


math::calculus::romberg(nTcl Math Libramath::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...?

_________________________________________________________________

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.





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math::calculus::romberg(nTcl Math Libramath::calculus::romberg(n)


       ::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;



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math::calculus::romberg(nTcl Math Libramath::calculus::romberg(n)


              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  pro-
       cess 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 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.



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math::calculus::romberg(nTcl Math Libramath::calculus::romberg(n)


       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:

              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 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.



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math::calculus::romberg(nTcl Math Libramath::calculus::romberg(n)


              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]



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math::calculus::romberg(nTcl Math Libramath::calculus::romberg(n)


       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/>







































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