diff --git a/doc/equations/polygamma3.mml b/doc/equations/polygamma3.mml index 215ecf2bc..3568763aa 100644 --- a/doc/equations/polygamma3.mml +++ b/doc/equations/polygamma3.mml @@ -9,202 +9,86 @@ - - - - - - - ∂ - n - - cot - - - π - x - - - - - ∂ - - x - n - - - - - - = - - - - - - π - n - - - - - sin - - n - + - 1 - - - - - π - x - - - - - - ∑ - - k - = - 0 - - - - - n - − - 2 - - 2 - - - - - C - - n - , - k - - - cos - - - - - 1 - + - 2 - k - - - π - x - - - - - ; - - + + + + ∂ + n + + cot + + + π + x + + + + + ∂ + + x + n + + + + = + + + + π + n + + + + + sin + n - 2 - - ∈ - ℕ - - - - - - - - = - - - - - - π - n - - - - - sin - - n - + - 1 - - - - - π - x - - - - - - - - C - - n - , - 0 - - - + - - ∑ - - k - = - 1 - - - - n - 2 - - - - - C - - n - , - k - - - cos - - - 2 - k - π - x - - - - - - - ; - - - n - 2 - - ∉ - ℕ - - - + + + 1 + + + + + π + x + + + + + + + ∑ + + k + = + 0 + + + n + − + 1 + + + + C + + k + , + n + + + cos + + + k + + + + π + n + + diff --git a/doc/equations/polygamma3.png b/doc/equations/polygamma3.png index 156285b33..e3e38511f 100644 Binary files a/doc/equations/polygamma3.png and b/doc/equations/polygamma3.png differ diff --git a/doc/equations/polygamma3.svg b/doc/equations/polygamma3.svg index c364b7cf6..3e05baaef 100644 --- a/doc/equations/polygamma3.svg +++ b/doc/equations/polygamma3.svg @@ -1,2 +1,2 @@ -∂ncot(πx)∂xn=πnsinn+1(πx)∑k=0n−22Cn,kcos((1+2k)πx);n2∈ℕ=πnsinn+1(πx)(Cn,0+∑k=1n2Cn,kcos(2kπx));n2∉ℕ \ No newline at end of file +∂ncot(πx)∂xn=πnsinn+1(πx)∑k=0n−1Ck,ncosk(πn) \ No newline at end of file diff --git a/doc/equations/polygamma7.mml b/doc/equations/polygamma7.mml index f96ff49a9..d3ea70c85 100644 --- a/doc/equations/polygamma7.mml +++ b/doc/equations/polygamma7.mml @@ -19,10 +19,13 @@ cos + + + k + - k - x + θ @@ -33,61 +36,14 @@ - x + θ = - − 1 - 2 - - - - - - k - + - n - - - cos - - - - - k - − - 1 - - - x - - - + - - - n - − - k - - - cos - - - - - k - + - 1 - - - x - - - sin @@ -99,11 +55,52 @@ - x + θ + + + + + + k + − + n + + + + cos + + n + + + 1 + + + + + θ + + + − + k + + cos + + n + − + 1 + + + + + θ + + + + + diff --git a/doc/equations/polygamma7.png b/doc/equations/polygamma7.png index f8fdae91d..36a0c03c8 100644 Binary files a/doc/equations/polygamma7.png and b/doc/equations/polygamma7.png differ diff --git a/doc/equations/polygamma7.svg b/doc/equations/polygamma7.svg index e068a4d0a..2f83cc0b4 100644 --- a/doc/equations/polygamma7.svg +++ b/doc/equations/polygamma7.svg @@ -1,2 +1,2 @@ -∂∂xcos(kx)sinn(x)=−12(k+n)cos((k−1)x)+(n−k)cos((k+1)x)sinn+1(x) \ No newline at end of file +∂∂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/html/index.html b/doc/html/index.html index 2017c71c9..64be12973 100644 --- a/doc/html/index.html +++ b/doc/html/index.html @@ -111,7 +111,7 @@ This manual is also available in -

Last revised: November 14, 2014 at 17:06:36 GMT

+

Last revised: November 20, 2014 at 10:35:49 GMT

diff --git a/doc/html/indexes/s01.html b/doc/html/indexes/s01.html index d753f84c8..7ed2007d0 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

diff --git a/doc/html/indexes/s02.html b/doc/html/indexes/s02.html index 153e14e32..08a77ab5a 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 64efd0370..7bdc0e8cd 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 d212d8c6d..bc332ea5b 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 e83adeeb2..005563e7f 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

diff --git a/doc/html/math_toolkit/conventions.html b/doc/html/math_toolkit/conventions.html index 985f76e85..36b191315 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 01e2bc85c..4b912f43f 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/polygamma.html b/doc/html/math_toolkit/sf_gamma/polygamma.html index e16218e27..a5528f5ca 100644 --- a/doc/html/math_toolkit/sf_gamma/polygamma.html +++ b/doc/html/math_toolkit/sf_gamma/polygamma.html @@ -181,13 +181,9 @@

As shown above, error rates are generally very acceptable for moderately - sized arguments. Error rates generally increase with increasing n - - this is particularly true for negative x. Indeed by - the time n=170 the errors are so bad we can no longer - even tell the sign of the result at double - precision. It should also be noted that for large n - the function becomes increasingly badly behaved when x - is negative and is very sensitive to slight changes in input. + 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.

For these reasons results should be treated with extreme @@ -227,6 +223,10 @@

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: diff --git a/doc/sf/polygamma.qbk b/doc/sf/polygamma.qbk index 699a45972..1f44e02d2 100644 --- a/doc/sf/polygamma.qbk +++ b/doc/sf/polygamma.qbk @@ -48,11 +48,8 @@ than the one shown will have __zero_error. ] As shown above, error rates are generally very acceptable for moderately sized -arguments. Error rates generally increase with increasing /n/ - this is particularly true -for negative /x/. Indeed by the time -/n=170/ the errors are so bad we can no longer even tell the sign of the result at `double` precision. -It should also be noted that for large /n/ the function becomes increasingly badly behaved -when /x/ is negative and is very sensitive to slight changes in input. +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. [*For these reasons results should be treated with extreme caution when /n/ is large and x negative]. @@ -78,6 +75,9 @@ from ['C[sub 1,0] = -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: