844088315459958697481264632e-01) }, + { SC_(-3.6306295394897460937500000000000000000000e+00), SC_(9.1696736941200755880066612020982424508209e-01), SC_(3.9896220802680404308672276469459511697767e-01) }, + { SC_(-3.5902690887451171875000000000000000000000e+00), SC_(9.6005749263309399424762040094854912358485e-01), SC_(2.7980280706071672819250833169767675946625e-01) }, + { SC_(-3.5683994293212890625000000000000000000000e+00), SC_(9.7700131832151631709443635682201094394830e-01), SC_(2.1323326194104695723693260957232796074504e-01) }, + { SC_(-3.4592370986938476562500000000000000000000e+00), SC_(9.9181146279384676289373069342215326327259e-01), SC_(-1.2771069755791767012021871220455601680871e-01) }, + { SC_(-3.3931655883789062500000000000000000000000e+00), SC_(9.4420293088917725164777488803566691613860e-01), SC_(-3.2936427447476383305860665908806457784788e-01) }, + { SC_(-3.2229461669921875000000000000000000000000e+00), SC_(6.4452819092715720460681805414890459455338e-01), SC_(-7.6458054585515448444199304174329276576484e-01) }, + { SC_(-3.2196769714355468750000000000000000000000e+00), SC_(6.3664172619341055705211898635086791126371e-01), SC_(-7.7115971916943020671267760720429588807288e-01) }, + { SC_(-3.1211061477661132812500000000000000000000e+00), SC_(3.7135335780369037404310918499389657193681e-01), SC_(-9.2849161743546414936854741692037249771494e-01) }, + { SC_(-3.1002845764160156250000000000000000000000e+00), SC_(3.0986713733563102476788873073830476308206e-01), SC_(-9.5077986789762286248285770175256182955887e-01) }, + { SC_(-3.0480184555053710937500000000000000000000e+00), SC_(1.5028291058010493207025880578034599707430e-01), SC_(-9.8864303304457275939138181401436526701454e-01) }, + { SC_(-2.9985380172729492187500000000000000000000e+00), SC_(-4.5929380467587678038943727972886926572694e-03), SC_(-9.9998945240442343165400505621385600628710e-01) }, + { SC_(-2.9928274154663085937500000000000000000000e+00), SC_(-2.2531432037793477675469226940100283743575e-02), SC_(-9.9974613506145963677579891828378701312519e-01) }, + { SC_(-2.9841060638427734375000000000000000000000e+00), SC_(-4.9911526865301578262897706703502998396013e-02), SC_(-9.9875364304015145795419227904280421041712e-01) }, + { SC_(-2.9607505798339843750000000000000000000000e+00), SC_(-1.2299346485981660064237020203833067995992e-01), SC_(-9.9240748062566369342250467688663564344568e-01) }, + { SC_(-2.9161844253540039062500000000000000000000e+00), SC_(-2.6028213071204756629425186005846465917215e-01), SC_(-9.6553260557683736103737209395477391945707e-01) }, + { SC_(-2.8910045623779296875000000000000000000000e+00), SC_(-3.3576688840249427771925848349727492747208e-01), SC_(-9.4194511339701049149409128507465051072874e-01) }, + { SC_(-2.8649091720581054687500000000000000000000e+00), SC_(-4.1177440566510667850464728538235996067571e-01), SC_(-9.1128581621747419851467502037941438628419e-01) }, + { SC_(-2.8070888519287109375000000000000000000000e+00), SC_(-5.6962394780317258349095500647963019767726e-01), SC_(-8.2190544352080338079904161219679275645723e-01) }, + { SC_(-2.8056478500366210937500000000000000000000e+00), SC_(-5.7333889774547421097781817548901513478394e-01), SC_(-8.1931831929476876961507584991747574097795e-01) }, + { SC_(-2.7644929885864257812500000000000000000000e+00), SC_(-6.7418976093170810969556332422484681604555e-01), SC_(-7.3855816714382508099719065401369284800331e-01) }, + { SC_(-2.7390956878662109375000000000000000000000e+00), SC_(-7.3091046787338668451145170450133524796388e-01), SC_(-6.8247336061791230280379473053114594954268e-01) }, + { SC_(-2.7355394363403320312500000000000000000000e+00), SC_(-7.3848948635740579707953709312462199704324e-01), SC_(-6.7426499133469398680589790058582600474355e-01) }, + { SC_(-2.6991062164306640625000000000000000000000e+00), SC_(-8.1066424743162907541066892533920219267801e-01), SC_(-5.8551129616439550978746461056239261291598e-01) }, + { SC_(-2.6305065155029296875000000000000000000000e+00), SC_(-9.1712149632570388738239981145670570715552e-01), SC_(-3.9860777837029461434452340017415559695465e-01) }, + { SC_(-2.6090793609619140625000000000000000000000e+00), SC_(-9.4185655472943839830579836290050447836428e-01), SC_(-3.3601522333547986304807314632542163159655e-01) }, + { SC_(-2.5050191879272460937500000000000000000000e+00), SC_(-9.9987568381768953436701127766269045040788e-01), SC_(-1.5767590497845478262656414731631988934965e-02) }, + { SC_(-2.4929447174072265625000000000000000000000e+00), SC_(-9.9975437034566987067580804393808517747219e-01), SC_(2.2163009153388271557271132253243652915113e-02) }, + { SC_(-2.4272384643554687500000000000000000000000e+00), SC_(-9.7398753115843827533170990651639503684643e-01), SC_(2.2660160888195438795075340941984049712659e-01) }, + { SC_(-2.3354558944702148437500000000000000000000e+00), SC_(-8.6934023639191