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Subject: [Boost-commit] svn:boost r83363 - in trunk/libs/math/doc/sf_and_dist: . graphs
From: pbristow_at_[hidden]
Date: 2013-03-08 09:34:38

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Author: pbristow
Date: 2013-03-08 09:34:37 EST (Fri, 08 Mar 2013)
New Revision: 83363
URL: http://svn.boost.org/trac/boost/changeset/83363

Log:
Updates to qbk sources and graphs for version in my dropbox 8 Mar13.

(html not updated).
Removed:
   trunk/libs/math/doc/sf_and_dist/graphs/bessel_zeros.png
   trunk/libs/math/doc/sf_and_dist/graphs/bessel_zeros.svg
Text files modified:
   trunk/libs/math/doc/sf_and_dist/bessel_jy.qbk | 215 ++++++++++++++++++++++++++++++++++++---
   trunk/libs/math/doc/sf_and_dist/html4_symbols.qbk | 12 +
   trunk/libs/math/doc/sf_and_dist/issues.qbk | 30 ++++-
   trunk/libs/math/doc/sf_and_dist/math.qbk | 3
   trunk/libs/math/doc/sf_and_dist/references.qbk | 12 +-
   trunk/libs/math/doc/sf_and_dist/roadmap.qbk | 2
   6 files changed, 238 insertions(+), 36 deletions(-)

Modified: trunk/libs/math/doc/sf_and_dist/bessel_jy.qbk
==============================================================================
--- trunk/libs/math/doc/sf_and_dist/bessel_jy.qbk (original)
+++ trunk/libs/math/doc/sf_and_dist/bessel_jy.qbk 2013-03-08 09:34:37 EST (Fri, 08 Mar 2013)
@@ -284,7 +284,8 @@
`#include <boost/math/special_functions/bessel.hpp>`

Functions for obtaining both a single zero or root of the Bessel function,
-and placing multiple zeros into a container like `std::vector` by providing an output iterator.
+and placing multiple zeros into a container like `std::vector`
+by providing an output iterator.

The signature of the single value functions are:

@@ -308,7 +309,7 @@
                                 unsigned number_of_zeros,
                                 OutputIterator out_it); //

-There are also versions which allows control of the
+There are also versions which allow control of the
__policy_section for error handling and precision.

   template <class T>
@@ -335,48 +336,224 @@

[h4 Description]

-The zeros or roots (values of x where the function crosses the y = 0 axis)
+For every real order [nu] cylindrical Bessel and Neumann functions have an infinite
+number of zeros on the positive real axis. The real zeros on the positive real
+axis can be found by solving for the roots of
+
+[emquad] ['J[sub [nu]](j[sub [nu], m]) = 0]
+
+[emquad] ['Y[sub [nu]](y[sub [nu], m]) = 0]
+
+Here, ['j[sub [nu], m]] represents the ['m[super th]]
+root of the cylindrical Bessel function of order ['[nu]],
+and ['y[sub [nu], m]] represents the ['m[super th]]
+root of the cylindrical Neumann function of order ['[nu]].
+
+The zeros or roots (values of `x` where the function crosses the horizontal `y = 0` axis)
of the Bessel and Neumann functions are computed by two functions,
`cyl_bessel_j_zero` and `cyl_neumann_zero`.

-In each case the index of the zero
+In each case the index or rank of the zero
returned is 1-based, which is to say:

    cyl_bessel_j_zero(v, 1);

-returns the first zero of Bessel J, and
+returns the first zero of Bessel J,
+
+Passing a start_index <= 0

    cyl_bessel_j_zero(v, 0);

results in a __domain_error being raised.

