Merged changes from sandbox to Trunk:

New special functions for truncation and rounding, plus exponential integrals and zeta.
New non central distributions.
Updated equation png's so that they are all consistent.

[SVN r44091]

doc/sf_and_dist/Jamfile.v2

@@ -40,7 +40,7 @@ boostbook standalone
         
         # PDF Options:
         # TOC Generation: this is needed for FOP-0.9 and later:
-        # <xsl:param>fop1.extensions=1
+        <xsl:param>fop1.extensions=0
         <format>pdf:<xsl:param>xep.extensions=1
         # TOC generation: this is needed for FOP 0.2, but must not be set to zero for FOP-0.9!
         <format>pdf:<xsl:param>fop.extensions=0

doc/sf_and_dist/bessel_ik.qbk

@@ -44,11 +44,11 @@ when `x <= 0`.
 
 The following graph illustrates the exponential behaviour of I[sub v].
 
-[$../graphs/bessel_i.png]
+[graph cyl_bessel_i]
 
 The following graph illustrates the exponential decay of K[sub v].
 
-[$../graphs/bessel_k.png]
+[graph cyl_bessel_k]
 
 [h4 Testing]
 

doc/sf_and_dist/bessel_jy.qbk

@@ -44,12 +44,12 @@ when `x <= 0`.
 
 The following graph illustrates the cyclic nature of J[sub v]:
 
-[$../graphs/bessel_jn.png]
+[graph cyl_bessel_j]
 
 The following graph shows the behaviour of Y[sub v]: this is also
 cyclic for large /x/, but tends to -[infin][space] for small /x/:
 
-[$../graphs/bessel_yv.png]
+[graph cyl_neumann]
 
 [h4 Testing]
 

doc/sf_and_dist/bessel_spherical.qbk

@@ -39,12 +39,12 @@ undefined or complex: this occurs when `x < 0`.
 The j[sub v][space] function is cyclic like J[sub v][space] but differs
 in its behaviour at the origin:
 
-[$../graphs/sph_bessel_j.png]
+[graph sph_bessel]
 
 Likewise y[sub v][space] is also cyclic for large x, but tends to -[infin][space]
 for small /x/:
 
-[$../graphs/sph_bessel_y.png]
+[graph sph_neumann]
 
 [h4 Testing]
 

doc/sf_and_dist/beta.qbk

@@ -22,7 +22,7 @@ The beta function is defined by:
 
 [equation beta1]
 
-[$../graphs/beta.png]
+[graph beta]
 
 [optional_policy]
 

doc/sf_and_dist/concepts.qbk

@@ -174,13 +174,25 @@ In the following table /r/ is an object of type `RealType`,
 [[`asin(cr1)`][RealType]]
 [[`tan(cr1)`][RealType]]
 [[`atan(cr1)`][RealType]]
+[[`fmod(cr1)`][RealType]]
+[[`round(cr1)`][RealType]]
+[[`iround(cr1)`][int]]
+[[`trunc(cr1)`][RealType]]
+[[`itrunc(cr1)`][int]]
 ]
 
 Note that the table above lists only those standard library functions known to
 be used (or likely to be used in the near future) by this library.  
-The following functions: `acos`, `atan2`, `fmod`, `cosh`, `sinh`, `tanh`, `modf` and `log10`
+The following functions: `acos`, `atan2`, `fmod`, `cosh`, `sinh`, `tanh`, `log10`,
+`lround`, `llround`, ltrunc`, `lltrunc` and `modf`
 are not currently used, but may be if further special functions are added.
 
+Note that the `round`, `trunc` and `modf` functions are not part of the
+current C++ standard: they are part of the additions added to C99 which will
+likely be in the next C++ standard.  There are Boost versions of these provided
+as a backup, and the functions are always called unqualified so that 
+argument-dependent-lookup can take place.
+
 In addition, for efficient and accurate results, a __lanczos is highly desirable.
 You may be able to adapt an existing approximation from 
 [@../../../../../boost/math/special_functions/lanczos.hpp

doc/sf_and_dist/credits.qbk

@@ -33,6 +33,17 @@ his
 program used to generate the html and pdf versions
 of this document, adding several new features en route.
 
+Plots of the functions and distributions were prepared in
+[@http://www.w3.org/ W3C] standard
+[@http://www.svg.org/ Scalable Vector Graphic (SVG)] format
+using a program created by Jacob Voytko during a Google 'Summer of Code'.
+Since browser support for rendering SVG is still not universal
+(Microsoft Internet Explorer, even IE 8 beta, still lacks native SVG support
+but can be made to work with
+[@http://www.adobe.com/svg/viewer/install/ Adobe's free SVG viewer] plugin),
+so the SVG files were batch converted to JPEG using
+[@http://www.inkscape.org/ Inkscape]. 
+
 We are also indebted to Matthias Schabel for managing the formal Boost-review
 of this library, and to all the reviewers - including Guillaume Melquiond,
 Arnaldur Gylfason, John Phillips, Stephan Tolksdorf and Jeff Garland 
@@ -41,7 +52,7 @@ Arnaldur Gylfason, John Phillips, Stephan Tolksdorf and Jeff Garland
 [endsect][/section:roadmap Roadmap]
 
 [/ 
-  Copyright 2006 John Maddock and Paul A. Bristow.
+  Copyright 2006 - 2008 John Maddock and Paul A. Bristow.
   Distributed under the Boost Software License, Version 1.0.
   (See accompanying file LICENSE_1_0.txt or copy at
   http://www.boost.org/LICENSE_1_0.txt).

doc/sf_and_dist/digamma.qbk

@@ -23,7 +23,7 @@ derivative of the gamma function:
 
 [equation digamma1]
 
-[$../graphs/digamma.png]
+[graph digamma]
 
 [optional_policy]
 

doc/sf_and_dist/dist_reference.qbk

@@ -15,6 +15,10 @@
 [include distributions/gamma.qbk]
 [include distributions/lognormal.qbk]
 [include distributions/negative_binomial.qbk]
+[include distributions/nc_beta.qbk]
+[include distributions/nc_chi_squared.qbk]
+[include distributions/nc_f.qbk]
+[include distributions/nc_t.qbk]
 [include distributions/normal.qbk]
 [include distributions/pareto.qbk]
 [include distributions/poisson.qbk]

doc/sf_and_dist/dist_tutorial.qbk

@@ -158,13 +158,32 @@ example, we would write:
 
 `pdf(binomial_distribution<RealType>(n, p), k);`
 
-The distribution (effectively the random variate) is said to be 'supported' over a range that is
+The ranges of random variate values that are permitted and are supported can be 
+tested by using two functions `range` and `support`.
+
+The distribution (effectively the random variate) is said to be 'supported' 
+over a range that is
 [@http://en.wikipedia.org/wiki/Probability_distribution
  "the smallest closed set whose complement has probability zero"].
 MathWorld uses the word 'defined' for this range.
 Non-mathematicians might say it means the 'interesting' smallest range
 of random variate x that has the cdf going from zero to unity.
 Outside are uninteresting zones where the pdf is zero, and the cdf zero or unity.
+
+For most distributions, with probability distribution functions one might describe
+as 'well-behaved', we have decided that it is most useful for the supported range
+to exclude random variate values like exact zero *if the end point is discontinuous*.
+For example, the Weibull (scale 1, shape 1) distribution smoothly heads for unity
+as the random variate x declines towards zero.
+But at x = zero, the value of the pdf is suddenly exactly zero, by definition.
+If you are plotting the PDF, or otherwise calculating,
+zero is not the most useful value for the lower limit of supported, as we discovered.
+So for this, and similar distributions,
+we have decided it is most numerically useful to use
+the closest value to zero, min_value, for the limit of the supported range.  
+(The `range` remains from zero, so you will still get `pdf(weibull, 0) == 0`).
+(Exponential and gamma distributions have similarly discontinuous functions).
+
 Mathematically, the functions may make sense with an (+ or -) infinite value,
 but except for a few special cases (in the Normal and Cauchy distributions)
 this implementation limits random variates to finite values from the `max` 
@@ -172,8 +191,6 @@ to `min` for the `RealType`.
 (See [link math_toolkit.backgrounders.implementation.handling_of_floating_point_infinity 
 Handling of Floating-Point Infinity] for rationale).
 
-The range of random variate values that is permitted and supported can be 
-tested by using two functions `range` and `support`.
 
 [note
 
@@ -365,6 +382,7 @@ Now that you have the basics, the next section looks at some worked examples.
 [include distributions/binomial_example.qbk]
 [include distributions/negative_binomial_example.qbk]
 [include distributions/normal_example.qbk]
+[include distributions/nc_chi_squared_example.qbk]
 [include distributions/error_handling_example.qbk]
 [include distributions/find_location_and_scale.qbk]
 [include distributions/nag_library.qbk]

doc/sf_and_dist/distributions/bernoulli.qbk

@@ -42,11 +42,11 @@ The following graph illustrates how the
 [@http://en.wikipedia.org/wiki/Probability_density_function probability density function pdf]
 varies with the outcome of the single trial:
 
-[$../graphs/bernoulli_pdf.png]
+[graph bernoulli_pdf]
 
 and the [@http://en.wikipedia.org/wiki/Cumulative_Distribution_Function Cumulative distribution function]
 
-[$../graphs/bernoulli_cdf.png]
+[graph bernoulli_cdf]
 
 [h4 Member Functions]
 

doc/sf_and_dist/distributions/beta.qbk

@@ -84,7 +84,7 @@ of the shape parameters.  Note the [alpha] = [beta] = 2 (blue line)
 is dome-shaped, and might be approximated by a symmetrical triangular 
 distribution.
 
-[$../graphs/beta_dist.png]
+[graph beta_pdf]
 
 If [alpha] = [beta] = 1, then it is a __space
 [@http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29 uniform distribution],

doc/sf_and_dist/distributions/binomial.qbk

@@ -76,12 +76,12 @@ The following two graphs illustrate how the PDF changes depending
 upon the distributions parameters, first we'll keep the success
 fraction /p/ fixed at 0.5, and vary the sample size:
 
-[$../graphs/binomial_pdf_1.png]
+[graph binomial_pdf_1]
 
 Alternatively, we can keep the sample size fixed at N=20 and
 vary the success fraction /p/:
 
-[$../graphs/binomial_pdf_2.png]
+[graph binomial_pdf_2]
 
 [discrete_quantile_warning Binomial]
    

doc/sf_and_dist/distributions/cauchy.qbk

@@ -44,12 +44,12 @@ of spectral lines.
 The following graph shows how the distributions moves as the
 location parameter changes:
 
-[$../graphs/cauchy1.png]
+[graph cauchy_pdf1]
 
 While the following graph shows how the shape (scale) parameter alters
 the distribution:
 
-[$../graphs/cauchy2.png]
+[graph cauchy_pdf2]
 
 [h4 Member Functions]
 

doc/sf_and_dist/distributions/chi_squared.qbk

@@ -49,7 +49,7 @@ and has a single parameter [nu][space] that specifies the number of degrees of
 freedom.  The following graph illustrates how the distribution changes
 for different values of [nu]:
 
-[$../graphs/chi_square.png]
+[graph chi_squared_pdf]
 
 [h4 Member Functions]
 
@@ -105,6 +105,11 @@ See also section on Sample sizes required in
 All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions]
 that are generic to all distributions are supported: __usual_accessors.
 
+(We have followed the usual restriction of the mode to degrees of freedom >= 2,
+but note that the maximum of the pdf is actually zero for degrees of freedom from 2 down to 0,
+and provide an extended definition that would avoid a discontinuity in the mode
+as alternative code in a comment).
+
 The domain of the random variable is \[0, +[infin]\].
 
 [h4 Examples]

doc/sf_and_dist/distributions/exponential.qbk

@@ -33,7 +33,7 @@ events that happen at a constant average rate.
 The following graph shows how the distribution changes for different
 values of the rate parameter lambda:
 
-[$../graphs/exponential_dist.png]
+[graph exponential_pdf]
 
 [h4 Member Functions]
 
@@ -90,6 +90,13 @@ In the following table [lambda] is the parameter lambda of the distribution,
 * [@http://www.itl.nist.gov/div898/handbook/eda/section3/eda3667.htm NIST Exploratory Data Analysis]
 * [@http://en.wikipedia.org/wiki/Exponential_distribution Wikipedia Exponential distribution]
 
+(See also the reference documentation for the related __extreme_distrib.)
+
+* 
+[@http://www.worldscibooks.com/mathematics/p191.html Extreme Value Distributions, Theory and Applications
+Samuel Kotz & Saralees Nadarajah]
+discuss the relationship of the types of extreme value distributions.
+
 [endsect][/section:exp_dist Exponential]
 
 [/ exponential.qbk

doc/sf_and_dist/distributions/extreme_value.qbk

@@ -36,6 +36,11 @@ More information can be found on the
 and [@http://en.wikipedia.org/wiki/Extreme_value_theory Extreme value theory]
 websites.
 
+The relationship of the types of extreme value distributions, of which this is but one, is
+discussed by
+[@http://www.worldscibooks.com/mathematics/p191.html Extreme Value Distributions, Theory and Applications
+Samuel Kotz & Saralees Nadarajah].
+
 The distribution has a PDF given by:
 
 f(x) = (1/scale) e[super -(x-location)/scale] e[super -e[super -(x-location)/scale]]
@@ -46,11 +51,11 @@ f(x) = e[super -x]e[super -e[super -x]]
 
 The following graph illustrates how the PDF varies with the location parameter:
 
-[$../graphs/extreme_val_dist.png]
+[graph extreme_value_pdf1]
 
 And this graph illustrates how the PDF varies with the shape parameter:
 
-[$../graphs/extreme_val_dist2.png]
+[graph extreme_value_pdf2]
 
 [h4 Member Functions]
 

doc/sf_and_dist/distributions/fisher.qbk

@@ -41,7 +41,7 @@ has the PDF:
 The following graph illustrates how the PDF varies depending on the
 two degrees of freedom parameters.
 
-[$../graphs/fisher_f.png]
+[graph fisher_f_pdf]
 
 
 [h4 Member Functions]

doc/sf_and_dist/distributions/gamma.qbk

@@ -61,9 +61,9 @@ parameter], you should pass the reciprocal of the rate as the scale parameter.
 The following two graphs illustrate how the PDF of the gamma distribution
 varies as the parameters vary:
 
-[$../graphs/gamma_dist1.png]
+[graph gamma1_pdf]
 
-[$../graphs/gamma_dist2.png]
+[graph gamma2_pdf]
 
 The [*Erlang Distribution] is the same as the Gamma, but with the shape parameter
 an integer.  It is often expressed using a /rate/ rather than a /scale/ as the 

doc/sf_and_dist/distributions/lognormal.qbk

@@ -43,11 +43,11 @@ parameter on the PDF, note that the range of the random
 variable remains \[0,+[infin]\] irrespective of the value of the
 location parameter:
 
-[$../graphs/lognormal1.png]
+[graph lognormal_pdf1]
 
 The next graph illustrates the effect of the scale parameter on the PDF:
 
-[$../graphs/lognormal2.png]
+[graph lognormal_pdf2]
 
 [h4 Member Functions]
 

doc/sf_and_dist/distributions/nc_beta.qbk

@@ -0,0 +1,211 @@
+[section:nc_beta_dist Noncentral Beta Distribution]
+
+``#include <boost/math/distributions/non_central_beta.hpp>``
+
+   namespace boost{ namespace math{ 
+
+   template <class RealType = double, 
+             class ``__Policy``   = ``__policy_class`` >
+   class non_central_beta_distribution;
+
+   typedef non_central_beta_distribution<> non_central_beta;
+
+   template <class RealType, class ``__Policy``>
+   class non_central_beta_distribution
+   {
+   public:
+      typedef RealType  value_type;
+      typedef Policy    policy_type;
+
+      // Constructor:
+      non_central_beta_distribution(RealType alpha, RealType beta, RealType lambda);
+
+      // Accessor to shape parameters:
+      RealType alpha()const;
+      RealType beta()const;
+
+      // Accessor to non-centrality parameter lambda:
+      RealType non_centrality()const;
+   };
+   
+   }} // namespaces
+   
+The noncentral beta distribution is a generalization of the __beta_distrib.
+
+It is defined as the ratio 
+X = [chi][sub m][super 2]([lambda]) \/ ([chi][sub m][super 2]([lambda]) 
++ [chi][sub n][super 2])
+where [chi][sub m][super 2]([lambda]) is a noncentral [chi][super 2]
+random variable with /m/ degrees of freedom, and [chi][sub n][super 2]
+is a central [chi][super 2] random variable with /n/ degrees of freedom.
+
+This gives a PDF that can be expressed as a Poisson mixture
+of beta distribution PDFs:
+
+[equation nc_beta_ref1]
+
+where P(i;[lambda]\/2) is the discrete Poisson probablity at /i/, with mean 
+[lambda]\/2, and I[sub x][super ']([alpha], [beta]) is the derivative of
+the incomplete beta function.  This leads to the usual form of the CDF
+as:
+
+[equation nc_beta_ref2]
+
+The following graph illustrates how the distribution changes
+for different values of [lambda]:
+
+[graph nc_beta_pdf]
+
+[h4 Member Functions]
+
+      non_central_beta_distribution(RealType a, RealType b, RealType lambda);
+      
+Constructs a noncentral beta distribution with shape parameters /a/ and /b/
+and non-centrality parameter /lambda/.
+
+Requires a > 0, b > 0 and lambda >= 0, otherwise calls __domain_error.
+
+      RealType alpha()const;
+      
+Returns the parameter /a/ from which this object was constructed.
+
+      RealType beta()const;
+      
+Returns the parameter /b/ from which this object was constructed.
+
+      RealType non_centrality()const;
+      
+Returns the parameter /lambda/ from which this object was constructed.
+
+[h4 Non-member Accessors]
+
+Most of the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions]
+are supported: __cdf, __pdf, __quantile, 
+__median, __mode, __hazard, __chf, __range and __support.
+
+However, the following are not currently implemented: 
+__mean, __variance, __sd, __skewness, 
+   __kurtosis and __kurtosis_excess.
+
+The domain of the random variable is \[0, 1\].
+
+[h4 Accuracy]
+
+The following table shows the peak errors
+(in units of [@http://en.wikipedia.org/wiki/Machine_epsilon epsilon]) 
+found on various platforms with various floating point types. 
+No comparison to the [@http://www.r-project.org/ R-2.5.1 Math library],
+or to the FORTRAN implementations of AS226 or AS310 are given since these appear
+to only guarantee absolute error: this would causes our test harness
+to assign an /"infinite"/ error to these libraries for some of our
+test values when measuring /relative error/.
+Unless otherwise specified any floating-point type that is narrower
+than the one shown will have __zero_error.
+
+[table Errors In CDF of the Noncentral Beta
+[[Significand Size] [Platform and Compiler] [[alpha], [beta],[lambda] < 200]  [[alpha],[beta],[lambda] > 200]]
+[[53] [Win32, Visual C++ 8] [Peak=620 Mean=22]  [Peak=8670 Mean=1040]]
+[[64] [RedHat Linux IA32, gcc-4.1.1] [Peak=825 Mean=50]  [Peak=2.5x10[super 4] Mean=4000]]
+
+[[64] [Redhat Linux IA64, gcc-3.4.4] [Peak=825 Mean=30]  [Peak=1.7x10[super 4] Mean=2500]]
+
+[[113] [HPUX IA64, aCC A.06.06] [Peak=420 Mean=50]  [Peak=9200 Mean=1200]]
+]
+
+Error rates for the PDF, the complement of the CDF and for the quantile 
+functions are broadly similar.
+
+[h4 Tests]
+
+There are two sets of test data used to verify this implementation:
+firstly we can compare with a few sample values generated by the
+[@http://www.r-project.org/ R library].
+Secondly, we have tables of test data, computed with this
+implementation and using interval arithmetic - this data should
+be accurate to at least 50 decimal digits - and is the used for
+our accuracy tests.
+
+[h4 Implementation]
+
+The CDF and its complement are evaluated as follows:
+
+First we determine which of the two values (the CDF or its
+complement) is likely to be the smaller, the crossover point
+is taken to be the mean of the distribution: for this we use the
+approximation due to: R. Chattamvelli and R. Shanmugam, 
+"Algorithm AS 310: Computing the Non-Central Beta Distribution Function",
+Applied Statistics, Vol. 46, No. 1. (1997), pp. 146-156.
+
+[equation nc_beta_ref3]
+
+Then either the CDF or its complement is computed using the
+relations:
+
+[equation nc_beta_ref4]
+
+The summation is performed by starting at i = [lambda]/2, and then recursing
+in both directions, using the usual recurrence relations for the Poisson
+PDF and incomplete beta functions.  This is the "Method 2" described
+by:
+
+Denise Benton and K. Krishnamoorthy,
+"Computing discrete mixtures of continuous
+distributions: noncentral chisquare, noncentral t
+and the distribution of the square of the sample
+multiple correlation coefficient", 
+Computational Statistics & Data Analysis 43 (2003) 249-267.
+
+Specific applications of the above formulae to the noncentral
+beta distribution can be found in:
+
+Russell V. Lenth,
+"Algorithm AS 226: Computing Noncentral Beta Probabilities",
+Applied Statistics, Vol. 36, No. 2. (1987), pp. 241-244.
+
+H. Frick,
+"Algorithm AS R84: A Remark on Algorithm AS 226: Computing Non-Central Beta
+Probabilities", Applied Statistics, Vol. 39, No. 2. (1990), pp. 311-312.
+
+Ming Long Lam,
+"Remark AS R95: A Remark on Algorithm AS 226: Computing Non-Central Beta
+Probabilities", Applied Statistics, Vol. 44, No. 4. (1995), pp. 551-552.
+
+Harry O. Posten,
+"An Effective Algorithm for the Noncentral Beta Distribution Function", 
+The American Statistician, Vol. 47, No. 2. (May, 1993), pp. 129-131.
+
+R. Chattamvelli,
+"A Note on the Noncentral Beta Distribution Function",
+The American Statistician, Vol. 49, No. 2. (May, 1995), pp. 231-234.
+
+Of these, the Posten reference provides the most complete overview, 
+and includes the modification starting iteration at [lambda]/2.
+
+The main difference between this implementation and the above 
+references is the direct computation of the complement when most
+efficient to do so, and the accumulation of the sum to -1 rather
+than subtracting the result from 1 at the end: this can substantially
+reduce the number of iterations required when the result is near 1.
+
+The PDF is computed using the methodology of Benton and Krishnamoorthy
+and the relation:
+
+[equation nc_beta_ref1]
+
+Quantiles are computed using a specially modified version of
+[link math_toolkit.toolkit.internals1.roots2 bracket_and_solve_root],
+starting the search for the root at the mean of the distribution.
+(A Cornish-Fisher type expansion was also tried, but while this gets
+quite close to the root in many cases, when it is wrong it tends to
+introduce quite pathological behaviour: more investigation in this
+area is probably warranted).
+
+[endsect][/section:nc_beta_dist]
+
+[/ nc_beta.qbk
+  Copyright 2008 John Maddock and Paul A. Bristow.
+  Distributed under the Boost Software License, Version 1.0.
+  (See accompanying file LICENSE_1_0.txt or copy at
+  http://www.boost.org/LICENSE_1_0.txt).
+]
+