873450982490002851733105284e-01), SC_(4.9421407647905260119880040102899169441924e-01) }, + { SC_(-2.3183279037475585937500000000000000000000e+00), SC_(-8.4150156664834359869540818771524073982051e-01), SC_(5.4025467450859075039160413881361295523246e-01) }, + { SC_(-2.3046054840087890625000000000000000000000e+00), SC_(-8.1743642279419458878430396216469861867532e-01), SC_(5.7601883188783919247059390170720375367970e-01) }, + { SC_(-2.2620973587036132812500000000000000000000e+00), SC_(-7.3346328340507778941301945165878879887661e-01), SC_(6.7972907242271353377593468076196477559511e-01) }, + { SC_(-2.2317276000976562500000000000000000000000e+00), SC_(-6.6537326993316934952514665105248864416333e-01), SC_(7.4651082487693489365391928166376502681137e-01) }, + { SC_(-2.2095050811767578125000000000000000000000e+00), SC_(-6.1167775393828157540960701817140835841373e-01), SC_(7.9110702521025501171584436651855140566145e-01) }, + { SC_(-2.1587514877319335937500000000000000000000e+00), SC_(-4.7831282457156194256147826366717615468415e-01), SC_(8.7818952501744986937643074486430914859891e-01) }, + { SC_(-2.0518007278442382812500000000000000000000e+00), SC_(-1.6201943644875820325965658806034453314425e-01), SC_(9.8678756691236579534545882522215038541766e-01) }, + { SC_(-1.9913291931152343750000000000000000000000e+00), SC_(2.7236774521824119247640639457167240403283e-02), SC_(9.9962901024012268934784446000199714742073e-01) }, + { SC_(-1.9808664321899414062500000000000000000000e+00), SC_(6.0073684468702575262797776910860329709958e-02), SC_(9.9819394530038838689681575445113488209634e-01) }, + { SC_(-1.9657425880432128906250000000000000000000e+00), SC_(1.0741519398297458068845667640105099725089e-01), SC_(9.9421425060275611673890046532660283866692e-01) }, + { SC_(-1.9592390060424804687500000000000000000000e+00), SC_(1.2770475450980803702535188445773082653842e-01), SC_(9.9181222803290223591996402565002586050197e-01) }, + { SC_(-1.9399342536926269531250000000000000000000e+00), SC_(1.8758420016848789064877165024139046518307e-01), SC_(9.8224852651818657111887581810958489845723e-01) }, + { SC_(-1.8726687431335449218750000000000000000000e+00), SC_(3.8943947239907955191806267942993095630447e-01), SC_(9.2105206005823935504113082081651078396843e-01) }, + { SC_(-1.8622159957885742187500000000000000000000e+00), SC_(4.1946983932693600433549716080531291905775e-01), SC_(9.0776927349136162127391594039572935545573e-01) }, + { SC_(-1.8571534156799316406250000000000000000000e+00), SC_(4.3385385318065783138522312391833924596257e-01), SC_(9.0098325960047459614595694068784152978597e-01) }, + { SC_(-1.8542351722717285156250000000000000000000e+00), SC_(4.4209565798070871752154886197372379282014e-01), SC_(8.9696790867600398251070757116939705415091e-01) }, + { SC_(-1.8153147697448730468750000000000000000000e+00), SC_(5.4819603791581527487211750854339931561457e-01), SC_(8.3634986938087221868046812319445417842653e-01) }, + { SC_(-1.7951345443725585937500000000000000000000e+00), SC_(6.0008215809974023581434137516121683653367e-01), SC_(7.9993837483293571774722252288759286003132e-01) }, + { SC_(-1.7917995452880859375000000000000000000000e+00), SC_(6.0843019022618795009788527140260807985237e-01), SC_(7.9360739892047676462942185706066696498359e-01) }, + { SC_(-1.7846164703369140625000000000000000000000e+00), SC_(6.2618253814748614262259811608337148376852e-01), SC_(7.7967648991050901236982422793142962643534e-01) }, + { SC_(-1.7720141410827636718750000000000000000000e+00), SC_(6.5655224709393941166212925616015192588965e-01), SC_(7.5428054915654475424462469532824344031256e-01) }, + { SC_(-1.7132878303527832031250000000000000000000e+00), SC_(7.8378224352958754678835476318536431268280e-01), SC_(6.2103574351862093135376998317963401489884e-01) }, + { SC_(-1.6237645149230957031250000000000000000000e+00), SC_(9.2535791354740309366987559846725871864256e-01), SC_(3.7909462121638822065656109832689999162915e-01) }, + { 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SC_(8.2471301545686251460782779794050924721187e-01) }, + { SC_(2.2311811447143554687500000000000000000000e+00), SC_(6.6409072491386110431221716662816882524076e-01), SC_(7.4765199731117047178284416396271497606128e-01) }, + { SC_(2.2333374023437500000000000000000000000000e+00), SC_(6.6914010605322762479513348096476764142743e-01), SC_(7.4313627180421982616113592960004612209391e-01) }, + { SC_(2.3376579284667968750000000000000000000000e+00), SC_(8.7273832759493568470846274321787972310865e-01), SC_(4.8818829517594409686222473986620551444155e-01) }, + { SC_(2.3487806320190429687500000000000000000000e+00), SC_(8.8926086331413399471006293915063362270572e-01), SC_(4.5740039022480194691544525128597463590371e-01) }, + { SC_(2.3517999649047851562500000000000000000000e+00), SC_(8.9355947075069965065490920875297802645467e-01), SC_(4.