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| author | Lorry Tar Creator <lorry-tar-importer@baserock.org> | 2015-04-08 03:09:47 +0000 |
|---|---|---|
| committer | <> | 2015-05-05 14:37:32 +0000 |
| commit | f2541bb90af059680aa7036f315f052175999355 (patch) | |
| tree | a5b214744b256f07e1dc2bd7273035a7808c659f /libs/math/doc/html/math_toolkit/sf_gamma | |
| parent | ed232fdd34968697a68783b3195b1da4226915b5 (diff) | |
| download | boost-tarball-master.tar.gz | |
Imported from /home/lorry/working-area/delta_boost-tarball/boost_1_58_0.tar.bz2.HEADboost_1_58_0master
Diffstat (limited to 'libs/math/doc/html/math_toolkit/sf_gamma')
9 files changed, 575 insertions, 67 deletions
diff --git a/libs/math/doc/html/math_toolkit/sf_gamma/digamma.html b/libs/math/doc/html/math_toolkit/sf_gamma/digamma.html index 58fe9c37f..cf915ee8b 100644 --- a/libs/math/doc/html/math_toolkit/sf_gamma/digamma.html +++ b/libs/math/doc/html/math_toolkit/sf_gamma/digamma.html @@ -3,11 +3,11 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Digamma</title> <link rel="stylesheet" href="../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../sf_gamma.html" title="Gamma Functions"> <link rel="prev" href="lgamma.html" title="Log Gamma"> -<link rel="next" href="gamma_ratios.html" title="Ratios of Gamma Functions"> +<link rel="next" href="trigamma.html" title="Trigamma"> </head> <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> <table cellpadding="2" width="100%"><tr> @@ -20,7 +20,7 @@ </tr></table> <hr> <div class="spirit-nav"> -<a accesskey="p" href="lgamma.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="gamma_ratios.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> +<a accesskey="p" href="lgamma.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="trigamma.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> </div> <div class="section"> <div class="titlepage"><div><div><h3 class="title"> @@ -51,10 +51,10 @@ defined as the logarithmic derivative of the gamma function: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/digamma1.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/digamma1.svg"></span> </p> <p> - <span class="inlinemediaobject"><img src="../../../graphs/digamma.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../graphs/digamma.svg" align="middle"></span> </p> <p> The final <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can @@ -63,11 +63,6 @@ documentation for more details</a>. </p> <p> - 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. - </p> -<p> The return type of this function is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result type calculation rules</em></span></a>: the result is of type <code class="computeroutput"><span class="keyword">double</span></code> when T is an integer type, and type T otherwise. @@ -308,7 +303,7 @@ For arguments > BIG the asymptotic expansion: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/digamma2.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/digamma2.svg"></span> </p> <p> can be used. However, this expansion is divergent after a few terms: exactly @@ -320,6 +315,17 @@ small number of terms and evaluated as a polynomial in <code class="computeroutput"><span class="number">1</span><span class="special">/(</span><span class="identifier">x</span><span class="special">*</span><span class="identifier">x</span><span class="special">)</span></code>. </p> <p> + 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: + </p> +<p> + <span class="inlinemediaobject"><img src="../../../equations/digamma4.svg"></span> + </p> +<p> + <span class="inlinemediaobject"><img src="../../../equations/digamma5.svg"></span> + </p> +<p> The rational approximation <a class="link" href="../sf_implementation.html#math_toolkit.sf_implementation.rational_approximations_used">devised by JM</a> in the range [1,2] is derived as follows. </p> @@ -329,7 +335,7 @@ the form used is: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/digamma3.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/digamma3.svg"></span> </p> <p> Where P(x) and Q(x) are the polynomials from the rational form of the Lanczos @@ -379,7 +385,7 @@ </tr></table> <hr> <div class="spirit-nav"> -<a accesskey="p" href="lgamma.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="gamma_ratios.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> +<a accesskey="p" href="lgamma.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="trigamma.