modules/math/romberg.man

[manpage_begin math::calculus::romberg n 0.6]
[copyright "2004 Kevin B. Kenny <kennykb@acm.org>. All rights\
reserved. Redistribution permitted under the terms of the Open\
Publication License <http://www.opencontent.org/openpub/>"]
[moddesc {Tcl Math Library}]
[titledesc {Romberg integration}]
[category  Mathematics]
[require Tcl 8.2]
[require math::calculus 0.6]
[description]
[para]
The [cmd romberg] procedures in the [cmd math::calculus] package
perform numerical integration of a function of one variable.  They
are intended to be of "production quality" in that they are robust,
precise, and reasonably efficient in terms of the number of function
evaluations.
[section "PROCEDURES"]

The following procedures are available for Romberg integration:

[list_begin definitions]
[call [cmd ::math::calculus::romberg] [arg f] [arg a] [arg b] [opt [arg "-option value"]...]]

Integrates an analytic function over a given interval.

[call [cmd ::math::calculus::romberg_infinity] [arg f] [arg a] [arg b] [opt [arg "-option value"]...]]

Integrates an analytic function over a half-infinite interval.

[call [cmd ::math::calculus::romberg_sqrtSingLower] [arg f] [arg a] [arg b] [opt [arg "-option value"]...]]

Integrates a function that is expected to be analytic over an interval
except for the presence of an inverse square root singularity at the
lower limit.

[call [cmd ::math::calculus::romberg_sqrtSingUpper] [arg f] [arg a] [arg b] [opt [arg "-option value"]...]]

Integrates a function that is expected to be analytic over an interval
except for the presence of an inverse square root singularity at the
upper limit.

[call [cmd ::math::calculus::romberg_powerLawLower] [arg gamma] [arg f] [arg a] [arg b] [opt [arg "-option value"]...]]

Integrates a function that is expected to be analytic over an interval
except for the presence of a power law singularity at the
lower limit.

[call [cmd ::math::calculus::romberg_powerLawUpper] [arg gamma] [arg f] [arg a] [arg b] [opt [arg "-option value"]...]]

Integrates a function that is expected to be analytic over an interval
except for the presence of a power law singularity at the
upper limit.

[call [cmd ::math::calculus::romberg_expLower] [arg f] [arg a] [arg b] [opt [arg "-option value"]...]]

Integrates an exponentially growing function; the lower limit of the
region of integration may be arbitrarily large and negative.

[call [cmd ::math::calculus::romberg_expUpper] [arg f] [arg a] [arg b] [opt [arg "-option value"]...]]

Integrates an exponentially decaying function; the upper limit of the
region of integration may be arbitrarily large.

[list_end]

[section PARAMETERS]

[list_begin definitions]

[def [arg f]]

Function to integrate.  Must be expressed as a single Tcl command,
to which will be appended a single argument, specifically, the
abscissa at which the function is to be evaluated. The first word
of the command will be processed with [cmd "namespace which"] in the
caller's scope prior to any evaluation. Given this processing, the
command may local to the calling namespace rather than needing to be
global.

[def [arg a]]

Lower limit of the region of integration.

[def [arg b]]

Upper limit of the region of integration.  For the
[cmd romberg_sqrtSingLower], [cmd romberg_sqrtSingUpper],
[cmd romberg_powerLawLower], [cmd romberg_powerLawUpper],
[cmd romberg_expLower], and [cmd romberg_expUpper] procedures,
the lower limit must be strictly less than the upper.  For
the other procedures, the limits may appear in either order.

[def [arg gamma]]

Power to use for a power law singularity; see section
[sectref "IMPROPER INTEGRALS"] for details.

[list_end]

[section OPTIONS]

[list_begin definitions]

[def "[option -abserror] [arg epsilon]"]

Requests that the integration machinery proceed at most until
the estimated absolute error of the integral is less than
[arg epsilon]. The error may be seriously over- or underestimated
if the function (or any of its derivatives) contains singularities;
see section [sectref "IMPROPER INTEGRALS"] for details. Default
is 1.0e-08.

[def "[option -relerror] [arg epsilon]"]

Requests that the integration machinery proceed at most until
the estimated relative error of the integral is less than
[arg epsilon]. The error may be seriously over- or underestimated
if the function (or any of its derivatives) contains singularities;
see section [sectref "IMPROPER INTEGRALS"] for details.  Default is
1.0e-06.