4894484319538579092387834408066355537480e-01) }, + { SC_(2.3615989685058593750000000000000000000000e+00), SC_(9.0695444786248140377665977375260325926430e-01), SC_(4.2122871400518448160209063533603877707564e-01) }, + { SC_(2.3782386779785156250000000000000000000000e+00), SC_(9.2772529803464095872112040596631014126442e-01), SC_(3.7326367541797662646784647430196344006644e-01) }, + { SC_(2.3834285736083984375000000000000000000000e+00), SC_(9.3368761060044476921711548941438701101882e-01), SC_(3.5808860050444529143114408314332600529137e-01) }, + { SC_(2.3848466873168945312500000000000000000000e+00), SC_(9.3527367258872278243326920322096935824487e-01), SC_(3.5392535563647114688274932903268095238901e-01) }, + { SC_(2.4022436141967773437500000000000000000000e+00), SC_(9.5321098636962397016546799166066099108051e-01), SC_(3.0230583101264941654714139135400476632369e-01) }, + { SC_(2.4602479934692382812500000000000000000000e+00), SC_(9.9221204663172793548513469089523554880392e-01), SC_(1.2456024453603865992531044703030770606574e-01) }, + { SC_(2.4654035568237304687500000000000000000000e+00), SC_(9.9409927898231117057693037228057679279072e-01), SC_(1.0847406845347446518601579735745017867104e-01) }, + { SC_(2.4778757095336914062500000000000000000000e+00), SC_(9.9758546444116512850079355986107988926734e-01), SC_(6.9449558211012868711701254228877324083109e-02) }, + { SC_(2.5142784118652343750000000000000000000000e+00), SC_(9.9899409553284540256810561838332194308754e-01), SC_(-4.4841912208470250395938017310281418704264e-02) }, + { SC_(2.5177888870239257812500000000000000000000e+00), SC_(9.9843881536284522828028348393187866490264e-01), SC_(-5.5856351266783040410469084720035114498915e-02) }, + { SC_(2.5667257308959960937500000000000000000000e+00), SC_(9.7810900468525295373101076810092704203059e-01), SC_(-2.0809318814805979949884912276499020086767e-01) }, + { SC_(2.5699672698974609375000000000000000000000e+00), SC_(9.7593918728194691849177504208954489276700e-01), SC_(-2.1804289194434414788759075840709254785401e-01) }, + { SC_(2.5752258300781250000000000000000000000000e+00), SC_(9.7220405411897038987674094446364189936340e-01), SC_(-2.3413516855576842900574271401719103593906e-01) }, + { SC_(2.5876626968383789062500000000000000000000e+00), SC_(9.6231637014349867626162609217881995793322e-01), SC_(-2.7193235142924949432619478747905995121995e-01) }, + { SC_(2.6466283798217773437500000000000000000000e+00), SC_(8.9576523522202297669181665064798731910975e-01), SC_(-4.4452743826183985868905617130138276115396e-01) }, + { SC_(2.6800680160522460937500000000000000000000e+00), SC_(8.4421341143838647980035969183880613057605e-01), SC_(-5.3600719766395076638299146307861267938126e-01) }, + { SC_(2.7257375717163085937500000000000000000000e+00), SC_(7.5889905709718526647373257399273129873174e-01), SC_(-6.5120827784742043835386853758773237310974e-01) }, + { SC_(2.7877416610717773437500000000000000000000e+00), SC_(6.1849755782607421827792609946160229156438e-01), SC_(-7.8578672104024637717491890266603742140897e-01) }, + { SC_(2.7930345535278320312500000000000000000000e+00), SC_(6.0534650679100039131411264058534198908238e-01), SC_(-7.9596206361605784580057896592926540608297e-01) }, + { SC_(2.9794301986694335937500000000000000000000e+00), SC_(6.4576969325134380719184170853709332046455e-02), SC_(-9.9791272916662439637917978556650177608583e-01) }, + { SC_(3.0140590667724609375000000000000000000000e+00), SC_(-4.4153501845982175189175058527762162423716e-02), SC_(-9.9902475858946401221242382128074456077456e-01) }, + { SC_(3.0274448394775390625000000000000000000000e+00), SC_(-8.6113718924577264256191741572510303127778e-02), SC_(-9.9628531426142125159845398506889259793638e-01) }, + { SC_(3.1272258758544921875000000000000000000000e+00), SC_(-3.8913452368727790960356345180278976524450e-01), SC_(-9.2118093905305886299976636088054353344837e-01) }, + { SC_(3.2463350296020507812500000000000000000000e+00), SC_(-6.9891857428318090957573573027947012586588e-01), SC_(-7.1520124896560809131726883061795792563693e-01) }, + { SC_(3.2589178085327148437500000000000000000000e+00), SC_(-7.2663709244263817401414242437628248351839e-01), SC_(-6.8702149594209125055751180411261051580649e-01) }, + { SC_(3.3070068359375000000000000000000000000000e+00), SC_(-8.2175864645735172534612332054328975554008e-01), SC_(-5.6983570173566802821049780901183552862792e-01) }, + { SC_(3.3258838653564453125000000000000000000000e+00), SC_(-8.5408772066117639250002441486943704634544e-01), SC_(-5.2012898920921176087604522292013122428088e-01) }, + { SC_(3.3375492095947265625000000000000000000000e+00), SC_(-8.7257153578052091358315534289731125877922e-01), SC_(-4.8848635082837587004720613584405617142606e-01) }, + { SC_(3.3669977188110351562500000000000000000000e+00), SC_(-9.1396798220297561033487383312195293377728e-