doc/sf_and_dist/distributions/nc_chi_squared.qbk

@@ -0,0 +1,280 @@
+[section:nc_chi_squared_dist Noncentral Chi-Squared Distribution]
+
+``#include <boost/math/distributions/non_central_chi_squared.hpp>``
+
+   namespace boost{ namespace math{ 
+
+   template <class RealType = double, 
+             class ``__Policy``   = ``__policy_class`` >
+   class non_central_chi_squared_distribution;
+
+   typedef non_central_chi_squared_distribution<> non_central_chi_squared;
+
+   template <class RealType, class ``__Policy``>
+   class non_central_chi_squared_distribution
+   {
+   public:
+      typedef RealType  value_type;
+      typedef Policy    policy_type;
+
+      // Constructor:
+      non_central_chi_squared_distribution(RealType v, RealType lambda);
+
+      // Accessor to degrees of freedom parameter v:
+      RealType degrees_of_freedom()const;
+
+      // Accessor to non centrality parameter lambda:
+      RealType non_centrality()const;
+
+      // Parameter finders:
+      static RealType find_degrees_of_freedom(RealType lambda, RealType x, RealType p);
+      template <class A, class B, class C>
+      static RealType find_degrees_of_freedom(const complemented3_type<A,B,C>& c);
+      
+      static RealType find_non_centrality(RealType v, RealType x, RealType p);
+      template <class A, class B, class C>
+      static RealType find_non_centrality(const complemented3_type<A,B,C>& c);
+   };
+   
+   }} // namespaces
+   
+The noncentral chi-squared distribution is a generalization of the 
+__chi_squared_distrib. If X[sub i] are [nu] independent, normally 
+distributed random variables with means [mu][sub i] and variances 
+[sigma][sub i][super 2], then the random variable
+
+[equation nc_chi_squ_ref1]
+
+is distributed according to the noncentral chi-squared distribution. 
+
+The noncentral chi-squared distribution has two parameters: 
+[nu] which specifies the number of degrees of freedom 
+(i.e. the number of X[sub i]), and [lambda] which is related to the 
+mean of the random variables X[sub i] by:
+
+[equation nc_chi_squ_ref2]
+
+(Note that some references define [lambda] as one half of the above sum).
+
+This leads to a PDF of:
+
+[equation nc_chi_squ_ref3]
+
+where ['f(x;k)] is the central chi-squared distribution PDF, and 
+['I[sub v](x)] is a modified Bessel function of the first kind.
+
+The following graph illustrates how the distribution changes
+for different values of [lambda]:
+
+[graph nccs_pdf]
+
+[h4 Member Functions]
+
+      non_central_chi_squared_distribution(RealType v, RealType lambda);
+      
+Constructs a Chi-Squared distribution with /v/ degrees of freedom
+and non-centrality parameter /lambda/.
+
+Requires v > 0 and lambda >= 0, otherwise calls __domain_error.
+
+      RealType degrees_of_freedom()const;
+      
+Returns the parameter /v/ from which this object was constructed.
+
+      RealType non_centrality()const;
+      
+Returns the parameter /lambda/ from which this object was constructed.
+
+
+   static RealType find_degrees_of_freedom(RealType lambda, RealType x, RealType p);
+
+This function returns the number of degrees of freedom /v/ such that:
+`cdf(non_central_chi_squared<RealType, Policy>(v, lambda), x) == p`
+
+   template <class A, class B, class C>
+   static RealType find_degrees_of_freedom(const complemented3_type<A,B,C>& c);
+   
+When called with argument `boost::math::complement(lambda, x, q)` 
+this function returns the number of degrees of freedom /v/ such that:
+
+`cdf(complement(non_central_chi_squared<RealType, Policy>(v, lambda), x)) == q`.
+
+   static RealType find_non_centrality(RealType v, RealType x, RealType p);
+   
+This function returns the non centrality parameter /lambda/ such that:
+
+`cdf(non_central_chi_squared<RealType, Policy>(v, lambda), x) == p`
+
+   template <class A, class B, class C>
+   static RealType find_non_centrality(const complemented3_type<A,B,C>& c);
+
+When called with argument `boost::math::complement(v, x, q)`
+this function returns the non centrality parameter /lambda/ such that:
+
+`cdf(complement(non_central_chi_squared<RealType, Policy>(v, lambda), x)) == q`.
+
+[h4 Non-member Accessors]
+
+All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions]
+that are generic to all distributions are supported: __usual_accessors.
+
+The domain of the random variable is \[0, +[infin]\].
+
+[h4 Examples]
+
+There is a 
+[link math_toolkit.dist.stat_tut.weg.nccs_eg worked example] 
+for the noncentral chi-squared distribution.
+
+[h4 Accuracy]
+
+The following table shows the peak errors
+(in units of [@http://en.wikipedia.org/wiki/Machine_epsilon epsilon]) 
+found on various platforms with various floating-point types, 
+along with comparisons to the [@http://www.r-project.org/ R-2.5.1 Math library].
+Unless otherwise specified, any floating-point type that is narrower
+than the one shown will have __zero_error.
+
+[table Errors In CDF of the Noncentral Chi-Squared
+[[Significand Size] [Platform and Compiler] [[nu],[lambda] < 200]  [[nu],[lambda] > 200]]
+[[53] [Win32, Visual C++ 8] [Peak=50 Mean=9.9
+
+R Peak=685 Mean=109
+]  [Peak=9780 Mean=718 
+
+R Peak=3x10[super 8] Mean=2x10[super 7] ] ]
+[[64] [RedHat Linux IA32, gcc-4.1.1] [Peak=270 Mean=27]  [Peak=7900 Mean=900]]
+
+[[64] [Redhat Linux IA64, gcc-3.4.4] [Peak=107 Mean=17]  [Peak=5000 Mean=630]]
+
+[[113] [HPUX IA64, aCC A.06.06] [Peak=270 Mean=20]  [Peak=4600 Mean=560]]
+]
+
+Error rates for the complement of the CDF and for the quantile 
