diff options
Diffstat (limited to 'libs/math/doc/html/math_toolkit/sf_beta')
4 files changed, 40 insertions, 40 deletions
diff --git a/libs/math/doc/html/math_toolkit/sf_beta/beta_derivative.html b/libs/math/doc/html/math_toolkit/sf_beta/beta_derivative.html index 5b32df073..a323a47f4 100644 --- a/libs/math/doc/html/math_toolkit/sf_beta/beta_derivative.html +++ b/libs/math/doc/html/math_toolkit/sf_beta/beta_derivative.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Derivative of the Incomplete Beta 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_beta.html" title="Beta Functions"> <link rel="prev" href="ibeta_inv_function.html" title="The Incomplete Beta Function Inverses"> <link rel="next" href="../sf_erf.html" title="Error Functions"> @@ -53,7 +53,7 @@ beta function <a class="link" href="ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>. </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/derivative2.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/derivative2.svg"></span> </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 diff --git a/libs/math/doc/html/math_toolkit/sf_beta/beta_function.html b/libs/math/doc/html/math_toolkit/sf_beta/beta_function.html index 9087ca33e..e8aedecd3 100644 --- a/libs/math/doc/html/math_toolkit/sf_beta/beta_function.html +++ b/libs/math/doc/html/math_toolkit/sf_beta/beta_function.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Beta</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_beta.html" title="Beta Functions"> <link rel="prev" href="../sf_beta.html" title="Beta Functions"> <link rel="next" href="ibeta_function.html" title="Incomplete Beta Functions"> @@ -50,10 +50,10 @@ The beta function is defined by: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/beta1.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/beta1.svg"></span> </p> <p> - <span class="inlinemediaobject"><img src="../../../graphs/beta.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../graphs/beta.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 @@ -256,7 +256,7 @@ approximation</a>, then we can combine the power terms: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/beta2.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/beta2.svg"></span> </p> <p> which is almost the ideal solution, however almost all of the error occurs @@ -266,7 +266,7 @@ which immediately cuts the maximum error in half: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/beta3.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/beta3.svg"></span> </p> <p> This may not be the final solution, but it is very competitive compared to @@ -280,7 +280,7 @@ fraction parts are collected together into: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/beta8.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/beta8.svg"></span> </p> <p> where la, lb and lc are the integration limits used for a, b, and a+b. @@ -293,10 +293,10 @@ we can use the recurrence relations: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/beta4.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/beta4.svg"></span> </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/beta5.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/beta5.svg"></span> </p> <p> to move to a more favorable region where they are both greater than 1. @@ -305,7 +305,7 @@ In addition: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/beta7.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/beta7.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_beta/ibeta_function.html b/libs/math/doc/html/math_toolkit/sf_beta/ibeta_function.html index a050e4326..20a3340d5 100644 --- a/libs/math/doc/html/math_toolkit/sf_beta/ibeta_function.html +++ b/libs/math/doc/html/math_toolkit/sf_beta/ibeta_function.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Incomplete Beta 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_beta.html" title="Beta Functions"> <link rel="prev" href="beta_function.html" title="Beta"> <link rel="next" href="ibeta_inv_function.html" title="The Incomplete Beta Function Inverses"> @@ -109,10 +109,10 @@ Returns the normalised incomplete beta function of a, b and x: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/ibeta3.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/ibeta3.svg"></span> </p> <p> - <span class="inlinemediaobject"><img src="../../../graphs/ibeta.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../graphs/ibeta.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> <span class="keyword">class</span> <span class="identifier">T3</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">ibetac</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">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">x</span><span class="special">);</span> @@ -125,7 +125,7 @@ and x: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/ibeta4.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/ibeta4.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> <span class="keyword">class</span> <span class="identifier">T3</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">beta</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">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">x</span><span class="special">);</span> @@ -137,7 +137,7 @@ Returns the full (non-normalised) incomplete beta function of a, b and x: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/ibeta1.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/ibeta1.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> <span class="keyword">class</span> <span class="identifier">T3</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">betac</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">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">x</span><span class="special">);</span> @@ -150,7 +150,7 @@ of a, b and x: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/ibeta2.