01), SC_(-4.0578630768893774739693190146836884794326e-01) }, + { SC_(3.4341087341308593750000000000000000000000e+00), SC_(-9.7865116955588442390300932302401182353872e-01), SC_(-2.0552831514635533577558252681957801819126e-01) }, + { SC_(3.4719457626342773437500000000000000000000e+00), SC_(-9.9611862556712482137384040772481006307913e-01), SC_(-8.8020928183371141600457936465291857450722e-02) }, + { SC_(3.4720849990844726562500000000000000000000e+00), SC_(-9.9615703275215020366558951270835450836767e-01), SC_(-8.7585193374402881607363783226196334139273e-02) }, + { SC_(3.5205917358398437500000000000000000000000e+00), SC_(-9.9790827684402469941929089861198503088668e-01), SC_(6.4645734632606333231408358461321085923473e-02) }, + { SC_(3.6017761230468750000000000000000000000000e+00), SC_(-9.4931744975485974001547195314422483338195e-01), SC_(3.1431891382945658324672056757940637855560e-01) }, + { SC_(3.6401405334472656250000000000000000000000e+00), SC_(-9.0463898322145714242894381928150807960181e-01), SC_(4.2617873015443671099809962769068440492845e-01) }, + { SC_(3.6573352813720703125000000000000000000000e+00), SC_(-8.8030890857630174758369213697721090228597e-01), SC_(4.7440091218419933754908503670548458480608e-01) }, + { SC_(3.6600542068481445312500000000000000000000e+00), SC_(-8.7622462669288746421939513920418821059507e-01), SC_(4.8190289849399121987063030800400737550306e-01) }, + { SC_(3.6743307113647460937500000000000000000000e+00), SC_(-8.5373690109536976936966210865941858472063e-01), SC_(5.2070462232255516658865250812898441468120e-01) }, + { SC_(3.6759395599365234375000000000000000000000e+00), SC_(-8.5109418559293582989264089230338154739944e-01), SC_(5.2501303531426465035302709741556823160457e-01) }, + { SC_(3.6779518127441406250000000000000000000000e+00), SC_(-8.4775823766338649388282137669646806530795e-01), SC_(5.3038285273363532972430422102518080779833e-01) }, + { SC_(3.7191076278686523437500000000000000000000e+00), SC_(-7.7229721139647358113807359437680820809513e-01), SC_(6.3526137712695126271658979722161374772548e-01) }, + { SC_(3.7197303771972656250000000000000000000000e+00), SC_(-7.7105289310908725572881798363458383569997e-01), SC_(6.3677110175329601145198110228336758429305e-01) }, + { SC_(3.7415590286254882812500000000000000000000e+00), SC_(-7.2560709999357252492032097752516068559035e-01), SC_(6.8810924745923713178558423965711944178988e-01) }, + { SC_(3.7509422302246093750000000000000000000000e+00), SC_(-7.0501057704102466691892843105783269928843e-01), SC_(7.0919678951633828078444575818375506502661e-01) }, + { SC_(3.7647418975830078125000000000000000000000e+00), SC_(-6.7361202403588387903950332613071875859902e-01), SC_(7.3908513790650654594303944610931020837681e-01) }, + { SC_(3.8488769531250000000000000000000000000000e+00), SC_(-4.5713127745715697303348813361224351474208e-01), SC_(8.8939923272419555705330444258023019744461e-01) }, + { SC_(3.8996763229370117187500000000000000000000e+00), SC_(-3.0998392718517287188809579093624249170750e-01), SC_(9.5074179717042915066727903436561012864445e-01) }, + { SC_(3.9081726074218750000000000000000000000000e+00), SC_(-2.8449945044851010534076472484154609631230e-01), SC_(9.5867620325868929081685950779266605717363e-01) }, + { SC_(3.9430503845214843750000000000000000000000e+00), SC_(-1.7795953146963244992533370940904044176079e-01), SC_(9.8403780677325040351109357253557270668483e-01) }, + { SC_(3.9525480270385742187500000000000000000000e+00), SC_(-1.4852322763418675969604894254451481003482e-01), SC_(9.8890891939203558912519717829149622171011e-01) }, + { SC_(3.9716901779174804687500000000000000000000e+00), SC_(-8.8820726269070577371479191303757464259147e-02), SC_(9.9604762867296402989876530880328911976327e-01) } + }}; +//#undef SC_ + diff --git a/test/trig_data2.ipp b/test/trig_data2.ipp new file mode 100644 index 000000000..8f9953f4c --- /dev/null +++ b/test/trig_data2.ipp @@ -0,0 +1,155 @@ +#ifndef SC_ +# define SC_(x) static_cast(BOOST_JOIN(x, L)) +#endif + static const boost::array, 148> trig_data2 = {{ + { SC_(-1.9999995231628417968750000000000000000000e+00), SC_(1.4980281131690112288542788461553611206918e-06), SC_(9.9999999999887795588607701655175253650385e-01) }, + { SC_(-1.9999985694885253906250000000000000000000e+00), SC_(4.4940843394935868575478645466191072371308e-06), SC_(9.9999999998990160297470825676169589344081e-01) }, + { SC_(-1.9999966621398925781250000000000000000000e+00), SC_(1.0486196791994822995771188961609302512444e-05), SC_(9.9999999994501983841826733236935959894956e-01) }, + { SC_(-1.9999957084655761718750000000000000000000e+00), SC_(1.3482253018117696189242608854023945230621e-05), SC_(9.9999999990911442677359804232502397526992e-01) }, + { SC_(-1.9999852180480957031250000000000000000000e+00), SC_(4.6438871491565280117698474932339407631997e-05), SC_(9.9999999892171560671359413131460719431659e-01) }, + { SC_(-1.9999709129333496093750000000000000000000e+00), SC_(9.1379714776169916676504893028080336305096e-05), SC_(9.9999999582487385499707753413380040254131e-01) }, + { SC_(-1.9999628067016601562500000000000000000000e+00), SC_(1.1684619256134242811309917534624539649780e-04), SC_(9.9999999317348361865817975703384517397171e-01) }, + { SC_(-1.9999003410339355468750000000000000000000e+00), SC_(3.1308787053741855117950880780497680759570e-04), SC_(9.9999995098799146008382980990445170109932e-01) }, + { SC_(-1.9998402595520019531250000000000000000000e+00), SC_(5.0183939684769924809171253847476904900975e-04), SC_(9.9999987407860195766947052118236852970333e-01) }, + { SC_(-1.9997320175170898437500000000000000000000e+00), SC_(8.4189170014837154687588880235168421682168e-04), SC_(9.9999964560911981419424672295679903989178e-01) }, + { SC_(-1.9992446899414062500000000000000000000000e+00), SC_(2.3728743044973247838253172935228172467424e-03), SC_(9.9999718472980465503219274828524949063826e-01) }, + { SC_(-1.9987516403198242187500000000000000000000e+00), SC_(3.9218275467790664715140272043149223642317e-03), SC_(9.9999230960477557309064944107279076437187e-01) }, + { SC_(-1.9976792335510253906250000000000000000000e+00), SC_(7.2908382328935566307435101112900335063237e-03), SC_(9.9997342148572217817205233668312377823042e-01) }, + { SC_(-1.9939575195312500000000000000000000000000e+00), SC_(1.8981872167508178357172130364879478736824e-02), SC_(9.9981982803353944005287225251112677857740e-01) }, + { SC_(-1.9844307899475097656250000000000000000000e+00), SC_(4.8892615404945346627001407931204211546893e-02), SC_(9.9880404092037197813676426019506317879923e-01) }, + { SC_(-1.9694142341613769531250000000000000000000e+00), SC_(9.5940223552297220176859818946140456738044e-02), SC_(9.9538709731678521178505835566419054715492e-01) }, + { SC_(-1.9376106262207031250000000000000000000000e+00), SC_(1.9474944565077729586702134297397558908793e-01), SC_(9.8085302335197751084221022090414475952815e-01) }, + { SC_(-1.8771944046020507812500000000000000000000e+00), SC_(3.7630523255097276560855730503841419331357e-01), SC_(9.2649574848175008067512810410200333264846e-01) }, + { SC_(-1.7540383338928222656250000000000000000000e+00), SC_(6.9807919407529171997035782798891723621393e-01), SC_(7.1602055752554423261801142921491759740208e-01) }, + { SC_(-1.7105970382690429687500000000000000000000e+00), SC_(7.8900402338852276350219626277702904546800e-01), SC_(6.1438802973098639626496377371872238232214e-01) }, + { SC_(-1.4999995231628417968750000000000000000000e+00), SC_(9.9999999999887795588607701655175253650385e-01), SC_(-1.4980281131690112288542788461553611206918e-06) }, + { SC_(-1.4999985694885253906250000000000000000000e+00), SC_(9.9999999998990160297470825676169589344081e-01), SC_(-4.4940843394935868575478645466191072371308e-06) }, + { SC_(-1.4999966621398925781250000000000000000000e+00), SC_(9.9999999994501983841826733236935959894956e-01), SC_(-1.0486196791994822995771188961609302512444e-05) }, + { SC_(-1.4999957084655761718750000000000000000000e+00), SC_(9.9999999990911442677359804232502397526992e-01), SC_(-1.3482253018117696189242608854023945230621e-05) }, + { SC_(-1.4999852180480957031250000000000000000000e+00), SC_(9.9999999892171560671359413131460719431659e-01), SC_(-4.6438871491565280117698474932339407631997e-05) }, + { SC_(-1.4999709129333496093750000000000000000000e+00), 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SC_(9.9999230960477557309064944107279076437187e-01), SC_(-3.9218275467790664715140272043149223642317e-03) }, + { SC_(2.5023202896118164062500000000000000000000e+00), SC_(9.9997343240648080532241755401327069531063e-01), SC_(-7.2893402445875685718309310011223138924629e-03) }, + { SC_(2.5060424804687500000000000000000000000000e+00), SC_(9.9981982803353944005287225251112677857740e-01), SC_(-1.8981872167508178357172130364879478736824e-02) }, + { SC_(2.5155687332153320312500000000000000000000e+00), SC_(9.9880411416176367891009655338470382368524e-01), SC_(-4.8891119168357641426955981953471866397804e-02) }, + { SC_(2.5305852890014648437500000000000000000000e+00), SC_(9.9538724103682040861237580872061890490118e-01), SC_(-9.5938732434334304771023968784410054299194e-02) }, + { SC_(2.5623893737792968750000000000000000000000e+00), SC_(9.8085302335197751084221022090414475952815e-01), SC_(-1.9474944565077729586702134297397558908793e-01) }, + { SC_(2.6228055953979492187500000000000000000000e+00), SC_(9.2649574848175008067512810410200333264846e-01), SC_(-3.7630523255097276560855730503841419331357e-01) }, + { SC_(2.7459611892700195312500000000000000000000e+00), SC_(7.1602160326699876911924965932760594412897e-01), SC_(-6.9807812145558366410510314778327047985312e-01) }, + { SC_(2.7894029617309570312500000000000000000000e+00), SC_(6.1438802973098639626496377371872238232214e-01), SC_(-7.8900402338852276350219626277702904546800e-01) }, + { SC_(2.8629188537597656250000000000000000000000e+00), SC_(4.1746437897978552326073028034696674963028e-01), SC_(-9.0869328834487492706020241698554993000928e-01) }, + { SC_(3.3629188537597656250000000000000000000000e+00), SC_(-9.0869328834487492706020241698554993000928e-01), SC_(-4.1746437897978552326073028034696674963028e-01) } + }}; +//#undef