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> </div> </body> </html> diff --git a/libs/math/doc/html/math_toolkit/sf_gamma/gamma_derivatives.html b/libs/math/doc/html/math_toolkit/sf_gamma/gamma_derivatives.html index 8b1b5f36f..d24f43447 100644 --- a/libs/math/doc/html/math_toolkit/sf_gamma/gamma_derivatives.html +++ b/libs/math/doc/html/math_toolkit/sf_gamma/gamma_derivatives.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Derivative of the Incomplete Gamma Function</title> <link rel="stylesheet" href="../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../sf_gamma.html" title="Gamma Functions"> <link rel="prev" href="igamma_inv.html" title="Incomplete Gamma Function Inverses"> <link rel="next" href="../factorials.html" title="Factorials and Binomial Coefficients"> @@ -53,7 +53,7 @@ gamma function. </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/derivative1.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/derivative1.svg"></span> </p> <p> The final <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can diff --git a/libs/math/doc/html/math_toolkit/sf_gamma/gamma_ratios.html b/libs/math/doc/html/math_toolkit/sf_gamma/gamma_ratios.html index b2a44d905..2bb703977 100644 --- a/libs/math/doc/html/math_toolkit/sf_gamma/gamma_ratios.html +++ b/libs/math/doc/html/math_toolkit/sf_gamma/gamma_ratios.html @@ -3,10 +3,10 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Ratios of Gamma Functions</title> <link rel="stylesheet" href="../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../sf_gamma.html" title="Gamma Functions"> -<link rel="prev" href="digamma.html" title="Digamma"> +<link rel="prev" href="polygamma.html" title="Polygamma"> <link rel="next" href="igamma.html" title="Incomplete Gamma Functions"> </head> <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> @@ -20,7 +20,7 @@ </tr></table> <hr> <div class="spirit-nav"> -<a accesskey="p" href="digamma.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="igamma.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> +<a accesskey="p" href="polygamma.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="igamma.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> </div> <div class="section"> <div class="titlepage"><div><div><h3 class="title"> @@ -58,7 +58,7 @@ Returns the ratio of gamma functions: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/gamma_ratio0.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/gamma_ratio0.svg"></span> </p> <p> The final <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can @@ -80,7 +80,7 @@ Returns the ratio of gamma functions: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/gamma_ratio1.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/gamma_ratio1.svg"></span> </p> <p> The final <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can @@ -100,7 +100,7 @@ otherwise the result type is simple T1. </p> <p> - <span class="inlinemediaobject"><img src="../../../graphs/tgamma_delta_ratio.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../graphs/tgamma_delta_ratio.svg" align="middle"></span> </p> <h5> <a name="math_toolkit.sf_gamma.gamma_ratios.h1"></a> @@ -349,7 +349,7 @@ </tr></table> <hr> <div class="spirit-nav"> -<a accesskey="p" href="digamma.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="igamma.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> +<a accesskey="p" href="polygamma.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="igamma.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> </div> </body> </html> diff --git a/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html b/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html index 095c79798..c7118eaf0 100644 --- a/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html +++ b/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Incomplete Gamma Functions</title> <link rel="stylesheet" href="../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../sf_gamma.html" title="Gamma Functions"> <link rel="prev" href="gamma_ratios.html" title="Ratios of Gamma Functions"> <link rel="next" href="igamma_inv.html" title="Incomplete Gamma Function Inverses"> @@ -97,13 +97,13 @@ Returns the normalised lower incomplete gamma function of a and z: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/igamma4.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/igamma4.svg"></span> </p> <p> This function changes rapidly from 0 to 1 around the point z == a: </p> <p> - <span class="inlinemediaobject"><img src="../../../graphs/gamma_p.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../graphs/gamma_p.svg" align="middle"></span> </p> <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">></span> <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_q</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">z</span><span class="special">);</span> @@ -115,13 +115,13 @@ Returns the normalised upper incomplete gamma function of a and z: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/igamma3.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/igamma3.svg"></span> </p> <p> This function changes rapidly from 1 to 0 around the point z == a: </p> <p> - <span class="inlinemediaobject"><img src="../../../graphs/gamma_q.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../graphs/gamma_q.svg" align="middle"></span> </p> <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">></span> <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_lower</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">z</span><span class="special">);</span> @@ -134,7 +134,7 @@ z: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/igamma2.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/igamma2.svg"></span> </p> <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">></span> <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">z</span><span class="special">);</span> @@ -147,7 +147,7 @@ z: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/igamma1.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/igamma1.svg"></span> </p> <h5> <a name="math_toolkit.sf_gamma.igamma.h2"></a> @@ -842,19 +842,19 @@ via: </p> <p> - 1) <span class="inlinemediaobject"><img src="../../../equations/igamma5.png"></span> + 1) <span class="inlinemediaobject"><img src="../../../equations/igamma5.svg"></span> </p> <p> - 2) <span class="inlinemediaobject"><img src="../../../equations/igamma6.png"></span> + 2) <span class="inlinemediaobject"><img src="../../../equations/igamma6.svg"></span> </p> <p> - 3) <span class="inlinemediaobject"><img src="../../../equations/igamma7.png"></span> + 3) <span class="inlinemediaobject"><img src="../../../equations/igamma7.svg"></span> </p> <p> The lower incomplete gamma is computed from its series representation: </p> <p> - 4) <span class="inlinemediaobject"><img src="../../../equations/igamma8.png"></span> + 4) <span class="inlinemediaobject"><img src="../../../equations/igamma8.svg"></span> </p> <p> Or by subtraction of the upper integral from either Γ(a) or 1 when <span class="emphasis"><em>x @@ -864,7 +864,7 @@ The upper integral is computed from Legendre's continued fraction representation: </p> <p> - 5) <span class="inlinemediaobject"><img src="../../../equations/igamma9.png"></span> + 5) <span class="inlinemediaobject"><img src="../../../equations/igamma9.svg"></span> </p> <p> When <span class="emphasis"><em>(x > 1.1)</em></span> or by subtraction of the lower integral @@ -876,14 +876,14 @@ area. However there is another series representation for the lower integral: </p> <p> - 6) <span class="inlinemediaobject"><img src="../../../equations/igamma10.png"></span> + 6) <span class="inlinemediaobject"><img src="../../../equations/igamma10.svg"></span> </p> <p> That lends itself to calculation of the upper integral via rearrangement to: </p> <p> - 7) <span class="inlinemediaobject"><img src="../../../equations/igamma11.png"></span> + 7) <span class="inlinemediaobject"><img src="../../../equations/igamma11.svg"></span> </p> <p> Refer to the documentation for <a class="link" href="../powers/powm1.html" title="powm1">powm1</a> @@ -911,14 +911,14 @@ 30</em></span> then the following finite sum is used: </p> <p> - 9) <span class="inlinemediaobject"><img src="../../../equations/igamma1f.png"></span> + 9) <span class="inlinemediaobject"><img src="../../../equations/igamma1f.svg"></span> </p> <p> While for half integers in the range <span class="emphasis"><em>0.5 <= a < 30</em></span> then the following finite sum is used: </p> <p> - 10) <span class="inlinemediaobject"><img src="../../../equations/igamma2f.png"></span> + 10) <span class="inlinemediaobject"><img src="../../../equations/igamma2f.svg"></span> </p> <p> These are both more stable and more efficient than the continued fraction @@ -930,16 +930,16 @@ In this area an expansion due to Temme is