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[Polygamma] Add equations and graphs.

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jzmaddock
2014-11-08 19:08:00 +00:00
parent 1beedd6487
commit 8147bb92d8
40 changed files with 623 additions and 55 deletions
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<?xml version='1.0'?>
<!DOCTYPE html PUBLIC '-//W3C//DTD XHTML 1.1 plus MathML 2.0//EN'
'http://www.w3.org/TR/MathML2/dtd/xhtml-math11-f.dtd'
[<!ENTITY mathml 'http://www.w3.org/1998/Math/MathML'>]>
<html xmlns='http://www.w3.org/1999/xhtml'>
<head><title>polygamma7</title>
<!-- MathML created with MathCast Equation Editor version 0.89 -->
</head>
<body>
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mfrac>
<mo>&#x2202;</mo>
<mrow>
<mo>&#x2202;</mo>
<mi>x</mi>
</mrow>
</mfrac>
<mfrac>
<mrow>
<mi>cos</mi>
<mfenced>
<mrow>
<mi>k</mi>
<mi>x</mi>
</mrow>
</mfenced>
</mrow>
<mrow>
<msup>
<mi>sin</mi>
<mi>n</mi>
</msup>
<mfenced>
<mrow>
<mi>x</mi>
</mrow>
</mfenced>
</mrow>
</mfrac>
<mo>=</mo>
<mo>&#x2212;</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mfrac>
<mrow>
<mfenced>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mi>n</mi>
</mrow>
</mfenced>
<mi>cos</mi>
<mfenced>
<mrow>
<mfenced>
<mrow>
<mi>k</mi>
<mo>&#x2212;</mo>
<mn>1</mn>
</mrow>
</mfenced>
<mi>x</mi>
</mrow>
</mfenced>
<mo>+</mo>
<mfenced>
<mrow>
<mi>n</mi>
<mo>&#x2212;</mo>
<mi>k</mi>
</mrow>
</mfenced>
<mi>cos</mi>
<mfenced>
<mrow>
<mfenced>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfenced>
<mi>x</mi>
</mrow>
</mfenced>
</mrow>
<mrow>
<msup>
<mi>sin</mi>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mfenced>
<mrow>
<mi>x</mi>
</mrow>
</mfenced>
</mrow>
</mfrac>
</mrow>
</math>
</body>
</html>

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</g>
<g id="plotPoints" clip-path="url(#plot_window)"></g>
<g id="title">
<text x="300" y="40" text-anchor="middle" font-size="20" font-family="Verdana">Polygamma</text></g>
<g id="plotXValues"></g>
<g id="plotYValues"></g>
</svg>
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98
doc/graphs/sf_graphs.cpp
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@@ -220,10 +220,10 @@ double find_end_point(F f, double x0, double target, bool rising, double x_off =
{
boost::math::tools::eps_tolerance<double> 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>(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;
#if 0
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;
@@ -401,7 +402,7 @@ int main()
max_val = f(4);
plot.clear();
plot.add(f, find_end_point(f, 0.1, -max_val, true), 4, "");
plot.add(f, -3, find_end_point(f, -0.1, -max_val, false), "");
plot.add(f, find_end_point(f, -0.1, -max_val, false), "");
plot.plot("Exponential Integral Ei", "expint_i.svg", "z", "expint(z)");
f2u = boost::math::expint;
@@ -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<double>() / 2),
find_end_point(
boost::bind(f3, _1, 0.25, boost::math::constants::pi<double>() / 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<double>() / 2),
-0.5, 4, true, 1) + 1, "n=0.25 &#x3C6;=&#x3C0;/2");
-0.5, 4, true, 1), "n=0.25 &#x3C6;=&#x3C0;/2");
plot.add(boost::bind(f3, _1, 0.75, boost::math::constants::pi<double>() / 2),
find_end_point(
boost::bind(f3, _1, 0.75, boost::math::constants::pi<double>() / 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<double>() / 2),
-0.5, 4, true, 1) + 1, "n=0.75 &#x3C6;=&#x3C0;/2");
-0.5, 4, true, 1), "n=0.75 &#x3C6;=&#x3C0;/2");
plot.plot("Elliptic Of the Third Kind", "ellint_3.svg", "k", "ellint_3(k, n, phi)");
f2 = boost::math::jacobi_sn;
@@ -695,7 +696,40 @@ int main()
plot.clear();
plot.add(f, -20, 3, "");
plot.plot("Bi'", "airy_bip.svg", "z", "airy_bi_prime(z)");
#endif
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;
}

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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"><p><small>Last revised: November 08, 2014 at 11:40:24 GMT</small></p></td>
<td align="left"><p><small>Last revised: November 08, 2014 at 18:57:03 GMT</small></p></td>
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<a name="idm1504580752"></a>Function Index</h2></div></div></div>
<a name="idm1504600816"></a>Function Index</h2></div></div></div>
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</div>
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<a name="idm1502735136"></a>Class Index</h2></div></div></div>
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<p><a class="link" href="s02.html#idx_id_25">B</a> <a class="link" href="s02.html#idx_id_26">C</a> <a class="link" href="s02.html#idx_id_27">D</a> <a class="link" href="s02.html#idx_id_28">E</a> <a class="link" href="s02.html#idx_id_29">F</a> <a class="link" href="s02.html#idx_id_30">G</a> <a class="link" href="s02.html#idx_id_31">H</a> <a class="link" href="s02.html#idx_id_32">I</a> <a class="link" href="s02.html#idx_id_35">L</a> <a class="link" href="s02.html#idx_id_36">M</a> <a class="link" href="s02.html#idx_id_37">N</a> <a class="link" href="s02.html#idx_id_38">O</a> <a class="link" href="s02.html#idx_id_39">P</a> <a class="link" href="s02.html#idx_id_40">Q</a> <a class="link" href="s02.html#idx_id_41">R</a> <a class="link" href="s02.html#idx_id_42">S</a> <a class="link" href="s02.html#idx_id_43">T</a> <a class="link" href="s02.html#idx_id_44">U</a> <a class="link" href="s02.html#idx_id_46">W</a></p>
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</div>
<div class="section">
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="idm1502620064"></a>Typedef Index</h2></div></div></div>
<a name="idm1502671520"></a>Typedef Index</h2></div></div></div>
<p><a class="link" href="s03.html#idx_id_48">A</a> <a class="link" href="s03.html#idx_id_49">B</a> <a class="link" href="s03.html#idx_id_50">C</a> <a class="link" href="s03.html#idx_id_51">D</a> <a class="link" href="s03.html#idx_id_52">E</a> <a class="link" href="s03.html#idx_id_53">F</a> <a class="link" href="s03.html#idx_id_54">G</a> <a class="link" href="s03.html#idx_id_55">H</a> <a class="link" href="s03.html#idx_id_56">I</a> <a class="link" href="s03.html#idx_id_59">L</a> <a class="link" href="s03.html#idx_id_61">N</a> <a class="link" href="s03.html#idx_id_62">O</a> <a class="link" href="s03.html#idx_id_63">P</a> <a class="link" href="s03.html#idx_id_65">R</a> <a class="link" href="s03.html#idx_id_66">S</a> <a class="link" href="s03.html#idx_id_67">T</a> <a class="link" href="s03.html#idx_id_68">U</a> <a class="link" href="s03.html#idx_id_69">V</a> <a class="link" href="s03.html#idx_id_70">W</a></p>
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</div>
<div class="section">
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="idm1502409248"></a>Macro Index</h2></div></div></div>
<a name="idm1502395040"></a>Macro Index</h2></div></div></div>
<p><a class="link" href="s04.html#idx_id_73">B</a> <a class="link" href="s04.html#idx_id_77">F</a></p>
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</div>
<div class="section">
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="idm1502002336"></a>Index</h2></div></div></div>
<a name="idm1502250400"></a>Index</h2></div></div></div>
<p><a class="link" href="s05.html#idx_id_96">A</a> <a class="link" href="s05.html#idx_id_97">B</a> <a class="link" href="s05.html#idx_id_98">C</a> <a class="link" href="s05.html#idx_id_99">D</a> <a class="link" href="s05.html#idx_id_100">E</a> <a class="link" href="s05.html#idx_id_101">F</a> <a class="link" href="s05.html#idx_id_102">G</a> <a class="link" href="s05.html#idx_id_103">H</a> <a class="link" href="s05.html#idx_id_104">I</a> <a class="link" href="s05.html#idx_id_105">J</a> <a class="link" href="s05.html#idx_id_106">K</a> <a class="link" href="s05.html#idx_id_107">L</a> <a class="link" href="s05.html#idx_id_108">M</a> <a class="link" href="s05.html#idx_id_109">N</a> <a class="link" href="s05.html#idx_id_110">O</a> <a class="link" href="s05.html#idx_id_111">P</a> <a class="link" href="s05.html#idx_id_112">Q</a> <a class="link" href="s05.html#idx_id_113">R</a> <a class="link" href="s05.html#idx_id_114">S</a> <a class="link" href="s05.html#idx_id_115">T</a> <a class="link" href="s05.html#idx_id_116">U</a> <a class="link" href="s05.html#idx_id_117">V</a> <a class="link" href="s05.html#idx_id_118">W</a> <a class="link" href="s05.html#idx_id_119">Z</a></p>
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<a name="math_toolkit.conventions"></a><a class="link" href="conventions.html" title="Document Conventions">Document Conventions</a>
</h2></div></div></div>
<p>
<a class="indexterm" name="idm1568176912"></a>
<a class="indexterm" name="idm1568337152"></a>
</p>
<p>
This documentation aims to use of the following naming and formatting conventions.

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<a name="math_toolkit.navigation"></a><a class="link" href="navigation.html" title="Navigation">Navigation</a>
</h2></div></div></div>
<p>
<a class="indexterm" name="idm1568190608"></a>
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<p>
Boost.Math documentation is provided in both HTML and PDF formats.

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<span class="inlinemediaobject"><img src="../../../equations/polygamma1.png"></span>
</p>
<p>
<span class="inlinemediaobject"><img src="../../../graphs/polygamma.png" align="middle"></span>
The following graphs illustrate the behaviour of the function for odd and
even order:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../graphs/polygamma2.png" align="middle"></span>
<span class="inlinemediaobject"><img src="../../../graphs/polygamma3.png" align="middle"></span>
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;14.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
@@ -217,7 +222,13 @@
</p>
<p>
The coefficients of the cosine terms can be calculated iteratively starting
from <span class="emphasis"><em>C<sub>1,0</sub> = -1</em></span>: see polygamma.hpp for the full details.
from <span class="emphasis"><em>C<sub>1,0</sub> = -1</em></span> and then using
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/polygamma7.png"></span>
</p>
<p>
to generate coefficients for n+1.
</p>
<p>
Once x is positive then we have two methods available to us, for small x

2
doc/html/math_toolkit/sf_gamma/trigamma.html
View File

@@ -54,7 +54,7 @@
<span class="inlinemediaobject"><img src="../../../equations/trigamma1.png"></span>
</p>
<p>
<span class="inlinemediaobject"><img src="../../../graphs/digamma.png" align="middle"></span>
<span class="inlinemediaobject"><img src="../../../graphs/trigamma.png" align="middle"></span>
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;14.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can

17
doc/html/math_toolkit/zetas/zeta.html
View File

@@ -292,6 +292,23 @@
of: &#949;/R(0). This saves us quite a few digits when dealing with large z, especially
when &#949; is small.
</p>
<p>
Finally, there are some special cases for integer arguments, there are closed
forms for negative or even integers:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/zeta7.png"></span>
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/zeta8.png"></span>
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/zeta9.png"></span>
</p>
<p>
and for positive odd integers we simply cache pre-computed values as these
are of great benefit to some infinite series calculations.
</p>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>

11
doc/sf/polygamma.qbk
View File

@@ -23,7 +23,10 @@ derivative of the digamma function:
[equation polygamma1]
[graph polygamma]
The following graphs illustrate the behaviour of the function for odd and even order:
[graph polygamma2]
[graph polygamma3]
[optional_policy]
@@ -70,7 +73,11 @@ has the general form:
[equation polygamma3]
The coefficients of the cosine terms can be calculated iteratively starting
from ['C[sub 1,0] = -1]: see polygamma.hpp for the full details.
from ['C[sub 1,0] = -1] and then using
[equation polygamma7]
to generate coefficients for n+1.
Once x is positive then we have two methods available to us, for small x
we use the series expansion:

2
doc/sf/trigamma.qbk
View File

@@ -23,7 +23,7 @@ derivative of the digamma function:
[equation trigamma1]
[graph digamma]
[graph trigamma]
[optional_policy]

12
doc/sf/zeta.qbk
View File

@@ -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]