Provided by: libmath-gsl-perl_0.44-1build3_amd64 
NAME
Math::GSL::Roots - Find roots of arbitrary 1-D functions
SYNOPSIS
use Math::GSL::Roots qw/:all/;
DESCRIPTION
• gsl_root_fsolver_alloc($T) -
This function returns a pointer to a newly allocated instance of a solver of type $T. $T must be one
of the constant included with this module. If there is insufficient memory to create the solver then
the function returns a null pointer and the error handler is invoked with an error code of
$GSL_ENOMEM.
• gsl_root_fsolver_free($s) -
Don't call this function explicitly. It will be called automatically in DESTROY for fsolver.
• "gsl_root_fsolver_set($s, $fspec, $x_lower, $x_upper)" -
This function initializes, or reinitializes, an existing solver $s to use the function described by
$fspec and the initial search interval [$x_lower, $x_upper]. $fspec may either be
• a coderef, e.g.
$fspec = sub { ... };
or
sub f { ... };
$fspec = \&f;
• an arrayref with elements [ $coderef, $params ]
The coderef is called as
&$coderef( $x, $params );
and should return the function evaluated at "$x, $params". For example, to find the root of a
quadratic with run-time specified coefficients "3, 2, 22",
$f = sub {
my ( $x, $params ) = @_;
return $params->[0] + $x * $params->[1] + $x**2 * $params->[2];
};
$fspec = [ $f, [ 3, 2, 22 ];
gsl_root_fsolver_set( $s, $fspec, $x_lower, $x_upper );
If there are no extra parameters, set $fspec to the function to be evaluated:
$fspec = sub {
my ( $x ) = shift;
return $x + $x**2;
};
gsl_root_fsolver_set( $s, $fspec, $x_lower, $x_upper );
Don't apply "gsl_root_fsolver_set" twice to the same fsolver. It will cause a memory leak. Instead
of this you should create new fsolver.
• gsl_root_fsolver_iterate($s) -
This function performs a single iteration of the solver $s. If the iteration encounters an unexpected
problem then an error code will be returned (the Math::GSL::Errno has to be included),
$GSL_EBADFUNC - The iteration encountered a singular point where the function or its derivative
evaluated to Inf or NaN.
$GSL_EZERODIV - The derivative of the function vanished at the iteration point, preventing the
algorithm from continuing without a division by zero.
• gsl_root_fsolver_name($s) -
This function returns the name of the solver use within the $s solver.
• gsl_root_fsolver_root($s) -
This function returns the current estimate of the root for the solver $s.
• gsl_root_fsolver_x_lower($s) -
This function returns the current lower value of the bracketing interval for the solver $s.
• gsl_root_fsolver_x_upper($s) -
This function returns the current lower value of the bracketing interval for the solver $s.
• gsl_root_fdfsolver_alloc($T) -
This function returns a pointer to a newly allocated instance of a derivative-based solver of type
$T. If there is insufficient memory to create the solver then the function returns a null pointer and
the error handler is invoked with an error code of $GSL_ENOMEM.
• "gsl_root_fdfsolver_set($s, $fspec, $root)" -
This function initializes, or reinitializes, an existing fdfsolver $s to use the function and its
derivatives specified by $fspec and the initial guess "$root."
$fspec may either be:
• A hashref with elements "f", "df", "fdf".
• An arrayref with elements "[ $hashref, $params ]"
where $hashref has elements "f", "df", "fdf";
The hashref elements are
• "f"
A coderef returning the value of the function at a given "x". It is called as "&$f($x, $params)".
• "df"
A coderef returning the value of the derivative of the function with respect to "x". It is called
as "&$df($x, $params)".
• "fdf"
A coderef returning the value of the function and its derivative with respect to "x". It is
called as "&$fdf($x, $params)".