used: </p> <p> - 11) <span class="inlinemediaobject"><img src="../../../equations/igamma16.png"></span> + 11) <span class="inlinemediaobject"><img src="../../../equations/igamma16.svg"></span> </p> <p> - 12) <span class="inlinemediaobject"><img src="../../../equations/igamma17.png"></span> + 12) <span class="inlinemediaobject"><img src="../../../equations/igamma17.svg"></span> </p> <p> - 13) <span class="inlinemediaobject"><img src="../../../equations/igamma18.png"></span> + 13) <span class="inlinemediaobject"><img src="../../../equations/igamma18.svg"></span> </p> <p> - 14) <span class="inlinemediaobject"><img src="../../../equations/igamma19.png"></span> + 14) <span class="inlinemediaobject"><img src="../../../equations/igamma19.svg"></span> </p> <p> The double sum is truncated to a fixed number of terms - to give a specific @@ -974,7 +974,7 @@ approximation</a> gives the greatest accuracy: </p> <p> - 15) <span class="inlinemediaobject"><img src="../../../equations/igamma12.png"></span> + 15) <span class="inlinemediaobject"><img src="../../../equations/igamma12.svg"></span> </p> <p> In the event that this causes underflow/overflow then the exponent can be @@ -988,7 +988,7 @@ can be avoided by using: </p> <p> - 16) <span class="inlinemediaobject"><img src="../../../equations/igamma13.png"></span> + 16) <span class="inlinemediaobject"><img src="../../../equations/igamma13.svg"></span> </p> <p> when <span class="emphasis"><em>a-x</em></span> is small and a and x are large. There is still diff --git a/libs/math/doc/html/math_toolkit/sf_gamma/igamma_inv.html b/libs/math/doc/html/math_toolkit/sf_gamma/igamma_inv.html index c315b98f2..6d9fa71bc 100644 --- a/libs/math/doc/html/math_toolkit/sf_gamma/igamma_inv.html +++ b/libs/math/doc/html/math_toolkit/sf_gamma/igamma_inv.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Incomplete Gamma Function Inverses</title> <link rel="stylesheet" href="../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../sf_gamma.html" title="Gamma Functions"> <link rel="prev" href="igamma.html" title="Incomplete Gamma Functions"> <link rel="next" href="gamma_derivatives.html" title="Derivative of the Incomplete Gamma Function"> diff --git a/libs/math/doc/html/math_toolkit/sf_gamma/lgamma.html b/libs/math/doc/html/math_toolkit/sf_gamma/lgamma.html index a8df176c2..5be633960 100644 --- a/libs/math/doc/html/math_toolkit/sf_gamma/lgamma.html +++ b/libs/math/doc/html/math_toolkit/sf_gamma/lgamma.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Log Gamma</title> <link rel="stylesheet" href="../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../sf_gamma.html" title="Gamma Functions"> <link rel="prev" href="tgamma.html" title="Gamma"> <link rel="next" href="digamma.html" title="Digamma"> @@ -57,7 +57,7 @@ is defined by: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/lgamm1.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/lgamm1.svg"></span> </p> <p> The second form of the function takes a pointer to an integer, which if non-null @@ -70,7 +70,7 @@ documentation for more details</a>. </p> <p> - <span class="inlinemediaobject"><img src="../../../graphs/lgamma.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../graphs/lgamma.svg" align="middle"></span> </p> <p> There are effectively two versions of this function internally: a fully generic @@ -359,7 +359,7 @@ for large arguments: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/gamma6.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/gamma6.svg"></span> </p> <p> For small arguments, the logarithm of tgamma is used. @@ -369,7 +369,7 @@ formula is used: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/lgamm3.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/lgamm3.svg"></span> </p> <p> For types of known precision, the <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos @@ -378,7 +378,7 @@ <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos approximation</a> is: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/lgamm4.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/lgamm4.svg"></span> </p> <p> Where L<sub>e,g</sub>   is the Lanczos sum, scaled by e<sup>g</sup>. @@ -447,7 +447,7 @@ -> 1</em></span>: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/lgamm5.