+functions are broadly similar.  Special mention should go to
+the `mode` function: there is no closed form for this function, 
+so it is evaluated numerically by finding the maxima of the PDF:
+in principal this can not produce an accuracy greater than the
+square root of the machine epsilon.
+
+[h4 Tests]
+
+There are two sets of test data used to verify this implementation:
+firstly we can compare with published data, for example with
+Table 6 of "Self-Validating Computations of Probabilities for 
+Selected Central and Noncentral Univariate Probability Functions",
+Morgan C. Wang and William J. Kennedy,
+Journal of the American Statistical Association, 
+Vol. 89, No. 427. (Sep., 1994), pp. 878-887.
+Secondly, we have tables of test data, computed with this
+implementation and using interval arithmetic - this data should
+be accurate to at least 50 decimal digits - and is the used for
+our accuracy tests.
+
+[h4 Implementation]
+
+The CDF and its complement are evaluated as follows:
+
+First we determine which of the two values (the CDF or its
+complement) is likely to be the smaller: for this we can use the
+relation due to Temme (see "Asymptotic and Numerical Aspects of the
+Noncentral Chi-Square Distribution", N. M. Temme, Computers Math. Applic.
+Vol 25, No. 5, 55-63, 1993) that:
+
+F([nu],[lambda];[nu]+[lambda]) [asymp] 0.5
+
+and so compute the CDF when the random variable is less than
+[nu]+[lambda], and its complement when the random variable is
+greater than [nu]+[lambda].  If necessary the computed result
+is then subtracted from 1 to give the desired result (the CDF or its
+complement).
+
+For small values of the non centrality parameter, the CDF is computed
+using the method of Ding (see "Algorithm AS 275: Computing the Non-Central 
+#2 Distribution Function", Cherng G. Ding, Applied Statistics, Vol. 41, 
+No. 2. (1992), pp. 478-482).  This uses the following series representation:
+
+[equation nc_chi_squ_ref4]
+
+which requires just one call to __gamma_p_derivative with the subsequent
+terms being computed by recursion as shown above.
+
+For larger values of the non-centrality parameter, Ding's method can take
+an unreasonable number of terms before convergence is achieved.  Furthermore,
+the largest term is not the first term, so in extreme cases the first term may
+be zero, leading to a zero result, even though the true value may be non-zero.
+
+Therefore, when the non-centrality parameter is greater than 200, the method due
+to Krishnamoorthy (see "Computing discrete mixtures of continuous distributions: 
+noncentral chisquare, noncentral t and the distribution of the 
+square of the sample multiple correlation coefficient", 
+Denise Benton and K. Krishnamoorthy, Computational Statistics & 
+Data Analysis, 43, (2003), 249-267) is used.
+
+This method uses the well known sum:
+
+[equation nc_chi_squ_ref5]
+
+Where P[sub a](x) is the incomplete gamma function.
+
+The method starts at the [lambda]th term, which is where the Poisson weighting
+function achieves its maximum value, although this is not necessarily
+the largest overall term.  Subsequent terms are calculated via the normal
+recurrence relations for the incomplete gamma function, and iteration proceeds
+both forwards and backwards until sufficient precision has been achieved.  It
+should be noted that recurrence in the forwards direction of P[sub a](x) is
+numerically unstable.  However, since we always start /after/ the largest
+term in the series, numeric instability is introduced more slowly than the
+series converges.
+
+Computation of the complement of the CDF uses an extension of Krishnamoorthy's
+method, given that:
+
+[equation nc_chi_squ_ref6]
+
+we can again start at the [lambda]'th term and proceed in both directions from
+there until the required precision is achieved.  This time it is backwards
+recursion on the incomplete gamma function Q[sub a](x) which is unstable.
+However, as long as we start well /before/ the largest term, this is not an
+issue in practice.
+
+The PDF is computed directly using the relation:
+
+[equation nc_chi_squ_ref3]
+
+Where ['f(x; v)] is the PDF of the central __chi_squared_distrib and 
+['I[sub v](x)] is a modified Bessel function, see __cyl_bessel_i.  
+For small values of the
+non-centrality parameter the relation in terms of __cyl_bessel_i
+is used.  However, this method fails for large values of the
+non-centrality parameter, so in that case the infinite sum is 
+evaluated using the method of Benton and Krishnamoorthy, and 
+the usual recurrence relations for successive terms.
+
+The quantile functions are computed by numeric inversion of the CDF.
+
+There is no [@http://en.wikipedia.org/wiki/Closed_form closed form]
+for the mode of the noncentral chi-squared
+distribution: it is computed numerically by finding the maximum
+of the PDF.  Likewise, the median is computed numerically via
+the quantile.
+
+The remaining non-member functions use the following formulas:
+
+[equation nc_chi_squ_ref7]
+
+Some analytic properties of noncentral distributions
+(particularly unimodality, and monotonicity of their modes)
+are surveyed and summarized by:
+
+Andrea van Aubel & Wolfgang Gawronski, Applied Mathematics and Computation, 141 (2003) 3-12.
+
+[endsect] [/section:nc_chi_squared_dist]
+
+[/ nc_chi_squared.qbk
+  Copyright 2008 John Maddock and Paul A. Bristow.
+  Distributed under the Boost Software License, Version 1.0.
+  (See accompanying file LICENSE_1_0.txt or copy at
+  http://www.boost.org/LICENSE_1_0.txt).
+]
+

