From 693f597348cf9df4beafa4fff2d048ebdd69ad2c Mon Sep 17 00:00:00 2001
From: "Paul A. Bristow" <pbristow@hetp.u-net.com>
Date: Tue, 2 Dec 2008 16:34:41 +0000
Subject: [PATCH] Placeholder only for laplace doc - needs serious work, and
 not added to the math.qbk, so won't be processed yet.

[SVN r50074]
---
 doc/sf_and_dist/distributions/laplace.qbk | 130 ++++++++++++++++++++++
 1 file changed, 130 insertions(+)
 create mode 100644 doc/sf_and_dist/distributions/laplace.qbk

diff --git a/doc/sf_and_dist/distributions/laplace.qbk b/doc/sf_and_dist/distributions/laplace.qbk
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+[section:laplace_dist Laplace Distribution]
+
+``#include <boost/math/distributions/laplace.hpp>``
+
+   namespace boost{ namespace math{
+
+   template <class RealType = double,
+             class ``__Policy``   = ``__policy_class`` >
+   class laplace_distribution;
+
+   typedef laplace_distribution<> laplace;
+
+   template <class RealType, class ``__Policy``>
+   class laplace_distribution
+   {
+   public:
+      typedef RealType value_type;
+      typedef Policy   policy_type;
+      // Construct:
+      laplace_distribution(RealType location = 0, RealType scale = 1);
+      // Accessors:
+      RealType location()const;
+      RealType scale()const;
+   };
+
+   }} // namespaces
+
+The laplace distribution is the distribution that is the difference betweeen
+two independent identically distributed exponential distributed variables.
+It is also called the double exponential distribution.
+
+TODO - this need complete revision - it has been subjected to a dirty find and replace
+so is plain wrong.  Also need pictures.
+
+and added to math.qbk to be used.
+
+
+For location and scale parameters /m/ and /s/ it is defined by the
+probability density function:
+
+[equation laplace_ref]
+
+The location and scale parameters are equivalent to the mean and
+standard deviation of the logarithm of the random variable.
+
+The following graph illustrates the effect of the location
+parameter on the PDF, note that the range of the random
+variable remains \[0,+[infin]\] irrespective of the value of the
+location parameter:
+
+[graph laplace_pdf1]
+
+The next graph illustrates the effect of the scale parameter on the PDF:
+
+[graph laplace_pdf2]
+
+[h4 Member Functions]
+
+   laplace_distribution(RealType location = 0, RealType scale = 1);
+
+Constructs a laplace distribution with location /location/ and
+scale /scale/.
+
+The location parameter is the same as the mean of the logarithm of the
+random variate.
+
+The scale parameter is the same as the standard deviation of the
+logarithm of the random variate.
+
+Requires that the scale parameter is greater than zero, otherwise calls
+__domain_error.
+
+   RealType location()const;
+
+Returns the /location/ parameter of this distribution.
+
+   RealType scale()const;
+
+Returns the /scale/ parameter of this distribution.
+
+[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]
+
+The laplace distribution is implemented in terms of the
+standard library log and exp functions, plus the
+[link math_toolkit.special.sf_erf.error_function error function],
+and as such should have very low error rates.
+
+[h4 Implementation]
+
+In the following table /m/ is the location parameter of the distribution,
+/s/ is it's scale parameter, /x/ is the random variate, /p/ is the probability
+and /q = 1-p/.
+
+[table
+[[Function][Implementation Notes]]
+[[pdf][Using the relation: pdf = e[super -(ln(x) - m)[super 2 ] \/ 2s[super 2 ] ] \/ (x * s * sqrt(2pi)) ]]
+[[cdf][Using the relation: p = cdf(normal_distribtion<RealType>(m, s), log(x)) ]]
+[[cdf complement][Using the relation: q = cdf(complement(normal_distribtion<RealType>(m, s), log(x))) ]]
+[[quantile][Using the relation: x = exp(quantile(normal_distribtion<RealType>(m, s), p))]]
+[[quantile from the complement][Using the relation: x = exp(quantile(complement(normal_distribtion<RealType>(m, s), q)))]]
+[[mean][e[super m + s[super 2 ] / 2 ] ]]
+[[variance][(e[super s[super 2] ] - 1) * e[super 2m + s[super 2 ] ] ]]
+[[mode][e[super m + s[super 2 ] ] ]]
+[[skewness][sqrt(e[super s[super 2] ] - 1) * (2 + e[super s[super 2] ]) ]]
+[[kurtosis][e[super 4s[super 2] ] + 2e[super 3s[super 2] ] + 3e[super 2s[super 2] ] - 3]]
+[[kurtosis excess][e[super 4s[super 2] ] + 2e[super 3s[super 2] ] + 3e[super 2s[super 2] ] - 6 ]]
+]
+
+[h4 References]
+
+[*http://mathworld.wolfram.com/LaplaceDistribution.html Weisstein, Eric W. "Laplace Distribution."] From MathWorld--A Wolfram Web Resource.
+[*http://en.wikipedia.org/wiki/Laplace_distribution Laplace Distribution]
+Abramowitz and Stegun 1972, p. 930.
+
+[endsect][/section:laplace_dist laplace]
+
+[/
+  Copyright 2008 John Maddock, Paul A. Bristow and M.A. (Thijs) van den Berg.
+  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).
+]
+