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/lgamm5.svg"></span> </p> <p> The C<sub>k</sub>   terms in the above are the same as in the <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos @@ -457,7 +457,7 @@ A similar rearrangement can be performed at <span class="emphasis"><em>z = 2</em></span>: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/lgamm6.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/lgamm6.svg"></span> </p> </div> <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> diff --git a/libs/math/doc/html/math_toolkit/sf_gamma/polygamma.html b/libs/math/doc/html/math_toolkit/sf_gamma/polygamma.html new file mode 100644 index 000000000..de3526403 --- /dev/null +++ b/libs/math/doc/html/math_toolkit/sf_gamma/polygamma.html @@ -0,0 +1,285 @@ +<html> +<head> +<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> +<title>Polygamma</title> +<link rel="stylesheet" href="../../math.css" type="text/css"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../index.html" title="Math Toolkit 2.2.0"> +<link rel="up" href="../sf_gamma.html" title="Gamma Functions"> +<link rel="prev" href="trigamma.html" title="Trigamma"> +<link rel="next" href="gamma_ratios.html" title="Ratios of Gamma Functions"> +</head> +<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> +<table cellpadding="2" width="100%"><tr> +<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> +<td align="center"><a href="../../../../../../index.html">Home</a></td> +<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td> +<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> +<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> +<td align="center"><a href="../../../../../../more/index.htm">More</a></td> +</tr></table> +<hr> +<div class="spirit-nav"> +<a accesskey="p" href="trigamma.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="gamma_ratios.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> +</div> +<div class="section"> +<div class="titlepage"><div><div><h3 class="title"> +<a name="math_toolkit.sf_gamma.polygamma"></a><a class="link" href="polygamma.html" title="Polygamma">Polygamma</a> +</h3></div></div></div> +<h5> +<a name="math_toolkit.sf_gamma.polygamma.h0"></a> + <span class="phrase"><a name="math_toolkit.sf_gamma.polygamma.synopsis"></a></span><a class="link" href="polygamma.html#math_toolkit.sf_gamma.polygamma.synopsis">Synopsis</a> + </h5> +<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">polygamma</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> +</pre> +<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> + +<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> +<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">polygamma</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span> + +<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> +<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">polygamma</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> + +<span class="special">}}</span> <span class="comment">// namespaces</span> +</pre> +<h5> +<a name="math_toolkit.sf_gamma.polygamma.h1"></a> + <span class="phrase"><a name="math_toolkit.sf_gamma.polygamma.description"></a></span><a class="link" href="polygamma.html#math_toolkit.sf_gamma.polygamma.description">Description</a> + </h5> +<p> + Returns the polygamma function of <span class="emphasis"><em>x</em></span>. Polygamma is defined + as the n'th derivative of the digamma function: + </p> +<p> + <span class="inlinemediaobject"><img src="../../../equations/polygamma1.svg"></span> + </p> +<p> + The following graphs illustrate the behaviour of the function for odd and + even order: + </p> +<p> + <span class="inlinemediaobject"><img src="../../../graphs/polygamma2.svg" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../graphs/polygamma3.svg" align="middle"></span> + </p> +<p> + The final <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a> 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 <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">policy + documentation for more details</a>. + </p> +<p> + The return type of this function is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result + type calculation rules</em></span></a>: the result is of type <code class="computeroutput"><span class="keyword">double</span></code> when T is an integer type, and type + T otherwise. + </p> +<h5> +<a name="math_toolkit.sf_gamma.polygamma.h2"></a> + <span class="phrase"><a name="math_toolkit.sf_gamma.polygamma.accuracy"></a></span><a class="link" href="polygamma.html#math_toolkit.sf_gamma.polygamma.accuracy">Accuracy</a> + </h5> +<p> + 