[def "[option -maxiter] [arg m]"]

Requests that integration terminate after at most [arg n] triplings of
the number of evaluations performed.  In other words, given [arg n]
for [option -maxiter], the integration machinery will make at most
3**[arg n] evaluations of the function.  Default is 14, corresponding
to a limit approximately 4.8 million evaluations. (Well-behaved
functions will seldom require more than a few hundred evaluations.)

[def "[option -degree] [arg d]"]

Requests that an extrapolating polynomial of degree [arg d] be used
in Romberg integration; see section [sectref "DESCRIPTION"] for
details. Default is 4.  Can be at most [arg m]-1.

[list_end]

[section DESCRIPTION]

The [cmd romberg] procedure performs Romberg integration using
the modified midpoint rule. Romberg integration is an iterative
process.  At the first step, the function is evaluated at the
midpoint of the region of integration, and the value is multiplied by
the width of the interval for the coarsest possible estimate.
At the second step, the interval is divided into three parts,
and the function is evaluated at the midpoint of each part; the
sum of the values is multiplied by three.  At the third step,
nine parts are used, at the fourth twenty-seven, and so on,
tripling the number of subdivisions at each step.

[para]

Once the interval has been divided at least [arg d] times,
a polynomial is fitted to the integrals estimated in the last
[arg d]+1 divisions.  The integrals are considered to be a
function of the square of the width of the subintervals
(any good numerical analysis text will discuss this process
under "Romberg integration").  The polynomial is extrapolated
to a step size of zero, computing a value for the integral and
an estimate of the error.

[para]

This process will be well-behaved only if the function is analytic
over the region of integration; there may be removable singularities
at either end of the region provided that the limit of the function
(and of all its derivatives) exists as the ends are approached.
Thus, [cmd romberg] may be used to integrate a function like
f(x)=sin(x)/x over an interval beginning or ending at zero.

[para]

Note that [cmd romberg] will either fail to converge or else return
incorrect error estimates if the function, or any of its derivatives,
has a singularity anywhere in the region of integration (except for
the case mentioned above).  Care must be used, therefore, in
integrating a function like 1/(1-x**2) to avoid the places
where the derivative is singular.

[section "IMPROPER INTEGRALS"]

Romberg integration is also useful for integrating functions over
half-infinite intervals or functions that have singularities.
The trick is to make a change of variable to eliminate the
singularity, and to put the singularity at one end or the other
of the region of integration.  The [cmd math::calculus] package
supplies a number of [cmd romberg] procedures to deal with the
commoner cases.

[list_begin definitions]

[def [cmd romberg_infinity]]

Integrates a function over a half-infinite interval; either
[arg a] or [arg b] may be infinite.  [arg a] and [arg b] must be
of the same sign; if you need to integrate across the axis,
say, from a negative value to positive infinity,
use [cmd romberg] to integrate from the negative
value to a small positive value, and then [cmd romberg_infinity]
to integrate from the positive value to positive infinity.  The
[cmd romberg_infinity] procedure works by making the change of
variable u=1/x, so that the integral from a to b of f(x) is
evaluated as the integral from 1/a to 1/b of f(1/u)/u**2.

[def "[cmd romberg_powerLawLower] and [cmd romberg_powerLawUpper]"]

Integrate a function that has an integrable power law singularity
at either the lower or upper bound of the region of integration
(or has a derivative with a power law singularity there).
These procedures take a first parameter, [arg gamma], which gives
the power law.  The function or its first derivative are presumed to diverge as
(x-[arg a])**(-[arg gamma]) or ([arg b]-x)**(-[arg gamma]).  [arg gamma]
must be greater than zero and less than 1.

[para]

These procedures are useful not only in integrating functions
that go to infinity at one end of the region of integration, but
also functions whose derivatives do not exist at the end of
the region.  For instance, integrating f(x)=pow(x,0.25) with the
origin as one end of the region will result in the [cmd romberg]
procedure greatly underestimating the error in the integral.
The problem can be fixed by observing that the first derivative
of f(x), f'(x)=x**(-3/4)/4, goes to infinity at the origin.  Integrating
using [cmd romberg_powerLawLower] with [arg gamma] set to 0.75
gives much more orderly convergence.