-[graph bessel_zeros]
+The order `v` of `J` can be positive, negative and zero for the `cyl_bessel_j`
+and `cyl_neumann` functions, but not infinite nor NaN.
+
+[graph bessel_j_zeros]
+
+[graph neumann_y_zeros]

[h4 Examples of finding Bessel and Neumann zeros]

-[import ../../example/bessel_zeros_example.cpp]
+[import ../../example/bessel_zeros_example_1.cpp]
+
+[bessel_zeros_example_1]
+[bessel_zeros_example_2]
+
+[import ../../example/bessel_zeros_interator_example.cpp]
+
+[bessel_zeros_iterator_example_1]
+[bessel_zeros_iterator_example_2]
+
+[import ../../example/neumann_zeros_example_1.cpp]
+
+[neumann_zeros_example_1]
+[neumann_zeros_example_2]
+
+[import ../../example/bessel_errors_example.cpp]
+
+[bessel_errors_example_1]
+[bessel_errors_example_2]
+
+The full code (and output) for these examples is at
+[@../../../example/bessel_zeros_example_1.cpp Bessel zeros],
+[@../../../example/bessel_zeros_interator_example.cpp Bessel zeros iterator],
+[@../../../example/neumann_zeros_example_1.cpp Neumann zeros],
+[@../../../example/bessel_errors_example.cpp Bessel error messages].
+
+[h3 Implementation]
+
+Various methods are used to compute initial estimates
+for ['j[sub [nu], m]] and ['y[sub [nu], m]] ; these are described in detail below.
+
+After finding the initial estimate of a given root,
+its precision is subsequently refined to the desired level
+using Newton-Raphson iteration from Boost.Math's __root_finding_with_derivatives
+utilities combined with the functions __cyl_bessel_j and __cyl_neumann.
+
+Newton iteration requires both ['J[sub [nu]](x)] or ['Y[sub [nu]](x)]
+as well as its derivative. The derivatives of ['J[sub [nu]](x)] and ['Y[sub [nu]](x)]
+with respect to ['x] are given by __Abramowitz_Stegun.
+In particular,
+
+[emquad] ['d/[sub dx] ['J[sub [nu]](x)] = ['J[sub [nu]-1](x)] - [nu]/[sub x] [sdot] J[sub [nu]](x)]
+
+[emquad] ['d/[sub dx] ['Y[sub [nu]](x)] = ['Y[sub [nu]-1](x)] - [nu]/[sub x] [sdot] Y[sub [nu]](x)]
+
+Enumeration of the rank of a root (in other words the index of a root)
+begins with one and counts up, in other words
+['m,=1,2,3,[ellipsis]] The value of the first root is always greater than zero.
+
+For certain special parameters, cylindrical Bessel functions
+and cylindrical Neumann functions have a root at the origin. For example,
+['J[sub [nu]](x)] has a root at the origin for every positive order
+['[nu] > 0], and for every negative integer order
+['[nu] = -n] with ['n [isin] [negative] [super +]] and ['n [ne] 0].
+
+In addition, ['Y[sub [nu]](x)] has a root at the origin
+for every negative half-integer order ['[nu] = -n/2], with ['n [isin] [negative] [super +]] and
+and ['n [ne] 0].
+
+For these special parameter values, the origin with
+a value of ['x = 0] is provided as the ['0[super th]]
+root generated by `cyl_bessel_j_zero()`
+and `cyl_neumann_zero()`.
+
+When calculating initial estimates for the roots
+of Bessel functions, a distinction is made between
+positive order and negative order, and different
+methods are used for these. In addition, different algorithms