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 <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively zero error</a>. + </p> +<div class="informaltable"><table class="table"> +<colgroup> +<col> +<col> +<col> +<col> +</colgroup> +<thead><tr> +<th> + <p> + Significand Size + </p> + </th> +<th> + <p> + Platform and Compiler + </p> + </th> +<th> + <p> + Small-medium positive arguments + </p> + </th> +<th> + <p> + Small-medium negative x + </p> + </th> +</tr></thead> +<tbody> +<tr> +<td> + <p> + 53 + </p> + </td> +<td> + <p> + Win32 Visual C++ 12 + </p> + </td> +<td> + <p> + Peak=5.0 Mean=1 + </p> + </td> +<td> + <p> + Peak=1200 Mean=65 + </p> + </td> +</tr> +<tr> +<td> + <p> + 64 + </p> + </td> +<td> + <p> + Win64 Mingw GCC + </p> + </td> +<td> + <p> + Peak=16 Mean=3 + </p> + </td> +<td> + <p> + Peak=33 Mean=3 + </p> + </td> +</tr> +<tr> +<td> + <p> + 113 + </p> + </td> +<td> + <p> + Win64 Mingw GCC __float128 + </p> + </td> +<td> + <p> + Peak=6.5 Mean=1 + </p> + </td> +<td> + <p> + Peak=30 Mean=4 + </p> + </td> +</tr> +</tbody> +</table></div> +<p> + 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 <span class="emphasis"><em>n!</em></span> + would overflow, the function changes directly from -∞ to +∞ somewhere between + each negative integer - <span class="emphasis"><em>these cases are not handled correctly</em></span>. + </p> +<p> + <span class="bold"><strong>For these reasons results should be treated with extreme + caution when <span class="emphasis"><em>n</em></span> is large and x negative</strong></span>. + </p> +<h5> +<a name="math_toolkit.sf_gamma.polygamma.h3"></a> + <span class="phrase"><a name="math_toolkit.sf_gamma.polygamma.testing"></a></span><a class="link" href="polygamma.html#math_toolkit.sf_gamma.polygamma.testing">Testing</a> + </h5> +<p> + Testing is against Mathematica generated spot values to 35 digit precision. + </p> +<h5> +<a name="math_toolkit.sf_gamma.polygamma.h4"></a> + <span class="phrase"><a name="math_toolkit.sf_gamma.polygamma.implementation"></a></span><a class="link" href="polygamma.html#math_toolkit.sf_gamma.polygamma.implementation">Implementation</a> + </h5> +<p> + For x < 0 the following reflection formula is used: + </p> +<p> + <span class="inlinemediaobject"><img src="../../../equations/polygamma2.svg"></span> + </p> +<p> + The n'th derivative of <span class="emphasis"><em>cot(x)</em></span> is tabulated for small + <span class="emphasis"><em>n</em></span>, and for larger n has the general form: + </p> +<p> + <span class="inlinemediaobject"><img src="../../../equations/polygamma3.svg"></span> + </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> and then using + </p> +<p> + <span class="inlinemediaobject"><img src="../../../equations/polygamma7.svg"></span> + </p> +<p> + to generate coefficients for n+1. + </p> +<p> + 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. + </p> +<p> + Once x is positive then we have two methods available to us, for small x + we use the series expansion: + </p> +<p> + <span class="inlinemediaobject"><img src="../../../equations/polygamma4.svg"></span> + </p> +<p> + Note that the evaluation of zeta functions at integer values is essentially + a table lookup as <a class="link" href="../zetas/zeta.html" title="Riemann Zeta Function">zeta</a> is + optimized for those cases. + </p> +<p> + For large x we use the asymptotic expansion: + </p> +<p> + <span class="inlinemediaobject"><img src="../../../equations/polygamma5.svg"></span> + </p> +<p> + For x in-between the two extremes we use the relation: + </p> +<p> + <span class="inlinemediaobject"><img src="../../../equations/polygamma6.svg"></span> + </p> +<p> + to make x large enough for the asymptotic expansion to be used. + </p> +<p> + There are also two special cases: + </p> +<p> + <span class="inlinemediaobject"><img src="../../../equations/polygamma8.svg"></span> + </p> +<p> + <span class="inlinemediaobject"><img src="../../../equations/polygamma9.svg"></span> + </p> +</div> +<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> +<td align="left"></td> +<td align="right"><div class="copyright-footer">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<p> + Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) + </p> +</div></td> +</tr></table> +<hr> +<div class="spirit-nav"> +<a accesskey="p" href="trigamma.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="gamma_ratios.