SC_ + diff --git a/tools/doc/cstdfloat.qbk b/tools/doc/cstdfloat.qbk new file mode 100644 index 000000000..5d9b2fd78 --- /dev/null +++ b/tools/doc/cstdfloat.qbk @@ -0,0 +1,249 @@ +[book Standardized Floating-Point typedefs for C and C++ + + [quickbook 1.7] + [copyright 2014 Christopher Kormanyos, John Maddock, Paul A. Bristow] + [license + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + [@http://www.boost.org/LICENSE_1_0.txt]) + ] + [authors [Kormanyos, Christopher], [Maddock, John], [Bristow, Paul A.] ] + [last-revision $Date$] + [/version 1.8.3] +] + +[template tr1[] [@http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2005/n1836.pdf Technical Report on C++ Library Extensions]] +[template C99[] [@http://www.open-std.org/JTC1/SC22/WG14/www/docs/n1256.pdf C99 Standard ISO/IEC 9899:1999]] + +[def __gsl [@http://www.gnu.org/software/gsl/ GSL-1.9]] +[def __glibc [@http://www.gnu.org/software/libc/ GNU C Lib]] +[def __hpc [@http://docs.hp.com/en/B9106-90010/index.html HP-UX C Library]] +[def __cephes [@http://www.netlib.org/cephes/ Cephes]] +[def __NTL [@http://www.shoup.net/ntl/ NTL A Library for doing Number Theory]] +[def __NTL_RR [@http://shoup.net/ntl/doc/RR.txt NTL::RR]] +[def __NTL_quad_float [@http://shoup.net/ntl/doc/quad_float.txt NTL::quad_float]] +[def __MPFR [@http://www.mpfr.org/ GNU MPFR library]] +[def __GMP [@http://gmplib.org/ GNU Multiple Precision Arithmetic Library]] +[def __multiprecision [@http://www.boost.org/doc/libs/1_53_0_beta1/libs/multiprecision/doc/html/index.html Boost.Multiprecision]] +[def __cpp_dec_float [@http://www.boost.org/doc/libs/1_53_0_beta1/libs/multiprecision/doc/html/boost_multiprecision/tut/floats/cpp_dec_float.html cpp_dec_float]] +[def __R [@http://www.r-project.org/ The R Project for Statistical Computing]] +[def __godfrey [link godfrey Godfrey]] +[def __pugh [link pugh Pugh]] +[def __NaN [@http://en.wikipedia.org/wiki/NaN NaN]] +[def __errno [@http://en.wikipedia.org/wiki/Errno `::errno`]] +[def __Mathworld [@http://mathworld.wolfram.com Wolfram MathWorld]] +[def __Mathematica [@http://www.wolfram.com/products/mathematica/index.html Wolfram Mathematica]] +[def __WolframAlpha [@http://www.wolframalpha.com/ Wolfram Alpha]] +[def __TOMS748 [@http://portal.acm.org/citation.cfm?id=210111 TOMS Algorithm 748: enclosing zeros of continuous functions]] +[def __TOMS910 [@http://portal.acm.org/citation.cfm?id=1916469 TOMS Algorithm 910: A Portable C++ Multiple-Precision System for Special-Function Calculations]] +[def __why_complements [link why_complements why complements?]] +[def __complements [link math_toolkit.stat_tut.overview.complements complements]] +[def __performance [link perf performance]] +[def __building [link math_toolkit.building building libraries]] +[def __e_float [@http://calgo.acm.org/910.zip e_float (TOMS Algorithm 910)]] +[def __Abramowitz_Stegun M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, NBS (1964)] +[def __DMLF [@http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]] +[def __IEEE754 [@http://en.wikipedia.org/wiki/IEEE_floating_point IEEE_floating_point]] +[def __N3626 [@http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2013/n3626.pdf N3626]] +[def __N1703 [@http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1703.pdf N1703]] + +[/ Some composite templates] +[template super[x]''''''[x]''''''] +[template sub[x]''''''[x]''''''] +[template floor[x]'''⌊'''[x]'''⌋'''] +[template floorlr[x][lfloor][x][rfloor]] +[template ceil[x] '''⌈'''[x]'''⌉'''] + +[/template header_file[file] [@../../../../[file] [file]]] + +[note A printer-friendly PDF version of this manual is also available.] + +[section:overview Overview] + +The header `` provides optional standardized +floating-point `typedef`s having specified widths. +These are useful for writing portable code because they +should behave identically on all platforms. +All `typedef`s are in `namespace boost`. + +The `typedef`s include `float16_t, float32_t, float64_t, float128_t`, +their corresponding least and fast types, +and the corresponding maximum-width type. +The `typedef`s are based on underlying built-in types +such as `float`, `double`, or `long double`, or based on other compiler-specific +non-standardized types such as `__float128`. +The underlying types of these typedef's must conform with +the corresponding