doc/sf_and_dist/distributions/nc_chi_squared_example.qbk

@@ -0,0 +1,20 @@
+[section:nccs_eg Non Central Chi Squared Example]
+
+(See also the reference documentation for the __non_central_chi_squared_distrib.)
+
+[section:nccs_power_eg Tables of the power function of the [chi][super 2] test.]
+
+[import ../../../example/nc_chi_sq_example.cpp]
+[nccs_eg]
+
+[endsect] [/nccs_power_eg Tables of the power function of the [chi][super 2] test.]
+
+[endsect] [/section:nccs_eg Non Central Chi Squared Example]
+
+[/ 
+  Copyright 2006 John Maddock and Paul A. Bristow.
+  Distributed under the Boost Software License, Version 1.0.
+  (See accompanying file LICENSE_1_0.txt or copy at
+  http://www.boost.org/LICENSE_1_0.txt).
+]
+

doc/sf_and_dist/distributions/nc_f.qbk

@@ -0,0 +1,193 @@
+[section:nc_f_dist Noncentral F Distribution]
+
+``#include <boost/math/distributions/non_central_f.hpp>``
+
+   namespace boost{ namespace math{ 
+
+   template <class RealType = double, 
+             class ``__Policy``   = ``__policy_class`` >
+   class non_central_f_distribution;
+
+   typedef non_central_f_distribution<> non_central_f;
+
+   template <class RealType, class ``__Policy``>
+   class non_central_f_distribution
+   {
+   public:
+      typedef RealType  value_type;
+      typedef Policy    policy_type;
+
+      // Constructor:
+      non_central_f_distribution(RealType v1, RealType v2, RealType lambda);
+
+      // Accessor to degrees_of_freedom parameters v1 & v2:
+      RealType degrees_of_freedom1()const;
+      RealType degrees_of_freedom2()const;
+
+      // Accessor to non-centrality parameter lambda:
+      RealType non_centrality()const;
+   };
+   
+   }} // namespaces
+   
+The noncentral F distribution is a generalization of the __F_distrib.
+It is defined as the ratio 
+
+   F = (X/v1) / (Y/v2)
+   
+where X is a noncentral [chi][super 2]
+random variable with /v1/ degrees of freedom and non-centrality parameter [lambda], 
+and Y is a central [chi][super 2] random variable with /v2/ degrees of freedom.
+
+This gives the following PDF:
+
+[equation nc_f_ref1]
+
+where L[sub a][super b](c) is a generalised Laguerre polynomial and B(a,b) is the 
+__beta function, or
+
+[equation nc_f_ref2]
+
+The following graph illustrates how the distribution changes
+for different values of [lambda]:
+
+[graph nc_f_pdf]
+
+[h4 Member Functions]
+
+      non_central_f_distribution(RealType v1, RealType v2, RealType lambda);
+      
+Constructs a non-central beta distribution with parameters /v1/ and /v2/
+and non-centrality parameter /lambda/.
+
+Requires v1 > 0, v2 > 0 and lambda >= 0, otherwise calls __domain_error.
+
+      RealType degrees_of_freedom1()const;
+      
+Returns the parameter /v1/ from which this object was constructed.
+
+      RealType degrees_of_freedom2()const;
+      
+Returns the parameter /v2/ from which this object was constructed.
+
+      RealType non_centrality()const;
+      
+Returns the non-centrality parameter /lambda/ from which this object was constructed.
+
+[h4 Non-member Accessors]
+
+All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions]
+that are generic to all distributions are supported: __usual_accessors.
+
+The domain of the random variable is \[0, +[infin]\].
+
+[h4 Accuracy]
+
+This distribution is implemented in terms of the
+__non_central_beta_distrib: refer to that distribution for accuracy data.
+
+[h4 Tests]
+
+Since this distribution is implemented by adapting another distribution, 
+the tests consist of basic sanity checks computed by the
+[@http://www.r-project.org/ R-2.5.1 Math library statistical
+package] and its pbeta and dbeta functions.
+
+[h4 Implementation]
+
+In the following table /v1/ and /v2/ are the first and second
+degrees of freedom parameters of the distribution, [lambda]
+is the non-centrality parameter,
+/x/ is the random variate, /p/ is the probability, and /q = 1-p/.
+
+[table
+[[Function][Implementation Notes]]
+[[pdf][Implemented in terms of the non-central beta PDF using the relation:
+
+f(x;v1,v2;[lambda]) = (v1\/v2) / ((1+y)*(1+y)) * g(y\/(1+y);v1\/2,v2\/2;[lambda])
+
+where g(x; a, b; [lambda]) is the non central beta PDF, and:
+
+y = x * v1 \/ v2
+]]
+[[cdf][Using the relation:
+
+p = B[sub y](v1\/2, v2\/2; [lambda])
+
+where B[sub x](a, b; [lambda]) is the noncentral beta distribution CDF and
+
+y = x * v1 \/ v2
+
+]]
+
+[[cdf complement][Using the relation:
+
+q = 1 - B[sub y](v1\/2, v2\/2; [lambda])
+
+where 1 - B[sub x](a, b; [lambda]) is the complement of the 
+noncentral beta distribution CDF and
+
+y = x * v1 \/ v2
+
+]]
+[[quantile][Using the relation:
+
+x = (bx \/ (1-bx)) * (v1 \/ v2)
+
+where
+
+bx = Q[sub p][super -1](v1\/2, v2\/2; [lambda])
+
+and 
+
+Q[sub p][super -1](v1\/2, v2\/2; [lambda])
+
+is the noncentral beta quantile.
+
+]]
+[[quantile
+
+from the complement][
+Using the relation:
+
+x = (bx \/ (1-bx)) * (v1 \/ v2)
+
+where
+
+bx = QC[sub q][super -1](v1\/2, v2\/2; [lambda])
+
+and 
+
+QC[sub q][super -1](v1\/2, v2\/2; [lambda])
+
+is the noncentral beta quantile from the complement.]]
+[[mean][v2 * (v1 + l) \/ (v1 * (v2 - 2))]]
+[[mode][By numeric maximalisation of the PDF.]]
+[[variance][Refer to, [@http://mathworld.wolfram.com/NoncentralF-Distribution.html
+    Weisstein, Eric W. "Noncentral F-Distribution." From MathWorld--A Wolfram Web Resource.]  ]]
+[[skewness][Refer to, [@http://mathworld.wolfram.com/NoncentralF-Distribution.html
+    Weisstein, Eric W. "Noncentral F-Distribution." From MathWorld--A Wolfram Web Resource.],
+    and to the [@http://reference.wolfram.com/mathematica/ref/NoncentralFRatioDistribution.html
+    Mathematica documentation]  ]]
+[[kurtosis and kurtosis excess]
+    [Refer to, [@http://mathworld.wolfram.com/NoncentralF-Distribution.html
+    Weisstein, Eric W. "Noncentral F-Distribution." From MathWorld--A Wolfram Web Resource.],
+    and to the [@http://reference.wolfram.com/mathematica/ref/NoncentralFRatioDistribution.html
+    Mathematica documentation]  ]]
+]
+
+Some analytic properties of noncentral distributions
+(particularly unimodality, and monotonicity of their modes)
+are surveyed and summarized by:
+
+Andrea van Aubel & Wolfgang Gawronski, Applied Mathematics and Computation, 141 (2003) 3-12.
+
+[endsect] [/section:nc_f_dist]
+
+[/ nc_f.qbk
+  Copyright 2008 John Maddock and Paul A. Bristow.
+  Distributed under the Boost Software License, Version 1.0.
+  (See accompanying file LICENSE_1_0.txt or copy at
+  http://www.boost.org/LICENSE_1_0.txt).
+]
+