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> +</div> +</body> +</html> diff --git a/libs/math/doc/html/math_toolkit/sf_gamma/tgamma.html b/libs/math/doc/html/math_toolkit/sf_gamma/tgamma.html index 937d4e55b..5f2e09031 100644 --- a/libs/math/doc/html/math_toolkit/sf_gamma/tgamma.html +++ b/libs/math/doc/html/math_toolkit/sf_gamma/tgamma.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Gamma</title> <link rel="stylesheet" href="../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../sf_gamma.html" title="Gamma Functions"> <link rel="prev" href="../sf_gamma.html" title="Gamma Functions"> <link rel="next" href="lgamma.html" title="Log Gamma"> @@ -62,10 +62,10 @@ Returns the "true gamma" (hence name tgamma) of value z: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/gamm1.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/gamm1.svg"></span> </p> <p> - <span class="inlinemediaobject"><img src="../../../graphs/tgamma.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../graphs/tgamma.svg" align="middle"></span> </p> <p> The final <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can @@ -393,7 +393,7 @@ function is implemented Sterling's approximation for lgamma for large z: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/gamma6.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/gamma6.svg"></span> </p> <p> Following exponentiation, downward recursion is then used for small values @@ -409,19 +409,19 @@ > 1 via: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/gamm3.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/gamm3.svg"></span> </p> <p> For very small z, this helps to preserve the identity: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/gamm4.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/gamm4.svg"></span> </p> <p> For z < -20 the reflection formula: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/gamm5.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/gamm5.svg"></span> </p> <p> is used. Particular care has to be taken to evaluate the <code class="literal">z * sin(π   * diff --git a/libs/math/doc/html/math_toolkit/sf_gamma/trigamma.html b/libs/math/doc/html/math_toolkit/sf_gamma/trigamma.html new file mode 100644 index 000000000..b7d758b85 --- /dev/null +++ b/libs/math/doc/html/math_toolkit/sf_gamma/trigamma.html @@ -0,0 +1,217 @@ +<html> +<head> +<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> +<title>Trigamma</title> +<link rel="stylesheet" href="../../math.css" type="text/css"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../index.html" title="Math Toolkit 2.2.0"> +<link rel="up" href="../sf_gamma.html" title="Gamma Functions"> +<link rel="prev" href="digamma.html" title="Digamma"> +<link rel="next" href="polygamma.html" title="Polygamma"> +</head> +<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> +<table cellpadding="2" width="100%"><tr> +<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> +<td align="center"><a href="../../../../../../index.html">Home</a></td> +<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td> +<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> +<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> +<td align="center"><a href="../../../../../../more/index.htm">More</a></td> +</tr></table> +<hr> +<div class="spirit-nav"> +<a accesskey="p" href="digamma.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="polygamma.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> +</div> +<div class="section"> +<div class="titlepage"><div><div><h3 class="title"> +<a name="math_toolkit.sf_gamma.trigamma"></a><a class="link" href="trigamma.html" title="Trigamma">Trigamma</a> +</h3></div></div></div> +<h5> +<a name="math_toolkit.sf_gamma.trigamma.h0"></a> + <span class="phrase"><a name="math_toolkit.sf_gamma.trigamma.synopsis"></a></span><a class="link" href="trigamma.html#math_toolkit.sf_gamma.trigamma.synopsis">Synopsis</a> + </h5> +<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">trigamma</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> +</pre> +<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> + +<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> +<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">trigamma</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span> + +<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> +<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">trigamma</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> + +<span