[para]

These procedures operate by making the change of variable
u=(x-a)**(1-gamma) ([cmd romberg_powerLawLower]) or
u=(b-x)**(1-gamma) ([cmd romberg_powerLawUpper]).

[para]

To summarize the meaning of gamma:

[list_begin itemized]
[item]
If f(x) ~ x**(-a) (0 < a < 1), use gamma = a
[item]
If f'(x) ~ x**(-b) (0 < b < 1), use gamma = b
[list_end]

[def "[cmd romberg_sqrtSingLower] and [cmd romberg_sqrtSingUpper]"]

These procedures behave identically to [cmd romberg_powerLawLower] and
[cmd romberg_powerLawUpper] for the common case of [arg gamma]=0.5;
that is, they integrate a function with an inverse square root
singularity at one end of the interval.  They have a simpler
implementation involving square roots rather than arbitrary powers.

[def "[cmd romberg_expLower] and [cmd romberg_expUpper]"]

These procedures are for integrating a function that grows or
decreases exponentially over a half-infinite interval.
[cmd romberg_expLower] handles exponentially growing functions, and
allows the lower limit of integration to be an arbitrarily large
negative number.  [cmd romberg_expUpper] handles exponentially
decaying functions and allows the upper limit of integration to
be an arbitrary large positive number.  The functions make the
change of variable u=exp(-x) and u=exp(x) respectively.

[list_end]

[section "OTHER CHANGES OF VARIABLE"]

If you need an improper integral other than the ones listed here,
a change of variable can be written in very few lines of Tcl.
Because the Tcl coding that does it is somewhat arcane,
we offer a worked example here.

[para]

Let's say that the function that we want to integrate is
f(x)=exp(x)/sqrt(1-x*x) (not a very natural
function, but a good example), and we want to integrate
it over the interval (-1,1).  The denominator falls to zero
at both ends of the interval. We wish to make a change of variable
from x to u
so that dx/sqrt(1-x**2) maps to du.  Choosing x=sin(u), we can
find that dx=cos(u)*du, and sqrt(1-x**2)=cos(u).  The integral
from a to b of f(x) is the integral from asin(a) to asin(b)
of f(sin(u))*cos(u).

[para]

We can make a function [cmd g] that accepts an arbitrary function
[cmd f] and the parameter u, and computes this new integrand.

[example {
proc g { f u } {
    set x [expr { sin($u) }]
    set cmd $f; lappend cmd $x; set y [eval $cmd]
    return [expr { $y / cos($u) }]
}
}]

Now integrating [cmd f] from [arg a] to [arg b] is the same
as integrating [cmd g] from [arg asin(a)] to [arg asin(b)].
It's a little tricky to get [cmd f] consistently evaluated in
the caller's scope; the following procedure does it.

[example {
proc romberg_sine { f a b args } {
    set f [lreplace $f 0 0\
               [uplevel 1 [list namespace which [lindex $f 0]]]]
    set f [list g $f]
    return [eval [linsert $args 0\
                      romberg $f\
                      [expr { asin($a) }] [expr { asin($b) }]]]
}
}]

This [cmd romberg_sine] procedure will do any function with
sqrt(1-x*x) in the denominator. Our sample function is
f(x)=exp(x)/sqrt(1-x*x):

[example {
proc f { x } {
    expr { exp($x) / sqrt( 1. - $x*$x ) }
}
}]

Integrating it is a matter of applying [cmd romberg_sine]
as we would any of the other [cmd romberg] procedures:

[example {
foreach { value error } [romberg_sine f -1.0 1.0] break
puts [format "integral is %.6g +/- %.6g" $value $error]

integral is 3.97746 +/- 2.3557e-010
}]


[section {BUGS, IDEAS, FEEDBACK}]

This document, and the package it describes, will undoubtedly contain
bugs and other problems.

Please report such in the category [emph {math :: calculus}] of the
[uri {http://sourceforge.net/tracker/?group_id=12883} {Tcllib SF Trackers}].

Please also report any ideas for enhancements you may have for either
package and/or documentation.


[see_also math::calculus math::interpolate]
[manpage_end]