doc/sf_and_dist/distributions/nc_t.qbk

@@ -0,0 +1,195 @@
+[section:nc_t_dist Noncentral T Distribution]
+
+``#include <boost/math/distributions/non_central_t.hpp>``
+
+   namespace boost{ namespace math{ 
+
+   template <class RealType = double, 
+             class ``__Policy``   = ``__policy_class`` >
+   class non_central_t_distribution;
+
+   typedef non_central_t_distribution<> non_central_t;
+
+   template <class RealType, class ``__Policy``>
+   class non_central_t_distribution
+   {
+   public:
+      typedef RealType  value_type;
+      typedef Policy    policy_type;
+
+      // Constructor:
+      non_central_t_distribution(RealType v, RealType delta);
+
+      // Accessor to degrees_of_freedom parameter v:
+      RealType degrees_of_freedom()const;
+
+      // Accessor to non-centrality parameter lambda:
+      RealType non_centrality()const;
+   };
+   
+   }} // namespaces
+   
+The noncentral T distribution is a generalization of the __students_t_distrib.
+Let X have a normal distribution with mean [delta] and variance 1, and let 
+[nu] S[super 2] have
+a chi-squared distribution with degrees of freedom [nu]. Assume that 
+X and S[super 2] are independent. The
+distribution of t[sub [nu]]([delta])=X/S is called a 
+noncentral t distribution with degrees of freedom [nu] and noncentrality
+parameter [delta].
+
+This gives the following PDF:
+
+[equation nc_t_ref1]
+
+where [sub 1]F[sub 1](a;b;x) is a confluent hypergeometric function.
+
+The following graph illustrates how the distribution changes
+for different values of [delta]:
+
+[graph nc_t_pdf]
+
+[h4 Member Functions]
+
+      non_central_t_distribution(RealType v, RealType lambda);
+      
+Constructs a non-central t distribution with degrees of freedom
+parameter /v/ and non-centrality parameter /delta/.
+
+Requires v > 0 and finite delta, otherwise calls __domain_error.
+
+      RealType degrees_of_freedom()const;
+      
+Returns the parameter /v/ from which this object was constructed.
+
+      RealType non_centrality()const;
+      
+Returns the non-centrality parameter /delta/ from which this object was constructed.
+
+[h4 Non-member Accessors]
+
+All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions]
+that are generic to all distributions are supported: __usual_accessors.
+
+The domain of the random variable is \[-[infin], +[infin]\].
+
+[h4 Accuracy]
+
+The following table shows the peak errors
+(in units of [@http://en.wikipedia.org/wiki/Machine_epsilon 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 __zero_error.
+
+[table Errors In CDF of the Noncentral T Distribution
+[[Significand Size] [Platform and Compiler] [[nu],[delta] < 600]]
+[[53] [Win32, Visual C++ 8] [Peak=120 Mean=26 ] ]
+[[64] [RedHat Linux IA32, gcc-4.1.1] [Peak=121 Mean=26] ]
+
+[[64] [Redhat Linux IA64, gcc-3.4.4] [Peak=122 Mean=25] ]
+
+[[113] [HPUX IA64, aCC A.06.06] [Peak=115 Mean=24] ]
+]
+
+[caution The complexity of the current algorithm is dependent upon
+[delta][super 2]: consequently the time taken to evaluate the CDF
+increases rapidly for [delta] > 500, likewise the accuracy decreases
+rapidly for very large [delta].]
+
+Accuracy for the quantile and PDF functions should be broadly similar,
+note however that the /mode/ is determined numerically and can not
+in principal be more accurate than the square root of machine epsilon.
+
+[h4 Tests]
+
+There are two sets of tests of this distribution: basic sanity checks
+compare this implementation to the test values given in
+"Computing discrete mixtures of continuous
+distributions: noncentral chisquare, noncentral t
+and the distribution of the square of the sample
+multiple correlation coefficient."
+Denise Benton, K. Krishnamoorthy, 
+Computational Statistics & Data Analysis 43 (2003) 249-267.
+While accuracy checks use test data computed with this
+implementation and arbitary precision interval arithmetic:
+this test data is believed to be accurate to at least 50
+decimal places.
+
+
+[h4 Implementation]
+
+The CDF is computed using a modification of the method
+described in
+"Computing discrete mixtures of continuous
+distributions: noncentral chisquare, noncentral t
+and the distribution of the square of the sample
+multiple correlation coefficient."
+Denise Benton, K. Krishnamoorthy, 
+Computational Statistics & Data Analysis 43 (2003) 249-267.
+
+This uses the following formula for the CDF:
+
+[equation nc_t_ref2]
+
+Where I[sub x](a,b) is the incomplete beta function, and
+[Phi](x) is the normal CDF at x.
+
+Iteration starts at the largest of the Poisson weighting terms
+(at i = [delta][super 2] / 2) and then proceeds in both directions
+as per Benton and Krishnamoorthy's paper.
+
+Alternatively, by considering what happens when t = [infin], we have
+x = 1, and therefore I[sub x](a,b) = 1 and:
+
+[equation nc_t_ref3]
+
+From this we can easily show that:
+
+[equation nc_t_ref4]
+
+and therefore we have a means to compute either the probability or its
+complement directly without the risk of cancellation error.  The
+crossover criterion for choosing whether to calculate the CDF or
+it's complement is the same as for the 
+__non_central_beta_distrib.
+
+The PDF can be computed by a very similar method using:
+
+[equation nc_t_ref5]
+
+Where I[sub x][super '](a,b) is the derivative of the incomplete beta function.
+
+The quantile is calculated via the usual 
+[link math_toolkit.toolkit.internals1.roots2
+derivative-free root-finding techniques],
+with the initial guess taken as the quantile of a normal approximation
+to the noncentral T.
+
+There is no closed form for the mode, so this is computed via
+functional maximisation of the PDF.
+
+The remaining functions (mean, variance etc) are implemented
+using the formulas given in 
+Weisstein, Eric W. "Noncentral Student's t-Distribution." 
+From MathWorld--A Wolfram Web Resource. 
+[@http://mathworld.wolfram.com/NoncentralStudentst-Distribution.html
+http://mathworld.wolfram.com/NoncentralStudentst-Distribution.html]
+and in the 
+[@http://reference.wolfram.com/mathematica/ref/NoncentralStudentTDistribution.html 
+Mathematica documentation].
+
+Some analytic properties of noncentral distributions
+(particularly unimodality, and monotonicity of their modes)
+are surveyed and summarized by:
+
+Andrea van Aubel & Wolfgang Gawronski, Applied Mathematics and Computation, 141 (2003) 3-12.
+
+[endsect] [/section:nc_t_dist]
+
+[/ nc_t.qbk
+  Copyright 2008 John Maddock and Paul A. Bristow.
+  Distributed under the Boost Software License, Version 1.0.
+  (See accompanying file LICENSE_1_0.txt or copy at
+  http://www.boost.org/LICENSE_1_0.txt).
+]
+

doc/sf_and_dist/distributions/negative_binomial.qbk

@@ -72,12 +72,12 @@ It has the PDF:
 The following graph illustrate how the PDF varies as the success fraction 
 /p/ changes:
 
