Provided by: libmath-gsl-perl_0.43-4build1_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 Xdelta-squaredX 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) 2008-2021 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.