+are used for the first root ['m = 1] and
+for subsequent roots with higher rank ['m [ge] 2].
+Furthermore, estimates of the roots for Bessel functions
+with order above and below a cutoff at ['[nu] = 2.2]
+are calculated with different methods.
+
+Calculations of the estimates of ['j[sub [nu],1]] and ['y[sub [nu],1]]
+with ['0 [le] [nu] < 2.2] use empirically tabulated values.
+The coefficients for these have been generated by a
+computer algebra system.
+
+Calculations of the estimates of ['j[sub [nu],1]] and ['y[sub [nu],1]]
+with ['[nu][ge] 2.2] use Eqs.9.5.14 and 9.5.15 in __Abramowitz_Stegun.
+
+In particular,
+
+[emquad] ['j[sub [nu],1] [cong] [nu] + 1.85575 [nu][super [frac13]] + 1.033150 [nu][super -[frac13]] - 0.00397 [nu][super -1] - 0.0908 [nu][super -5/3] + 0.043 [nu][super -7/3] + [ellipsis]]
+
+and
+
+[emquad] ['y[sub [nu],1] [cong] [nu] + 0.93157 [nu][super [frac13]] + 0.26035 [nu][super -[frac13]] + 0.01198 [nu][super -1] - 0.0060 [nu][super -5/3] - 0.001 [nu][super -7/3] + [ellipsis]]
+
+Calculations of the estimates of ['j[sub [nu], m]] and ['y[sub [nu], m]]
+with rank ['m > 2] and ['0 [le] [nu] < 2.2] use
+McMahon's approximation, as described in M. Abramowitz and I. A. Stegan, Section 9.5 and 9.5.12.
+In particular,
+
+[emquad] ['j[sub [nu],m], y[sub [nu],m] [cong] [beta] - ([mu]-1) / 8[beta]]
+
+[emquad] [emquad] [emquad] ['- 4([mu]-1)(7[mu] - 31) / 3(8[beta])[super 3]]
+
+[emquad] [emquad] [emquad] ['-32([mu]-1)(83[mu][sup2] - 982[mu] + 3779) / 15(8[beta])[super 5]]
+
+[emquad] [emquad] [emquad] ['-64([mu]-1)(6949[mu][super 3] - 153855[mu][sup2] + 1585743[mu]- 6277237) / 105(8a)[super 7]]
+
+[emquad] [emquad] [emquad] ['- [ellipsis]] [emquad] [emquad] (5)
+
+where ['[mu] = 4[nu][super 2]] and ['[beta] = (m + [frac12][nu] - [frac14])[pi]]
+for ['j[sub [nu],m]] and
+['[beta] = (m + [frac12][nu] -[frac34])[pi] for ['y[sub [nu],m]]].
+
+Calculations of the estimates of ['j[sub [nu], m]] and ['y[sub [nu], m]]
+with ['[nu] [ge] 2.2] use
+one term in the asymptotic expansion given in
+Eq.9.5.22 and top line of Eq.9.5.26 combined with Eq. 9.3.39,
+all in __Abramowitz_Stegun explicit and easy-to-understand treatment
+for asymptotic expansion of zeros.
+The latter two equations are expressed for argument ['(x)] greater than one.
+(Olver also gives the series form of the equations in
+[@http://dlmf.nist.gov/10.21#vi [sect]10.21(vi) McMahon's Asymptotic Expansions for Large Zeros] - using slightly difference variable names).
+
+In summary,
+
+[emquad] ['j[sub [nu], m] [sim] [nu]x(-[zeta]) + f[sub 1](-[zeta]/[nu])]
+
+where ['-[zeta] = [nu][super -2/3]a[sub m]] and ['a[sub m]] is
+the absolute value of the ['m[super th]] root of ['Ai(x)] on the negative real axis.
+
+Here ['x = x(-[zeta])] is the inverse of the function
+
+[emquad] ['[frac23](-[zeta])[super 3/2] = [radic](x[sup2] - 1) - cos[supminus][sup1](1/x)] [emquad] [emquad] (7)
+
+Furthermore,
+
+[emquad] ['f[sub 1](-[zeta]) = [frac12]x(-[zeta]) {h(-[zeta])}[sup2] [sdot] b[sub 0](-[zeta])]
+
+where
+
+[emquad] ['h(-[zeta]) = {4(-[zeta]) / (x[sup2] - 1)}[super 4]]
+
+and