-[$../graphs/neg_binomial_pdf1.png]
+[graph negative_binomial_pdf_1]
 
 Alternatively, this graph shows how the shape of the PDF varies as
 the number of successes changes:
 
-[$../graphs/neg_binomial_pdf2.png]
+[graph negative_binomial_pdf_2]
 
 [h4 Related Distributions]
 

doc/sf_and_dist/distributions/normal.qbk

@@ -40,7 +40,7 @@ Given mean [mu][space] and standard deviation [sigma][space] it has the PDF:
 The variation the PDF with its parameters is illustrated
 in the following graph:
 
-[$../graphs/normal.png]
+[graph normal_pdf]
 
 [h4 Member Functions]
 

doc/sf_and_dist/distributions/pareto.qbk

@@ -37,11 +37,14 @@ The [@http://mathworld.wolfram.com/paretoDistribution.html Pareto distribution]
 often describes the larger compared to the smaller.
 A classic example is that 80% of the wealth is owned by 20% of the population.
 
-The following graph illustrates how the PDF varies with the shape parameter [alpha]:
+The following graph illustrates how the PDF varies with the location parameter [beta]:
+
+[graph pareto_pdf1]
+
+And this graph illustrates how the PDF varies with the shape parameter [alpha]:
+
+[graph pareto_pdf2]
 
-[/$../graphs/paretoShape.png]
-[/ TODO produce a graph as png or svg]
-[@http://upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Pareto_distributionPDF.png/325px-Pareto_distributionPDF.png Pareto pdf]
 
 [h4 Related distributions]
 

doc/sf_and_dist/distributions/poisson.qbk

@@ -41,7 +41,7 @@ for k events, with an expected number of events [lambda].
 
 The following graph illustrates how the PDF varies with the parameter [lambda]:
 
-[$../graphs/poisson.png]
+[graph poisson_pdf_1]
 
 [discrete_quantile_warning Poisson]
 

doc/sf_and_dist/distributions/rayleigh.qbk

@@ -40,11 +40,11 @@ or real and imaginary components may have absolute values that are Rayleigh dist
 
 The following graph illustrates how the Probability density Function(pdf) varies with the shape parameter [sigma]:
 
-[$../graphs/rayleigh_pdf.png]
+[graph rayleigh_pdf]
 
 and the Cumulative Distribution Function (cdf)
 
-[$../graphs/rayleigh_cdf.png]
+[graph rayleigh_cdf]
 
 [h4 Related distributions]
 

doc/sf_and_dist/distributions/students_t.qbk

@@ -55,7 +55,7 @@ As the number of degrees of freedom tends towards infinity, then this
 distribution approaches the normal-distribution.  The following graph
 illustrates how the PDF varies with the degrees of freedom [nu]:
 
-[$../graphs/students_t.png]
+[graph students_t_pdf]
 
 [h4 Member Functions]
 

doc/sf_and_dist/distributions/triangular.qbk

@@ -72,11 +72,11 @@ The following graph illustrates how the
 [@http://en.wikipedia.org/wiki/Probability_density_function probability density function PDF]
 varies with the various parameters:
 
-[$../graphs/triangular_pdf.png]
+[graph triangular_pdf]
 
 and cumulative distribution function
 
-[$../graphs/triangular_cdf.png]
+[graph triangular_cdf]
 
 [h4 Member Functions]
 

doc/sf_and_dist/distributions/uniform.qbk

@@ -61,11 +61,11 @@ The following graph illustrates how the
 [@http://en.wikipedia.org/wiki/Probability_density_function probability density function PDF]
 varies with the shape parameter:
 
-[$../graphs/uniform_pdf.png]
+[graph uniform_pdf]
 
 Likewise for the CDF:
 
-[$../graphs/uniform_cdf.png]
+[graph uniform_cdf]
 
 [h4 Member Functions]
 

doc/sf_and_dist/distributions/weibull.qbk

@@ -45,11 +45,11 @@ If the failure rate is:
 
 The following graph illustrates how the PDF varies with the shape parameter [alpha]:
 
-[$../graphs/weibull.png]
+[graph weibull_pdf1]
 
 While this graph illustrates how the PDF varies with the scale parameter [beta]:
 
-[$../graphs/weibull2.png]
+[graph weibull_pdf2]
 
 [h4 Related distributions]
 
@@ -58,6 +58,11 @@ When [alpha][space] = 3, the
 [@http://en.wikipedia.org/wiki/Normal_distribution normal distribution].
 When [alpha][space] = 1, the Weibull distribution reduces to the
 [@http://en.wikipedia.org/wiki/Exponential_distribution exponential distribution].
+The relationship of the types of extreme value distributions, of which the Weibull is but one, is
+discussed by
+[@http://www.worldscibooks.com/mathematics/p191.html Extreme Value Distributions, Theory and Applications
+Samuel Kotz & Saralees Nadarajah].
+
    
 [h4 Member Functions]
 

doc/sf_and_dist/ellint_carlson.qbk

@@ -76,7 +76,7 @@ These functions return Carlson's symmetrical elliptic integrals, the functions
 have complicated behavior over all their possible domains, but the following
 graph gives an idea of their behavior:
 
-[$../graphs/ellint_c.png]
+[graph ellint_carlson]
 
 The return type of these functions is computed using the __arg_pomotion_rules
 when the arguments are of different types: otherwise the return is the same type 

doc/sf_and_dist/ellint_legendre.qbk

@@ -35,7 +35,7 @@ LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
 These two functions evaluate the incomplete elliptic integral of the first kind
 ['F([phi], k)] and its complete counterpart ['K(k) = F([pi]/2, k)].
 
-[$../graphs/ellint_1.png]
+[graph ellint_1]
 
 The return type of these functions is computed using the __arg_pomotion_rules
 when T1 and T2 are different types: when they are the same type then the result
@@ -135,7 +135,7 @@ and
 These two functions evaluate the incomplete elliptic integral of the second kind
 ['E([phi], k)] and its complete counterpart ['E(k) = E([pi]/2, k)].
 
-[$../graphs/ellint_2.png]
+[graph ellint_2]
 
 The return type of these functions is computed using the __arg_pomotion_rules
 when T1 and T2 are different types: when they are the same type then the result
@@ -235,7 +235,7 @@ and
 These two functions evaluate the incomplete elliptic integral of the third kind
 ['[Pi](n, [phi], k)] and its complete counterpart ['[Pi](n, k) = E(n, [pi]/2, k)].
 
-[$../graphs/ellint_3.png]
+[graph ellint_3]
 
 The return type of these functions is computed using the __arg_pomotion_rules
 when the arguments are of different types: when they are the same type then the result

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doc/sf_and_dist/equations/nc_chi_squ_ref7.mml

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+

doc/sf_and_dist/equations/nc_f_ref1.mml

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doc/sf_and_dist/equations/nc_f_ref2.mml

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