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author | jwe |
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date | Thu, 01 Jun 2006 19:05:32 +0000 |
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## Copyright (C) 1998 Walter Gautschi ## ## This file is part of Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 2, or (at your option) ## any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, write to the Free ## Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA ## 02110-1301, USA. ## -*- texinfo -*- ## @deftypefn {Function File} {@var{q} =} quadl (@var{f}, @var{a}, @var{b}) ## @deftypefnx {Function File} {@var{q} =} quadl (@var{f}, @var{a}, @var{b}, @var{tol}) ## @deftypefnx {Function File} {@var{q} =} quadl (@var{f}, @var{a}, @var{b}, @var{tol}, @var{trace}) ## @deftypefnx {Function File} {@var{q} =} quadl (@var{f}, @var{a}, @var{b}, @var{tol}, @var{trace}, @var{p1}, @var{p2}, @dots{}) ## ## Numerically evaluate integral using adaptive Lobatto rule. ## @code{quadl (@var{f}, @var{a}, @var{b})} approximates the integral of ## @code{@var{f}(@var{x})} to machine precision. @var{f} is either a ## function handle, inline function or string containing the name of ## the function to evaluate. The function @var{f} must return a vector ## of output values if given a vector of input values. ## ## If defined, @var{tol} defines the relative tolerance to which to ## which to integrate @code{@var{f}(@var{x})}. While if @var{trace} is ## defined, displays the left end point of the current interval, the ## interval length, and the partial integral. ## ## Additional arguments @var{p1}, etc, are passed directly to @var{f}. ## To use default values for @var{tol} and @var{trace}, one may pass ## empty matrices. ## ## Reference: W. Gander and W. Gautschi, 'Adaptive Quadrature - ## Revisited', BIT Vol. 40, No. 1, March 2000, pp. 84--101. ## @url{http://www.inf.ethz.ch/personal/gander/} ## ## @end deftypefn ## Author: Walter Gautschi ## Date: 08/03/98 ## Reference: Gander, Computermathematik, Birkhaeuser, 1992. ## 2003-08-05 Shai Ayal ## * permission from author to release as GPL ## 2004-02-10 Paul Kienzle ## * renamed to quadl for compatibility ## * replace global variable terminate2 with local function need_warning ## * add paper ref to docs function Q = quadl(f,a,b,tol,trace,varargin) need_warning(1); if (nargin < 4) tol=[]; endif if (nargin < 5) trace = []; endif if (isempty(tol)) tol = eps; endif if (isempty(trace)) trace=0; endif if (tol < eps) tol = eps; endif m = (a+b)/2; h = (b-a)/2; alpha = sqrt(2/3); beta = 1/sqrt(5); x1 = .942882415695480; x2 = .641853342345781; x3 = .236383199662150; x = [a,m-x1*h,m-alpha*h,m-x2*h,m-beta*h,m-x3*h,m,m+x3*h,... m+beta*h,m+x2*h,m+alpha*h,m+x1*h,b]; y = feval(f,x,varargin{:}); fa = y(1); fb = y(13); i2 = (h/6)*(y(1)+y(13)+5*(y(5)+y(9))); i1 = (h/1470)*(77*(y(1)+y(13))+432*(y(3)+y(11))+ ... 625*(y(5)+y(9))+672*y(7)); is = h*(.0158271919734802*(y(1)+y(13))+.0942738402188500 ... *(y(2)+y(12))+.155071987336585*(y(3)+y(11))+ ... .188821573960182*(y(4)+y(10))+.199773405226859 ... *(y(5)+y(9))+.224926465333340*(y(6)+y(8))... +.242611071901408*y(7)); s = sign(is); if (s == 0), s=1; endif erri1 = abs(i1-is); erri2 = abs(i2-is); R = 1; if (erri2 != 0) R = erri1/erri2; endif if (R > 0 && R < 1) tol=tol/R; endif is = s*abs(is)*tol/eps; if (is == 0) is = b-a; endif Q = adaptlobstp(f,a,b,fa,fb,is,trace,varargin{:}); endfunction ## ADAPTLOBSTP Recursive function used by QUADL. ## ## Q = ADAPTLOBSTP('F',A,B,FA,FB,IS,TRACE) tries to ## approximate the integral of F(X) from A to B to ## an appropriate relative error. The argument 'F' is ## a string containing the name of f. The remaining ## arguments are generated by ADAPTLOB or by recursion. ## ## Walter Gautschi, 08/03/98 function Q = adaptlobstp(f,a,b,fa,fb,is,trace,varargin) h = (b-a)/2; m = (a+b)/2; alpha = sqrt(2/3); beta = 1/sqrt(5); mll = m-alpha*h; ml = m-beta*h; mr = m+beta*h; mrr = m+alpha*h; x = [mll,ml,m,mr,mrr]; y = feval(f,x,varargin{:}); fmll = y(1); fml = y(2); fm = y(3); fmr = y(4); fmrr = y(5); i2 = (h/6)*(fa+fb+5*(fml+fmr)); i1 = (h/1470)*(77*(fa+fb)+432*(fmll+fmrr)+625*(fml+fmr) ... +672*fm); if ((is+(i1-i2) == is) || (mll <= a) || (b <= mrr)) if (((m <= a) || (b <= m)) && need_warning()) warning(['Interval contains no more machine number. ',... 'Required tolerance may not be met.']); need_warning(0); endif Q = i1; if (trace) disp([a b-a Q]); endif else Q = adaptlobstp(f,a,mll,fa,fmll,is,trace,varargin{:})+... adaptlobstp(f,mll,ml,fmll,fml,is,trace,varargin{:})+... adaptlobstp(f,ml,m,fml,fm,is,trace,varargin{:})+... adaptlobstp(f,m,mr,fm,fmr,is,trace,varargin{:})+... adaptlobstp(f,mr,mrr,fmr,fmrr,is,trace,varargin{:})+... adaptlobstp(f,mrr,b,fmrr,fb,is,trace,varargin{:}); endif endfunction function r = need_warning(v) persistent w = []; if (nargin == 0) r = w; else w = v; endif endfunction