specifications of binary16, binary32, binary64, +and binary128 in __IEEE754 floating-point format +[@http://en.wikipedia.org/wiki/IEEE_floating_point]. + +The typedef's are based on __N3626 +proposed for a new C++14 standard header `` and +__N1703 proposed for a new C language standard header ``. + +The 128-bit floating-point type, of great interest in scientific and +numeric programming, is not required in the boost header, +and may not be supplied for all platforms/compilers, because compiler +support for a 128-bit floating-point type is not mandated by either +the C standard or the C++ standard. + +The following code uses `` in combination with +`` to compute a simplified +version of the Jahnke-Emden-Lambda function. Here, we use +a floating-point type with exactly 64 bits (i.e., `float64_t`). +If we were to use, for instance, built-in `double`, +then there would be no guarantee that the code would +behave identically on all platforms. With `float64_t` from +``, however, this is very likely. +Using `float64_t`, we know that +this code is portable and uses a floating-point type +with approximately 15 decimal digits of precision. + + #include + #include + #include + + boost::float64_t jahnke_emden_lambda(boost::float64_t v, boost::float64_t x) + { + const boost::float64_t gamma_v_plus_one = boost::math::tgamma(v + 1); + const boost::float64_t x_half_pow_v = std::pow(x / 2, v); + + return gamma_v_plus_one * boost::math::cyl_bessel_j(x, v) / x_half_pow_v; + } + +See `cstdfloat_test.cpp` for a more detailed test program. + +[endsect] [/section:overview Overview] + +[section:rationale Rationale] + +The implementation of `` is designed to utilize ``, +defined in the 1989 C standard. The preprocessor is used to query certain +preprocessor definitions in `` such as FLT_MAX, DBL_MAX, etc. +Based on the results of these queries, an attempt is made to automatically +detect the presence of built-in floating-point types having specified widths. +An unequivocal test regarding conformance with __IEEE754 (IEC599) based on +[@ http://en.cppreference.com/w/cpp/types/numeric_limits/is_iec559 `std::numeric_limits<>::is_iec559`] +is performed with `BOOST_STATIC_ASSERT`. + +The header `` makes the standardized floating-point +`typedef`s safely available in `namespace boost` without placing any names +in `namespace std`. The intention is to complement rather than compete +with a potential future C++ Standard Library that may contain these `typedef`s. +Should some future C++ standard include `` and ``, +then `` will continue to function, but will become redundant +and may be safely deprecated. + +Because `` is a boost header, its name conforms to the +boost header naming conventions, not the C++ Standard Library header +naming conventions. + +[note + [*cannot synthesize or create +a `typedef` if the underlying type is not provided by the compiler]. +For example, if a compiler does not have an underlying floating-point +type with 128 bits (highly sought-after in scientific and numeric programming), +then `float128_t` and its corresponding least and fast types are not +provided by `.] + +[warning +As an implementation artifact, certain C macro names from `` +may possibly be visible to users of ``. +Don't rely on using these macros; they are not part of any Boost-specified interface. +Use `std::numeric_limits<>` for floating-point ranges, etc. instead.] + +[endsect] [/section:rationale Rationale] + +[section:exact_typdefs Exact-Width Floating-Point `typedef`s] + +The `typedef float#_t`, with # replaced by the width, designates a +floating-point type of exactly # bits. For example `float32_t` denotes +a single-precision floating-point type with approximately +7 decimal digits of precision (equivalent to binary32 in __IEEE754). + +Floating-point types specified in C and C++ are allowed to have +implementation-specific widths and formats. +However, if a platform supports underlying floating-point types +(conformant with __IEEE754) with widths of 16, 32, 64, 128 bits, +or any combination thereof, +then `` does provide the corresponding `typedef`s +`float16_t, float32_t, float64_t, float128_t,` +their corresponding least and fast types, +and the corresponding maximum-width type + +The absence of `float128_t` is indicated by the macro `BOOST_NO_FLOAT128_T`. + +[endsect] [/section:exact_typdefs Exact-Width Floating-Point `typedef`s] + + +[section:fastest_typdefs Fastest minimum-width floating-point `typedef`s] + +The `typedef float_least#_t`, with # replaced by the width, designates a +floating-point type with a [*width of at least # bits], such that no +floating-point type with lesser size has at least the specified width. +Thus, `float_least32_t` denotes the smallest floating-point type with +a width of at least 32 bits. + +Minimum-width floating-point