math::calculus::romberg(n)     Tcl Math Library     math::calculus::romberg(n)



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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...?

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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
              singularity 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
              singularity 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
              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
              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.

       a      Lower limit of the region of integration.

       b      Upper limit of the region of integration.  For the
              romberg_sqrtSingLower, romberg_sqrtSingUpper,
              romberg_powerLawLower, 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
              functions 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
       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.

       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
       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.

       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
       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.

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
              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.

       romberg_powerLawLower and 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, 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
              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 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:

              o      If f(x) ~ x**(-a) (0 < a < 1), use gamma = a

              o      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
              singularity at one end of the interval.  They have a simpler
              implementation 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
              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.

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.

       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).

       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://sourceforge.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(n)
