From octave-sources-request at bevo dot che dot wisc dot edu Mon Jun 1 12:02:37 1998 Subject: iterative solution of linear systems From: Mario Storti To: octave-sources at bevo dot che dot wisc dot edu cc: mstorti at minerva dot unl dot edu dot ar Date: Mon, 01 Jun 1998 14:05:19 -0300 ------- =_aaaaaaaaaa0 Content-Type: text/plain; charset="us-ascii" Content-ID: <19256 dot 896720658 dot 1 at minerva> Hi all! Here there are some Octave scripts for solving non-symmetric systems of linear equations iteratively with the Conjugate Gradient on Normal Equations method, and the Bi-Conjugate Gradient method. They have the following features: * Only real (not complex) systems allowed (this may be extended in the future). * Multiple right hand sides allowed * You can pass indistinctly both the coefficient matrix as well as the string naming a function that returns `A*x' when passed `x' as argument. * You can specify both a tolerance to the norm of the residual, as well as the maximum number of iterations allowed. * Preconditionings must be defined externally by the user * A short LaTeX paper with mode details is included. * Simple installation. Copy the script files bicg.m and cgne.m to somewhere in the LOADPATH * Some examples included. Below are the two scripts, and attached and encoded the full package (including doc and examples). Mario Storti %%%%%%<>%%%%%%<>%%%%%%<>%%%%%%<>%%%%%%<>%%%%%%<>%%%%%%<>%%%%%%<>%%%%%%<>% Mario Alberto Storti | Fax: (54)(42) 55.09.44 | Centro Internacional de Metodos Computacionales| Tel: (54)(42) 55.91.75 | en Ingenieria - CIMEC |........................| INTEC, Guemes 3450 - 3000 Santa Fe, Argentina | Reply: mstorti at minerva dot unl dot edu dot ar,| http://venus dot unl dot edu dot ar/gtm-eng dot html http://venus.unl.edu.ar/gtm-eng.html | ------- =_aaaaaaaaaa0 Content-Type: text/plain; name="cgne.m"; charset="us-ascii" Content-ID: <19256 dot 896720658 dot 2 at minerva> Content-Description: octave file ## Copyright (C) 1998, Mario A. Storti ## ## 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, 59 Temple Place - ## Suite 330, Boston, MA 02111-1307, USA. ##usage: [x,rhist]= cgne (A,b,x0,toler) ## ## Solve the linear system A*x=b iteratively by the method of Conjugate ## Gradient on Normal Equations. ## ## A (matrix/string) : The matrix of coefficients of the linear ## system. ## If it is a string, say "atimesx" ## then it should be ## the name of an octave function returning A*x ## when passed x as an argument. ## atimesx(x) should return A*x, and ## atimes(x,"t") should return A'*x. ## b (vector/matrix) : right hand side. If it is a matrix, then ## it represents multiple right hand sides, ## and x returns a matrix containing the ## corresponding solutions. ## x0 (vector, optional) : initial guess for x ## tolerance ([scalar scalar]) ## : tolerance on the residual ## (default: 1e-10), maximum number of ## iterations (default: infinity). ## x (vector) : solution of the linear system ## to prescribed tolerance ## rhist (vector) : history of convergence. rhist(k) returns the ## residual at iteration k. ## Author: Mario Storti ## Keywords: linear systems, iterative methods ## Created: Sun May 31 19:54:39 ART 1998 function [x,rhist]= cgne (rname_,b,x0,tolerance) #$Id: cgne.m,v 1.4 1998/06/01 16:12:41 mstorti Exp mstorti $ if nargin<=1 || nargin >4 usage ("[x,rhist]= cgne (\"atimesx\",b,x0,tolerance)"); return endif if nargin<=3 tolerance=[1e-10 -1]; endif if length(tolerance)==1 tolerance(2)=-1; endif if all(tolerance<0) error("one of the components of tolerance must be greater than zero") endif if nargin<=2 x0=zeros(size(b)); endif if any(size(x0)!=size(b)) error("x0 and b should be of the same size") endif ## Initialization s=x0; beta=sqrt(sum(b.*b)); if isstr(rname_) t1=feval(rname_,s); t2=b-t1; r1=feval(rname_,t2,"t"); else t1=rname_*s; t2=b-t1; r1=rname_'*t2; endif # r2=feval(preco_,r1); r2=r1; r3=-r2; D0=sum(r1.*r2); Ogb=sqrt(D0)/beta; ## Main loop iter=0; rhist=[]; if (max(Ogb)0) x=x0; rhist=Ogb; return endif while 1 iter++; if isstr(rname_) t1=feval(rname_,r3); t2=feval(rname_,t1,"t"); else t1=rname_*r3; t2=rname_'*t1; endif sigma=sum(r3.*t2); lambda=D0./sigma; s=s+rightscal(r3,lambda'); r1=r1+rightscal(t2,lambda'); # t2=feval(preco_,r1); t2=r1; D1=sum(r1.*t2); ## Energy norm of the residual Ogb=sqrt(D1)./beta; ## Save convergence history rhist=[rhist;Ogb]; ## Check convergence if (max(Ogb)0) || \ (iter>tolerance(2) && tolerance(2)>0) break; endif alpha=D1./D0; D0=D1; r3=rightscal(r3,alpha')-t2; endwhile x=-s; endfunction ------- =_aaaaaaaaaa0 Content-Type: text/plain; name="bicg.m"; charset="us-ascii" Content-ID: <19256 dot 896720658 dot 3 at minerva> Content-Description: octave file ## Copyright (C) 1998, Mario A. Storti ## ## 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, 59 Temple Place - ## Suite 330, Boston, MA 02111-1307, USA. ##usage: [x,rhist]= bicg (A,b,x0,tolerance) ## ## Solve the linear system A*x=b iteratively by the method ## of BI-Conjugate Gradient. ## ## A (matrix/string) : The matrix of coefficients of the linear ## system. ## If it is a string, say "atimesx" ## then it should be ## the name of an octave function returning A*x ## when passed x as an argument. ## atimesx(x) should return A*x, and ## atimes(x,"t") should return A'*x. ## b (vector/matrix) : right hand side. If it is a matrix, then ## it represents multiple right hand sides, ## and x returns a matrix containing the ## corresponding solutions. ## x0 (vector, optional) : initial guess for x ## tolerance ([scalar scalar]) ## : tolerance on the residual ## (default: 1e-10), maximum number of ## iterations (default: infinity). ## x (vector) : solution of the linear system ## to prescribed tolerance ## rhist (vector) : history of convergence. rhist(k) returns the ## residual at iteration k. ## Author: Mario Storti ## Keywords: linear systems, iterative methods ## Created: Sun May 31 19:54:39 ART 1998 function [s,rhist]= bicg (rname_,b,x0,tolerance) #$Id: bicg.m,v 1.2 1998/06/01 16:12:35 mstorti Exp mstorti $ ## Check consistency of arguments if nargin<=1 || nargin >4 usage ("[x,rhist]= bicg (\"atimesx\",b,x0,tolerance)"); return endif if nargin<=3 tolerance=1e-10; endif if nargin<=2 x0=zeros(size(b)); endif if any(size(x0)!=size(b)) error("x0 and b should be of the same size") endif ## 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