types are provided for all existing +exact-width floating-point types on a given platform. + +For example, if a platfrom supports `float32_t` and `float64_t`, +then `float_least32_t` and `float_least64_t` will also be supported, etc. + +[endsect] [/section:fastest_typdefs Fastest minimum-width floating-point `typedef`s] + +[section:fastest_typdefs Fastest minimum-width floating-point `typedef`s] + +The typedef `float_fast#_t`, with # replaced by the width, designates +the [*fastest] floating-point type with a width of at least # bits. + +There is no absolute guarantee that these types are the fastest for all purposes. +In any case, however, they satisfy the precision and width requirements. + +Fastest minimum-width floating-point types are provided for all existing +exact-width floating-point types on a given platform. + +For example, if a platform supports `float32_t` and `float64_t`, +then `float_fast32_t` and `float_fast64_t` will also be supported, etc. + +[endsect] [/section:fastest_typdefs Fastest minimum-width floating-point `typedef`s] + +[section:greatest_typdefs Greatest-width floating-point typedef] + +The `typedef floatmax_t` designates a floating-point type capable of representing +any value of any floating-point type in a given platform. + +The greatest-width typedef is provided for all platforms. + +[endsect] [/section:greatest_typdefs Greatest-width floating-point typedef] + +[section:macros Floating-Point Constant Macros] + +All macros of the type `BOOST_FLOAT16_C, BOOST_FLOAT32_C, BOOST_FLOAT64_C, +BOOST_FLOAT128_C, BOOST_FLOATMAX_C` are always defined after inclusion of +``. These allow floating-point constants of at +least the specified width to be declared. + +For example: + + #include + + // Declare Pythagoras' constant with approximately 7 decimal digits of precision. + static const boost::float32_t pi = BOOST_FLOAT32_C(3.1415926536); + + // Declare the Euler-gamma constant with approximately 34 decimal digits of precision. + static const boost::float128_t euler = BOOST_FLOAT128_C(0.57721566490153286060651209008240243104216); + +[endsect] [/section:macros Floating-Point Constant Macros] + + +[/ cstdfloat.qbk + Copyright 2014 Christopher Kormanyos, John Maddock and Paul A. Bristow. + Distributed under the Boost Software License, Version 1.0. + (See accompanying file LICENSE_1_0.txt or copy at + http://www.boost.org/LICENSE_1_0.txt). +] + + + + diff --git a/tools/trig_data.cpp b/tools/trig_data.cpp new file mode 100644 index 000000000..41233578f --- /dev/null +++ b/tools/trig_data.cpp @@ -0,0 +1,83 @@ +// (C) Copyright John Maddock 2006. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include "mp_t.hpp" + +using namespace boost::math::tools; +using namespace std; + +float external_f; +float force_truncate(const float* f) +{ + external_f = *f; + return external_f; +} + +float truncate_to_float(mp_t r) +{ + float f = boost::math::tools::real_cast(r); + return force_truncate(&f); +} + +struct trig_data_generator +{ + boost::math::tuple operator()(mp_t z) + { + return boost::math::make_tuple(sin(z * boost::math::constants::pi()), cos(z * boost::math::constants::pi())); + } +}; + + +int main(int argc, char*argv []) +{ + parameter_info arg1; + test_data data; + + bool cont; + std::string line; + + std::cout << "Welcome.\n" + "This program will generate spot tests for the cos_pi and sin_pi functions:\n"; + + do{ + if(0 == get_user_parameter_info(arg1, "a")) + return 1; + data.insert(trig_data_generator(), arg1); + + std::cout << "Any more data [y/n]?"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + cont = (line == "y"); + }while(cont); + + std::cout << "Enter name of test data file [default=trig_data.ipp]"; + std::getline(std::cin, line); + boost::algorithm::trim(line); + if(line == "") + line = "trig_data.ipp"; + std::ofstream ofs(line.c_str()); + ofs << std::scientific << std::setprecision(40); + write_code(ofs, data, "trig_data"); + + return 0; +} + +/* Output for asymptotic limits: + +Erf asymptotic limit for 24 bit numbers is 2.8 after approximately 6 terms. +Erfc asymptotic limit for 24 bit numbers is 4.12064 after approximately 17 terms. +Erf asymptotic limit for 53 bit numbers is 4.3 after approximately 11 terms. +Erfc asymptotic limit for 53 bit numbers is 6.19035 after approximately 29 terms. +Erf asymptotic limit for 64 bit numbers is 4.8 after approximately 12 terms. +Erfc asymptotic limit for 64 bit numbers is 7.06004 after approximately 29 terms. +Erf asymptotic limit for 106 bit numbers is 6.5 after approximately 14 terms. +Erfc asymptotic limit for 106 bit numbers is 11.6626 after approximately 29 terms. +Erf asymptotic limit for 113 bit numbers is 6.8 after approximately 14 terms. +Erfc asymptotic limit for 113 bit numbers is 12.6802 after approximately 29 terms. +*/ +