For example, to find the root of a quadratic with run-time specified coefficients "3, 2, 22",
$fdf = {
f => sub {
my ( $x, $params ) = @_;
return $params->[0] + $x * $params->[1] + $x**2 * $params->[2];
},
df => sub {
my ( $x, $params ) = @_;
$params->[1] + 2 * $x * $params->[2];
},
fdf => sub {
my ( $x, $params ) = @_;
return
$params->[0] + $x * $params->[1] + $x**2 * $params->[2],
$params->[1] + 2 * $x * $params->[2];
},
};
$fspec = [ $fdf, [ 3, 2, 22 ];
gsl_root_fdsolver_set( $s, $fspec );
If there are no extra parameters, set $fspec to $fdf:
$fdf = {
f => sub {
my $x = shift;
return $x + $x**2;
},
df => sub {
my $x = shift;
1 + 2 * $x;
},
fdf => sub {
my $x = shift;
return
$x + $x**2,
1 + 2 * $x;
},
};
gsl_root_fdfsolver_set( $s, $fdf );
Don't apply "gsl_root_fdffsolver_set" twice to the same fdfsolver. It will cause a memory leak.
Instead of this you should create new fdfsolver.
• gsl_root_fdfsolver_iterate($s) -
This function performs a single iteration of the solver $s. If the iteration encounters an unexpected
problem then an error code will be returned (the Math::GSL::Errno has to be included),
$GSL_EBADFUNC - The iteration encountered a singular point where the function or its derivative
evaluated to Inf or NaN. $GSL_EZERODIV - The derivative of the function vanished at the iteration
point, preventing the algorithm from continuing without a division by zero.
• gsl_root_fdfsolver_free($s) -
Don't call this function explicitly. It will be called automatically in DESTROY for fdfsolver.
• gsl_root_fdfsolver_name($s) -
This function returns the name of the solver use within the $s solver.
• gsl_root_fdfsolver_root($s) -
This function returns the current estimate of the root for the solver $s.
• "gsl_root_test_interval($x_lower, $x_upper, $epsabs, $epsrel)" -
This function tests for the convergence of the interval [$x_lower, $x_upper] with absolute error
epsabs and relative error $epsrel. The test returns $GSL_SUCCESS if the following condition is
achieved,
|a - b| < epsabs + epsrel min(|a|,|b|)
when the interval x = [a,b] does not include the origin. If the interval
includes the origin then \min(|a|,|b|) is replaced by zero (which is the
minimum value of |x| over the interval). This ensures that the relative error
is accurately estimated for roots close to the origin. This condition on the
interval also implies that any estimate of the root r in the interval
satisfies the same condition with respect to the true root r^*,
|r - r^*| < epsabs + epsrel r^*
assuming that the true root r^* is contained within the interval.
• "gsl_root_test_residual($f, $epsabs)" -
This function tests the residual value $f against the absolute error bound $epsabs. The test returns
$GSL_SUCCESS if the following condition is achieved,
|$f| < $epsabs
and returns $GSL_CONTINUE otherwise. This criterion is suitable for situations where the precise
location of the root, x, is unimportant provided a value can be found where the residual, |f(x)|, is
small enough.
• "gsl_root_test_delta($x1, $x0, $epsabs, $epsrel)" -
This function tests for the convergence of the sequence ..., $x0, $x1 with absolute error $epsabs and
relative error $epsrel. The test returns $GSL_SUCCESS if the following condition is achieved,
|x_1 - x_0| < epsabs + epsrel |x_1|
and returns $GSL_CONTINUE otherwise.
This module also includes the following constants :
• $gsl_root_fsolver_bisection -
The bisection algorithm is the simplest method of bracketing the roots of a function. It is the
slowest algorithm provided by the library, with linear convergence. On each iteration, the interval
is bisected and the value of the function at the midpoint is calculated. The sign of this value is
used to determine which half of the interval does not contain a root. That half is discarded to give
a new, smaller interval containing the root. This procedure can be continued indefinitely until the
interval is sufficiently small. At any time the current estimate of the root is taken as the midpoint
of the interval.
