math::calculus::romberg(3tcl) Tcl Math Library math::calculus::romberg(3tcl)______________________________________________________________________________NAME
math::calculus::romberg - Romberg integration
SYNOPSIS
package require Tcl 8.2
package require math::calculus 0.6
::math::calculus::romberg f a b ?-option value...?
::math::calculus::romberg_infinity f a b ?-option value...?
::math::calculus::romberg_sqrtSingLower f a b ?-option value...?
::math::calculus::romberg_sqrtSingUpper f a b ?-option value...?
::math::calculus::romberg_powerLawLower gamma f a b ?-option value...?
::math::calculus::romberg_powerLawUpper gamma f a b ?-option value...?
::math::calculus::romberg_expLower f a b ?-option value...?
::math::calculus::romberg_expUpper f a b ?-option value...?
_________________________________________________________________DESCRIPTION
The romberg procedures in the 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.
PROCEDURES
The following procedures are available for Romberg integration:
::math::calculus::romberg f a b ?-option value...?
Integrates an analytic function over a given interval.
::math::calculus::romberg_infinity f a b ?-option value...?
Integrates an analytic function over a half-infinite interval.
::math::calculus::romberg_sqrtSingLower f a b ?-option value...?
Integrates a function that is expected to be analytic over an
interval except for the presence of an inverse square root sin‐
gularity at the lower limit.
::math::calculus::romberg_sqrtSingUpper f a b ?-option value...?
Integrates a function that is expected to be analytic over an
interval except for the presence of an inverse square root sin‐
gularity at the upper limit.
::math::calculus::romberg_powerLawLower gamma f a b ?-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.
::math::calculus::romberg_powerLawUpper gamma f a b ?-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.
::math::calculus::romberg_expLower f a b ?-option value...?
Integrates an exponentially growing function; the lower limit of
the region of integration may be arbitrarily large and negative.
::math::calculus::romberg_expUpper f a b ?-option value...?
Integrates an exponentially decaying function; the upper limit
of the region of integration may be arbitrarily large.
PARAMETERS
f Function to integrate. Must be expressed as a single Tcl com‐
mand, 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 namespace which in
the caller's scope prior to any evaluation. Given this process‐
ing, the command may local to the calling namespace rather than
needing to be global.
a Lower limit of the region of integration.
b Upper limit of the region of integration. For the
romberg_sqrtSingLower, romberg_sqrtSingUpper, romberg_power‐
LawLower, romberg_powerLawUpper, romberg_expLower, and
romberg_expUpper procedures, the lower limit must be strictly
less than the upper. For the other procedures, the limits may
appear in either order.
gamma Power to use for a power law singularity; see section IMPROPER
INTEGRALS for details.
OPTIONS-abserror epsilon
Requests that the integration machinery proceed at most until
the estimated absolute error of the integral is less than
epsilon. The error may be seriously over- or underestimated if
the function (or any of its derivatives) contains singularities;
see section IMPROPER INTEGRALS for details. Default is 1.0e-08.
-relerror epsilon
Requests that the integration machinery proceed at most until
the estimated relative error of the integral is less than
epsilon. The error may be seriously over- or underestimated if
the function (or any of its derivatives) contains singularities;
see section IMPROPER INTEGRALS for details. Default is 1.0e-06.
-maxiter m
Requests that integration terminate after at most n triplings of
the number of evaluations performed. In other words, given n
for -maxiter, the integration machinery will make at most 3**n
evaluations of the function. Default is 14, corresponding to a
limit approximately 4.8 million evaluations. (Well-behaved func‐
tions will seldom require more than a few hundred evaluations.)
-degree d
Requests that an extrapolating polynomial of degree d be used in
Romberg integration; see section DESCRIPTION for details.
Default is 4. Can be at most m-1.
DESCRIPTION
The 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 mid‐
point 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.
Once the interval has been divided at least d times, a polynomial is
fitted to the integrals estimated in the last d+1 divisions. The inte‐
grals 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 esti‐
mate of the error.