class="special">}}</span> <span class="comment">// namespaces</span> +</pre> +<h5> +<a name="math_toolkit.sf_gamma.trigamma.h1"></a> + <span class="phrase"><a name="math_toolkit.sf_gamma.trigamma.description"></a></span><a class="link" href="trigamma.html#math_toolkit.sf_gamma.trigamma.description">Description</a> + </h5> +<p> + Returns the trigamma function of <span class="emphasis"><em>x</em></span>. Trigamma is defined + as the derivative of the digamma function: + </p> +<p> + <span class="inlinemediaobject"><img src="../../../equations/trigamma1.svg"></span> + </p> +<p> + <span class="inlinemediaobject"><img src="../../../graphs/trigamma.svg" align="middle"></span> + </p> +<p> + The final <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a> 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 <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">policy + documentation for more details</a>. + </p> +<p> + The return type of this function is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result + type calculation rules</em></span></a>: the result is of type <code class="computeroutput"><span class="keyword">double</span></code> when T is an integer type, and type + T otherwise. + </p> +<h5> +<a name="math_toolkit.sf_gamma.trigamma.h2"></a> + <span class="phrase"><a name="math_toolkit.sf_gamma.trigamma.accuracy"></a></span><a class="link" href="trigamma.html#math_toolkit.sf_gamma.trigamma.accuracy">Accuracy</a> + </h5> +<p> + 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 <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively zero error</a>. + </p> +<div class="informaltable"><table class="table"> +<colgroup> +<col> +<col> +<col> +</colgroup> +<thead><tr> +<th> + <p> + Significand Size + </p> + </th> +<th> + <p> + Platform and Compiler + </p> + </th> +<th> + <p> + Random Values + </p> + </th> +</tr></thead> +<tbody> +<tr> +<td> + <p> + 53 + </p> + </td> +<td> + <p> + Win32 Visual C++ 12 + </p> + </td> +<td> + <p> + Peak=1.0 Mean=0.4 + </p> + </td> +</tr> +<tr> +<td> + <p> + 64 + </p> + </td> +<td> + <p> + Win64 Mingw GCC + </p> + </td> +<td> + <p> + Peak=1.4 Mean=0.4 + </p> + </td> +</tr> +<tr> +<td> + <p> + 113 + </p> + </td> +<td> + <p> + Win64 Mingw GCC __float128 + </p> + </td> +<td> + <p> + Peak=1.0 Mean=0.5 + </p> + </td> +</tr> +</tbody> +</table></div> +<p> + 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. + </p> +<h5> +<a name="math_toolkit.sf_gamma.trigamma.h3"></a> + <span class="phrase"><a name="math_toolkit.sf_gamma.trigamma.testing"></a></span><a class="link" href="trigamma.html#math_toolkit.sf_gamma.trigamma.testing">Testing</a> + </h5> +<p> + Testing is against Mathematica generated spot values to 35 digit precision. + </p> +<h5> +<a name="math_toolkit.sf_gamma.trigamma.h4"></a> + <span class="phrase"><a name="math_toolkit.sf_gamma.trigamma.implementation"></a></span><a class="link" href="trigamma.html#math_toolkit.sf_gamma.trigamma.implementation">Implementation</a> + </h5> +<p> + The arbitrary precision version of this function simply calls <a class="link" href="polygamma.html" title="Polygamma">polygamma</a>. + </p> +<p> + For built in fixed precision types, negative arguments are first made positive + via: + </p> +<p> + <span class="inlinemediaobject"><img src="../../../equations/trigamma2.svg"></span> + </p> +<p> + Then arguments in the range [0, 1) are shifted to >= 1 via: + </p> +<p> + <span class="inlinemediaobject"><img src="../../../equations/trigamma3.svg"></span> + </p> +<p> + Then evaluation is via one of a number of rational approximations, for small + x these are of the form: + </p> +<p> + <span class="inlinemediaobject"><img src="../../../equations/trigamma4.svg"></span> + </p> +<p> + and for large x of the form: + </p> +<p> + <span class="inlinemediaobject"><img src="../../../equations/trigamma5.svg"></span> + </p> +</div> +<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> +<td align="left"></td> +<td align="right"><div class="copyright-footer">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<p> + Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) + </p> +</div></td> +</tr></table> +<hr> +<div class="spirit-nav"> +<a accesskey="p" href="digamma.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="polygamma.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> +</div> +</body> +</html> |