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[Maxima-commits] CVS: maxima/doc/info Numerical.texi,1.9,1.10
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From: Robert D. <rob...@us...> - 2004-11-04 15:32:34
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Update of /cvsroot/maxima/maxima/doc/info In directory sc8-pr-cvs1.sourceforge.net:/tmp/cvs-serv23209/doc/info Modified Files: Numerical.texi Log Message: Strike out description of obsolete function DCADRE, move it to None.texi. Index: Numerical.texi =================================================================== RCS file: /cvsroot/maxima/maxima/doc/info/Numerical.texi,v retrieving revision 1.9 retrieving revision 1.10 diff -u -d -r1.9 -r1.10 --- Numerical.texi 4 Nov 2004 05:12:25 -0000 1.9 +++ Numerical.texi 4 Nov 2004 15:32:19 -0000 1.10 @@ -1,151 +1,14 @@ @menu * Introduction to Numerical:: -* DCADRE:: * FOURIER:: * NDIFFQ:: * Definitions for Numerical:: @end menu -@node Introduction to Numerical, DCADRE, Numerical, Numerical +@node Introduction to Numerical, FOURIER, Numerical, Numerical @section Introduction to Numerical -@node DCADRE, FOURIER, Introduction to Numerical, Numerical -@section DCADRE -The following is obsolete and does not exist in Maxima 5.9. We leave -the documentation here for historical purposes. - -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 -Maxima. 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 -@example - LOADFILE("imsl")$ -@end example -For a demo of this package, do -@example - batch("dcadre.mc"); -@end example -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 Maxima 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: - -@example -| / B | [ | / B | ] -| [ | [ | [ | ] -| I F(x)dx - DCADRE | <= max [ ROMBERG_AERR, ROMBERG_RERR * | I F(x)dx | ] -| ] | [ | ] | ] -| / A | [ | / A | ] -@end example -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) - -@itemize @bullet -@item -1. DCADRE (the translated-Fortran base for IMSL_ROMBERG) can, in many cases, - handle jump discontinuities and certain algebraic discontinuities. See - reference for full details. -@item -2. The relative error parameter ROMBERG_RERR must be in the interval [0.0,0.1]. - For example, ROMBERG_RERR=0.1 indicates that the estimate of the intergral - is to be correct to one digit, where as ROMBERG_RERR=1.0E-4 calls for four - digits of accuracy. If DCADRE determines that the relative accuracy - requirement cannot be satisfied, IER is set to 133 (ROMBERG_RERR should be - large enough that, when added to 100.0, the result is a number greater than - 100.0 (this will not be true of very tiny floating point numbers due to - the nature of machine arithmetic)). -@item -3. The absolute error parameter, ROMBERG_AERR, should be nonnegative. In - order to give a reasonable value for ROMBERG_AERR, the user must know - the approximate magnitude of the integral being computed. In many cases, - it is satisfactory to use AERR=0.0. In this case, only the relative error - requirement is satisfied in the compuatation. -@item -4. We quote from the reference, ``A very cautious man would accept DCADRE - only if IER [the warning or error code] is 0 or 65. The merely reasonable - man would keep the faith even if IER is 66. The adventurous man is quite - often right in accepting DCADRE even if the IER is 131 or 132.'' Even when - IER is not 0, DCADRE returns the best estimate that has been computed. -@end itemize - -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. - -@node FOURIER, NDIFFQ, DCADRE, Numerical +@node FOURIER, NDIFFQ, Introduction to Numerical, Numerical @section FOURIER - There is a Fast Fourier Transform package, do DESCRIBE(FFT) for details. There is also a Fourier Series package. It may be |