• $gsl_root_fsolver_brent -
The Brent-Dekker method (referred to here as Brent's method) combines an interpolation strategy with
the bisection algorithm. This produces a fast algorithm which is still robust. On each iteration
Brent's method approximates the function using an interpolating curve. On the first iteration this is
a linear interpolation of the two endpoints. For subsequent iterations the algorithm uses an inverse
quadratic fit to the last three points, for higher accuracy. The intercept of the interpolating curve
with the x-axis is taken as a guess for the root. If it lies within the bounds of the current
interval then the interpolating point is accepted, and used to generate a smaller interval. If the
interpolating point is not accepted then the algorithm falls back to an ordinary bisection step. The
best estimate of the root is taken from the most recent interpolation or bisection.
• $gsl_root_fsolver_falsepos -
The false position algorithm is a method of finding roots based on linear interpolation. Its
convergence is linear, but it is usually faster than bisection. On each iteration a line is drawn
between the endpoints (a,f(a)) and (b,f(b)) and the point where this line crosses the x-axis taken as
a "midpoint". The value of the function at this point is calculated and its sign is used to determine
which side of the interval does not contain a root. That side is discarded to give a new, smaller
interval containing the root. This procedure can be continued indefinitely until the interval is
sufficiently small. The best estimate of the root is taken from the linear interpolation of the
interval on the current iteration.
• $gsl_root_fdfsolver_newton -
Newton's Method is the standard root-polishing algorithm. The algorithm begins with an initial guess
for the location of the root. On each iteration, a line tangent to the function f is drawn at that
position. The point where this line crosses the x-axis becomes the new guess. The iteration is
defined by the following sequence, x_{i+1} = x_i - f(x_i)/f'(x_i) Newton's method converges
quadratically for single roots, and linearly for multiple roots.
• $gsl_root_fdfsolver_secant -
The secant method is a simplified version of Newton's method which does not require the computation
of the derivative on every step. On its first iteration the algorithm begins with Newton's method,
using the derivative to compute a first step,
x_1 = x_0 - f(x_0)/f'(x_0)
Subsequent iterations avoid the evaluation of the derivative by replacing it with a numerical
estimate, the slope of the line through the previous two points,
x_{i+1} = x_i f(x_i) / f'_{est}
where
f'_{est} = (f(x_i) - f(x_{i-1})/(x_i - x_{i-1})
When the derivative does not change significantly in the vicinity of the root the secant method gives
a useful saving. Asymptotically the secant method is faster than Newton's method whenever the cost of
evaluating the derivative is more than 0.44 times the cost of evaluating the function itself. As with
all methods of computing a numerical derivative the estimate can suffer from cancellation errors if
the separation of the points becomes too small.
On single roots, the method has a convergence of order (1 + \sqrt 5)/2 (approximately 1.62). It
converges linearly for multiple roots.
• $gsl_root_fdfsolver_steffenson -
The Steffenson Method provides the fastest convergence of all the routines. It combines the basic
Newton algorithm with an Aitken “delta-squared” acceleration. If the Newton iterates are x_i then the
acceleration procedure generates a new sequence R_i:
R_i = x_i - (x_{i+1} - x_i)^2 / (x_{i+2} - 2 x_{i+1} + x_{i})
which converges faster than the original sequence under reasonable conditions. The new sequence
requires three terms before it can produce its first value so the method returns accelerated values
on the second and subsequent iterations. On the first iteration it returns the ordinary Newton
estimate. The Newton iterate is also returned if the denominator of the acceleration term ever
becomes zero.
As with all acceleration procedures this method can become unstable if the function is not well-
behaved.
For more information about these functions, we refer you to the official GSL documentation:
<http://www.gnu.org/software/gsl/manual/html_node/>
AUTHORS
Jonathan "Duke" Leto <jonathan@leto.net> and Thierry Moisan <thierry.moisan@gmail.com>
COPYRIGHT AND LICENSE
Copyright (C) 2008-2023 Jonathan "Duke" Leto and Thierry Moisan
This program is free software; you can redistribute it and/or modify it under the same terms as Perl
itself.