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/ibeta2.svg"></span> </p> <h5> <a name="math_toolkit.sf_beta.ibeta_function.h2"></a> @@ -891,7 +891,7 @@ both x and y, and can return either p or q where these are related by: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv5.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv5.svg"></span> </p> <p> so at any point we can swap a for b, x for y and p for q if this results @@ -905,7 +905,7 @@ than the alternatives - however it is used to generate test data: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/ibeta5.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/ibeta5.svg"></span> </p> <p> The following continued fraction is due to <a href="http://portal.acm.org/citation.cfm?doid=131766.131776" target="_top">Didonato @@ -913,13 +913,13 @@ greater than 1: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/ibeta6.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/ibeta6.svg"></span> </p> <p> For smallish b and x then a series representation can be used: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/ibeta7.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/ibeta7.svg"></span> </p> <p> When b << a then the transition from 0 to 1 occurs very close to x @@ -927,7 +927,7 @@ case the following series representation is used: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/ibeta8.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/ibeta8.svg"></span> </p> <p> @@ -942,17 +942,17 @@ the following finite sum: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/ibeta12.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/ibeta12.svg"></span> </p> <p> Finally we can sidestep difficult areas, or move to an area with a more efficient means of computation, by using the duplication formulae: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/ibeta10.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/ibeta10.svg"></span> </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/ibeta11.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/ibeta11.svg"></span> </p> <p> The domains of a, b and x for which the various methods are used are identical diff --git a/libs/math/doc/html/math_toolkit/sf_beta/ibeta_inv_function.html b/libs/math/doc/html/math_toolkit/sf_beta/ibeta_inv_function.html index f3d78ad4e..f18f6513c 100644 --- a/libs/math/doc/html/math_toolkit/sf_beta/ibeta_inv_function.html +++ b/libs/math/doc/html/math_toolkit/sf_beta/ibeta_inv_function.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>The Incomplete Beta 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_beta.html" title="Beta Functions"> <link rel="prev" href="ibeta_function.html" title="Incomplete Beta Functions"> <link rel="next" href="beta_derivative.html" title="Derivative of the Incomplete Beta Function"> @@ -337,7 +337,7 @@ Firstly recall that: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv5.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv5.svg"></span> </p> <p> We may wish to start from either p or q, and to calculate either x or y. @@ -356,7 +356,7 @@ The nearly symmetrical case (section 2 of the paper) is used for </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv2.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv2.svg"></span> </p> <p> and involves solving the inverse error function first. The method is accurate @@ -367,7 +367,7 @@ The general error function case (section 3 of the paper) is used for </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv3.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv3.svg"></span> </p> <p> and again expresses the inverse incomplete beta in terms of the inverse of @@ -378,7 +378,7 @@ it is better to use the following as an initial estimate: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv4.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv4.svg"></span> </p> <p> Finally the for all other cases where <code class="computeroutput"><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span> <span class="special">></span> @@ -394,7 +394,7 @@ (<code class="literal">q<sup>1/p</sup> < 10<sup>-3</sup></code>) we use: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv6.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv6.svg"></span> </p> <p> which is both cheaper to compute than the full method, and a more accurate @@ -417,7 +417,7 @@ and on a convex curve. An initial estimate for x is made with: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv7.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv7.svg"></span> </p> <p> which is provably below the true value for x: <a href="http://en.wikipedia.org/wiki/Newton%27s_method" target="_top">Newton @@ -435,13 +435,13 @@ and on a concave curve. An initial estimate for x is made with: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv4.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv4.svg"></span> </p> <p> which can be improved somewhat to: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv1.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv1.svg"></span> </p> <p> when b and x are both small (I've used b < a and x < 0.2). This actually @@ -456,7 +456,7 @@ points of inflection in [0,1]. For small p, x can be estimated using </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv4.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv4.svg"></span> </p> <p> which under-estimates x, and drops us on the right side of the true value @@ -470,7 +470,7 @@ quarter circle, and estimating y from: </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv8.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv8.svg"></span> </p> <p> This doesn't guarantee that we will drop in on the right side of x for monotonic |