-[bessel_zero_example_1]
+[emquad] ['b[sub 0](-[zeta]) = -5/(48[zeta][sup2]) + 1/(-[zeta])[super [frac12]] [sdot] { 5/(24(x[super 2]-1)[super 3/2]) + 1/(8(x[super 2]-1)[super [frac12])]}]

-[bessel_zero_example_2]
+When solving for ['x(-[zeta])] in Eq. 7 above,
+the right-hand-side is expanded to order -['x[sup2]] in
+a Taylor series for large ['x]. This results in

-[bessel_zero_example_iterator_1]
+[emquad] ['[frac23](-[zeta])[super 3/2] [approx] x + 1/2x - [pi]/2]

-[bessel_zero_example_iterator_2]
+The positive root of the resulting quadratic equation
+is used to find an initial estimate ['x(-[zeta])].
+This initial estimate is subsequently refined with
+several steps of Newton-Raphson iteration
+in Eq. 7.

-[bessel_zero_example_iterator_3]
+Estimates of the roots of cylindrical Bessel functions
+of negative order on the positive real axis are found
+using interlacing relations. For example, the ['m[super th]]
+root of the cylindrical Bessel function ['j[sub -[nu],m]]
+is bracketed by the ['m[super th]] root and the
+['(m+1)[super th]] root of the Bessel function of
+corresponding positive integer order. In other words,

-The full code (and output) for this example for Bessel `cyl_bessel_j_zeros` is at
-[@../../../example/bessel_zeros_example.cpp Bessel, Neumann and Airy zeros].
+[emquad] ['j[sub n[nu],m]] < ['j[sub -[nu],m]] < ['j[sub n[nu],m+1]]

+where ['m > 1] and ['n[sub [nu]]] represents the integral
+floor of the absolute value of ['|-[nu]|].

-[h4 Implementation]
+Similar bracketing relations are used to find estimates
+of the roots of Neumann functions of negative order,
+whereby a discontinuity at every negative half-integer
+order needs to be handled.

-The method uses Newton-Raphson method starting with McMahon's approximation.
-See "Handbook of Mathematical Functions", M. Abramowitz and I. A. Stegan, section 9.5.
-Also: NIST Digital Library of Mathematical Functions, [@http://dlmf.nist.gov/10.21#vi Bessel Function Zeros].
+Bracketing relations do not hold for the first root
+of cylindrical Bessel functions and cylindrical Neumann
+functions with negative order. Therefore, iterative algorithms
+combined with root-finding via bisection are used
+to localize ['j[sub -[nu],1]] and ['y[sub -[nu],1]].

-[h4 Testing]
+[h3 Testing]

The precision of evaluation of zeros was tested at 50 decimal digits using `cpp_dec_float_50`
and found identical with spot values computed by __WolframAlpha.

Deleted: trunk/libs/math/doc/sf_and_dist/graphs/bessel_zeros.png
==============================================================================
Binary file. No diff available.

Deleted: trunk/libs/math/doc/sf_and_dist/graphs/bessel_zeros.svg
==============================================================================
--- trunk/libs/math/doc/sf_and_dist/graphs/bessel_zeros.svg 2013-03-08 09:34:37 EST (Fri, 08 Mar 2013)
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Modified: trunk/libs/math/doc/sf_and_dist/html4_symbols.qbk
==============================================================================
--- trunk/libs/math/doc/sf_and_dist/html4_symbols.qbk (original)
+++ trunk/libs/math/doc/sf_and_dist/html4_symbols.qbk 2013-03-08 09:34:37 EST (Fri, 08 Mar 2013)
@@ -78,7 +78,8 @@
[template frasl[]'''⁄'''] [/ / fraction slash]
[template weierp[]'''℘'''] [/ P script capital P = power set = Weierstrass p]
[template image[]'''ℑ'''] [/ I blackletter capital I = imaginary part]
-[template real[]'''ℜ'''] [/ R blackletter capital R = real part symbol]
+[template real[]'''ℜ'''] [/ R blackletter capital R = real part math symbol]
+[template negative[]'''ℕ'''] [/ N blackletter capital N = Negative number math symbol]
[template trade[]'''™'''] [/ ™ trade mark sign]
[template alefsym[]'''ℵ'''] [/ ? alef symbol = first transfinite cardinal]
[template larr[]'''←'''] [/ ? leftwards arrow]
@@ -116,6 +117,7 @@
[template there4[]'''∴'''] [/ ? therefore]
[template sim[]'''∼'''] [/ ~ tilde operator = varies with = similar to]
[template cong[]'''≅'''] [/ ? approximately equal to]
+[template approx[]'''≈'''] [/ ? ~~ very approximately equal to]
[template asymp[]'''≈'''] [/ ˜ almost equal to = asymptotic to]
[template ne[]'''≠'''] [/ ? not equal to]
[template equiv[]'''≡'''] [/ = identical to]
@@ -157,6 +159,7 @@
[template dot[]'''․'''] [/ dot leader]
[template dot2[]'''‥'''] [/ ? dots leader]
[template ellipsis[]'''…'''] [/ horizontal ellipsis]
+[template sect[]'''§'''] [/ ? section or paragraph sign]

