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Diffstat (limited to 'libs/math/doc/html/math_toolkit/sf_gamma/digamma.html')
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1 files changed, 20 insertions, 14 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> |