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,
romberg may be used to integrate a function like f(x)=sin(x)/x over an
interval beginning or ending at zero.
Note that romberg will either fail to converge or else return incorrect
error estimates if the function, or any of its derivatives, has a sin‐
gularity anywhere in the region of integration (except for the case
mentioned above). Care must be used, therefore, in integrating a func‐
tion like 1/(1-x**2) to avoid the places where the derivative is singu‐
lar.
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 math::calculus package supplies a number of romberg procedures to
deal with the commoner cases.
romberg_infinity
Integrates a function over a half-infinite interval; either a or
b may be infinite. a and b must be of the same sign; if you
need to integrate across the axis, say, from a negative value to
positive infinity, use romberg to integrate from the negative
value to a small positive value, and then romberg_infinity to
integrate from the positive value to positive infinity. The
romberg_infinity procedure works by making the change of vari‐
able u=1/x, so that the integral from a to b of f(x) is evalu‐
ated as the integral from 1/a to 1/b of f(1/u)/u**2.
romberg_powerLawLower and romberg_powerLawUpper
Integrate a function that has an integrable power law singular‐
ity at either the lower or upper bound of the region of integra‐
tion (or has a derivative with a power law singularity there).
These procedures take a first parameter, gamma, which gives the
power law. The function or its first derivative are presumed to
diverge as (x-a)**(-gamma) or (b-x)**(-gamma). gamma must be
greater than zero and less than 1.
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 romberg pro‐
cedure 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. Inte‐
grating using romberg_powerLawLower with gamma set to 0.75 gives
much more orderly convergence.
These procedures operate by making the change of variable u=(x-
a)**(1-gamma) (romberg_powerLawLower) or u=(b-x)**(1-gamma)
(romberg_powerLawUpper).
To summarize the meaning of gamma:
· If f(x) ~ x**(-a) (0 < a < 1), use gamma = a
· If f'(x) ~ x**(-b) (0 < b < 1), use gamma = b
romberg_sqrtSingLower and romberg_sqrtSingUpper
These procedures behave identically to romberg_powerLawLower and
romberg_powerLawUpper for the common case of gamma=0.5; that is,
they integrate a function with an inverse square root singular‐
ity at one end of the interval. They have a simpler implementa‐
tion involving square roots rather than arbitrary powers.
romberg_expLower and romberg_expUpper
These procedures are for integrating a function that grows or
decreases exponentially over a half-infinite interval.
romberg_expLower handles exponentially growing functions, and
allows the lower limit of integration to be an arbitrarily large
negative number. romberg_expUpper handles exponentially decay‐
ing 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.
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 exam‐
ple here.
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 exam‐
ple), and we want to integrate it over the interval (-1,1). The denom‐
inator 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).
We can make a function g that accepts an arbitrary function f and the
parameter u, and computes this new integrand.
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 f from a to b is the same as integrating g from asin(a)
to asin(b). It's a little tricky to get f consistently evaluated in
the caller's scope; the following procedure does it.
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($b) }]]]
}
This romberg_sine procedure will do any function with sqrt(1-x*x) in
the denominator. Our sample function is f(x)=exp(x)/sqrt(1-x*x):
proc f { x } {
expr { exp($x) / sqrt( 1. - $x*$x ) }
}
Integrating it is a matter of applying romberg_sine as we would any of
the other romberg procedures:
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
BUGS, IDEAS, FEEDBACK
This document, and the package it describes, will undoubtedly contain
bugs and other problems. Please report such in the category math ::
calculus of the Tcllib SF Trackers [http://source‐
forge.net/tracker/?group_id=12883]. Please also report any ideas for
enhancements you may have for either package and/or documentation.
SEE ALSO
math::calculus, math::interpolate
CATEGORY
Mathematics
COPYRIGHT
Copyright (c) 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/>
math 0.6 math::calculus::romberg(3tcl)
