Introduction to Numerical Numerical ELLIPT
The following is obsolete. To make an interface to fortran libraries in the current MAXIMA look at the examples in "maxima/src/fortdef.lsp" - The IMSL version of Romberg integration is now available in Macsyma. For documentation, Do PRINTFILE(DCADRE,USAGE,IMSL1); . For a demo, do batch("dcadre.mc"); This is a numerical integration package using cautious, adaptive Romberg extrapolation. The DCADRE package is written to call the IMSL fortran library routine DCADRE. This is documentation for that program. Send bugs/comments to KMP To load this package, do
For a demo of this package, do
The worker function takes the following syntax: IMSL_ROMBERG(fn,low,hi) where fn is a function of 1 argument; low and hi should be the lower and upper bounds of integration. fn must return floating point values. IMSL_ROMBERG(exp,var,low,hi) where exp should be integrated over the range var=low to hi. The result of evaluating exp must always be a floating point number. FAST_IMSL_ROMBERG(fn,low,hi) This function does no error checking but may achieve a speed gain over the IMSL_ROMBERG function. It expects that fn is a Lisp function (or translated Macsyma function) which accepts a floating point argument and that it always returns a floating point value.
Returns either [SUCCESS, answer, error] where answer is the result of the integration and error is the estimated bound on the absolute error of the output, DCADRE, as described in PURPOSE below. or [WARNING, n, answer, error] where n is a warning code, answer is the answer, and error is the estimated bound on the absolute error of the output, DCADRE, as described in PURPOSE below. The following warnings may occur: 65 = One or more singularities were successfully handled. 66 = In some subinterval(s), the estimate of the integral was accepted merely because the estimated error was small, even though no regular behavior was recognized. or [ERROR, errorcode] where error code is the IMSL-generated error code. The following error codes may occur: 131 = Failure due to insufficient internal working storage. 132 = Failure. This may be due to too much noise in function (relative to the given error requirements) or due to an ill-behaved integrand. 133 = RERR is greater than 0.1 or less than 0.0 or is too small for the precision of the machine.
The following flags have an influence upon the operation of IMSL_ROMBERG --
ROMBERG_AERR [Default 1.0E-5] -- Desired absolute error in answer.
ROMBERG_RERR [Default 0.0] -- Desired relative error in the answer.
Note: If IMSL signals an error, a message will be printed on the user's console stating the nature of the error. (This error message may be supressed by setting IMSLVERBOSE to FALSE.)
Note: Because this uses a translated Fortran routine, it may not be recursively invoked. It does not call itself, but the user should be aware that he may not type ^A in the middle of an IMSL_ROMBERG computation, begin another calculation using the same package, and expect to win -- IMSL_ROMBERG will complain if it was already doing one project when you invoke it. This should cause minimal problems.
Purpose (modified version of the IMSL documentation) ----------------------------------------------------
DCADRE attempts to solve the following problem: Given a real-valued function F of one argument, two real numbers A and B, find a number
DCADRE such that:
| / B | [ | / B | ] | [ | [ | [ | ] | I F(x)dx - DCADRE | <= max [ ROMBERG_AERR, ROMBERG_RERR * | I F(x)dx | ] | ] | [ | ] | ] | / A | [ | / A | ]
Algorithm (modified version of the IMSL documentation)
This routine uses a scheme whereby DCADRE is computed as the sum of estimates for the integral of F(x) over suitably chosen subintervals of the given interval of integration. Starting with the interval of integration itself as the first such subinterval, cautious Romberg extrapolation is used to find an acceptable estimate on a given subinterval. If this attempt fails, the subinterval is divided into two subintervals of equal length, each of which is considered separately. Programming Notes (modified version of the IMSL documentation)
For references on this technique, see de Boor, Calr, ``CADRE: An Algorithm for Numerical Quadrature,'' Mathematical Software (John R. Rice, Ed.), New York, Academic Press, 1971, Chapter 7.Introduction to Numerical Numerical ELLIPT