[template dotover[]'''̇'''] [/ dot over symbol]
[template recur[]''' ̇'''] [/ math recurring symbol, eg after 0.333]
@@ -193,10 +196,15 @@
[template pow4[]'''⁴'''] [/ superscript 4 character]
[template pown[]'''ⁿ'''] [/ superscript n character]
[template frac12[]'''½'''] [/ fraction half]
+[template frac13[]'''⅓'''] [/ fraction third]
[template frac14[]'''¼'''] [/ fraction quarter]
[template frac34[]'''¾'''] [/ fraction three quarter]
+[template frac23[]'''⅔'''] [/ fraction two third]
[template sup1[]'''¹'''] [/ superscript one = superscript digit one ]
[template sup2[]'''²'''] [/ superscript two = superscript digit two = squared ]
+[template supminus[]'''⁻'''] [/ superscript minus]
+[template supplus[]'''⁺'''] [/ superscript plus]
+
[template cubed[]'''³'''] [/ superscript three = superscript digit three = cubed ]
[template macron[]'''¯'''] [/ macron = spacing macron = overline = APL overbar ]
[template deg[]'''°'''] [/ degree sign ]
@@ -208,7 +216,7 @@
[template laquo[]'''«'''] [/ left-pointing double angle quotation mark = left pointing guillemet ]
[template raquo[]'''»'''] [/ right-pointing double angle quotation mark = right pointing guillemet ]

-[/
+[/
Copyright 2007, 2010, 2012 Paul A. Bristow.
Distributed under the Boost Software License, Version 1.0.
(See accompanying file LICENSE_1_0.txt or copy at

Modified: trunk/libs/math/doc/sf_and_dist/issues.qbk
==============================================================================
--- trunk/libs/math/doc/sf_and_dist/issues.qbk (original)
+++ trunk/libs/math/doc/sf_and_dist/issues.qbk 2013-03-08 09:34:37 EST (Fri, 08 Mar 2013)
@@ -9,16 +9,30 @@
Items labled "Low Priority" should probably be investigated at
some point. Such classifications are obviously highly subjective.

-If you don't see a component listed here, then we don't have any known
+If you don't see a component listed here, then we don't have any known
issues with it.

+[h4 Derivatives of Bessel functions (and their zeros)]
+
+Potentially, there could be native support
+for `cyl_bessel_j_prime()` and `cyl_neumann_prime()`.
+One could also imagine supporting the zeros
+thereof, but they might be slower to calculate
+since root bracketing might be needed instead
+of Newton iteration (for the lack of 2nd derivatives).
+
+Since Boost.Math's Bessel functions are so excellent,
+the quick way to `cyl_bessel_j_prime()` and
+`cyl_neumann_prime()` would be via relationship with
+`cyl_bessel_j()` and `cyl_neumann()`.
+
[h4 tgamma]

* Can the __lanczos be optimized any further? (low priority)

[h4 Incomplete Beta]

-* Investigate Didonato and Morris' asymptotic expansion for large a and b
+* Investigate Didonato and Morris' asymptotic expansion for large a and b
(medium priority).

[h4 Inverse Gamma]
@@ -53,13 +67,13 @@
would it be more efficient evaluated this way, rather than by Carlson's
algorithms? (Low Priority).
* Should we add an implementation of Carlson's R[sub G]? It's not
-required for the Legendre form integrals, but some people may find it
+required for the Legendre form integrals, but some people may find it
useful (Low Priority).
-* There are a several other integrals: D([phi], k), Z([beta], k),
+* There are a several other integrals: D([phi], k), Z([beta], k),
[Lambda][sub 0]([beta], k) and Bulirsch's ['el] functions that could
be implemented using Carlson's integrals (Low Priority).
* The integrals K(k) and E(k) could be implemented using rational
-approximations (both for efficiency and accuracy),
+approximations (both for efficiency and accuracy),
assuming we can find them. (Medium Priority).
* There is a sub-domain of __ellint_3 that is unimplemented (see the docs
for details), currently
@@ -82,7 +96,7 @@

[h4 Statistical distributions]

-* Student's t Perhaps switch to normal distribution
+* Student's t Perhaps switch to normal distribution
as a better approximation for very large degrees of freedom?

[h4 Feature Requests]
@@ -136,7 +150,7 @@

Also asked for more than once:

-* Add support for interpolated distributions, possibly combine with numeric
+* Add support for interpolated distributions, possibly combine with numeric
integration and differentiation.
* Add support for bivariate and multivariate distributions: most especially the normal.
* Add support for the log of the cdf and pdf:
@@ -146,7 +160,7 @@

[endsect] [/section:issues Known Issues, and Todo List]

-[/
+[/
   Copyright 2006, 2010 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

Modified: trunk/libs/math/doc/sf_and_dist/math.qbk
==============================================================================
--- trunk/libs/math/doc/sf_and_dist/math.qbk (original)
+++ trunk/libs/math/doc/sf_and_dist/math.qbk 2013-03-08 09:34:37 EST (Fri, 08 Mar 2013)
@@ -89,6 +89,7 @@
[def __lanczos [link math_toolkit.backgrounders.lanczos Lanczos approximation]]
[def __zero_error [link zero_error effectively zero error]]
[def __relative_error [link math_toolkit.backgrounders.relative_error relative zero error]]
+[def __root_finding_with_derivatives [link math_toolkit.toolkit.internals1.roots root-finding with derivatives]]

[/gammas]
[def __lgamma [link math_toolkit.special.sf_gamma.lgamma lgamma]]
@@ -329,6 +330,8 @@
[def __performance [link math_toolkit.perf performance]]
[def __building [link math_toolkit.main_overview.building building libraries]]
[def __e_float [@http://calgo.acm.org/910.zip e_float (TOMS Algorithm 910)]]
+[def __Abramowitz_Stegun M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, NBS (1964)]
+[def _DMLF [@http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]]

[/ Some composite templates]
[template super[x]'''<superscript>'''[x]'''</superscript>''']

Modified: trunk/libs/math/doc/sf_and_dist/references.qbk
==============================================================================
--- trunk/libs/math/doc/sf_and_dist/references.qbk (original)
+++ trunk/libs/math/doc/sf_and_dist/references.qbk 2013-03-08 09:34:37 EST (Fri, 08 Mar 2013)
@@ -10,12 +10,12 @@
Abramowitz and Stegun's Handbook of Mathematical Functions (often called simply A&S),

M. Abramowitz and I. A. Stegun (Eds.) (1964)
-Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,
+Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,
National Bureau of Standards Applied Mathematics Series,
U.S. Government Printing Office, Washington, D.C.
+[/ __Abramowitz_Stegun]

NIST Handbook of Mathematical Functions
-
Edited by: Frank W. J. Olver, University of Maryland and National Institute of Standards and Technology, Maryland,
Daniel W. Lozier, National Institute of Standards and Technology, Maryland,
Ronald F. Boisvert, National Institute of Standards and Technology, Maryland,
@@ -48,8 +48,8 @@
We found (and used to create cross-check spot values - as far as their accuracy allowed).

[@http://functions.wolfram.com/ The Wolfram Functions Site]
-The Wolfram Functions Site - Providing
-the mathematical and scientific community with the world's largest
+The Wolfram Functions Site - Providing
+the mathematical and scientific community with the world's largest
(and most authorititive) collection of formulas and graphics about mathematical functions.

[@http://www.moshier.net/cephes28.zip 100-decimal digit calculator] provided some spot values.
@@ -95,7 +95,7 @@

Jim Albert, Bayesian Computation with R, ISBN 978-0-387-71384-7.

-[@http://www.quantnet.com/cplusplus-statistical-distributions-boost
+[@http://www.quantnet.com/cplusplus-statistical-distributions-boost
C++ Statistical Distributions in Boost - QuantNetwork forum]
discusses using Boost.Math in finance.

@@ -106,7 +106,7 @@
discusses application of Boost.Math, especially in finance.]

[endsect] [/section:references References]
-[/
+[/
   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

Modified: trunk/libs/math/doc/sf_and_dist/roadmap.qbk
==============================================================================
--- trunk/libs/math/doc/sf_and_dist/roadmap.qbk (original)
+++ trunk/libs/math/doc/sf_and_dist/roadmap.qbk 2013-03-08 09:34:37 EST (Fri, 08 Mar 2013)
@@ -256,7 +256,7 @@

]
[/
- Copyright 2006 - 2010 John Maddock and Paul A. Bristow.
+ Copyright 2006 - 2013 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).

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