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// Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
// Copyright (C) 2013 - Scilab Enterprises - Paul Bignier
//
// This file must be used under the terms of the CeCILL.
// This source file is licensed as described in the file COPYING, which
// you should have received as part of this distribution. The terms
// are also available at
// http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt
//
// bicgstab --
// BICG solves the linear system %Ax=b using the BiConjugate Gradient Stabilized method.
// If M is given, it is used as a preconditionning matrix.
// If both M and M2 are given, the matrix M * M2 is used as a preconditionning
// matrix.
//
// input %A REAL matrix or a function y=Ax(x) which computes y=%A*x for a given x
// b REAL right hand side vector
// tol, optional REAL error tolerance (default: 1e-8)
// maxIter, optional INTEGER maximum number of iterations (default: size(%b))
// %M, optional REAL preconditioner matrix (default: none)
// %M2, optional REAL preconditioner matrix (default: none)
// x0, optional REAL initial guess vector (default: the zero vector)
// verbose, optional INTEGER set to 1 to enable verbose logging (default : 0)
//
// output x REAL solution vector
// resNorm REAL final relative norm of the residual
// iter INTEGER number of iterations performed
// resVec REAL residual vector
//
// References
//
// "Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems"
// by Henk van der Vorst.
//
// http://epubs.siam.org/doi/abs/10.1137/0913035
// http://dl.acm.org/citation.cfm?id=131916.131930&coll=DL&dl=GUIDE&CFID=372773884&CFTOKEN=56630250
// http://mathworld.wolfram.com/BiconjugateGradientStabilizedMethod.html
//
// Notes
// This script was originally a matlab > scilab translation of the bicgstab.m
// script from http://www.netlib.org/templates/matlab
//
// The input / output arguments of this command are the same as Matlab's bicgstab command.
//
function [x, resNorm, iter, resVec] = %bicgstab(%A, %b, tol, maxIter, %M, %M2, x0, verbose )
// Initialization
bnrm2 = norm(%b);
if (verbose==1) then
printf(gettext("Norm of right-hand side : %s\n"), string(bnrm2));
end
if (bnrm2 == 0) then
if (verbose==1) then
printf(gettext("Special processing where the right-hand side is zero.\n"));
end
// When rhs is 0, there is a trivial solution : x=0
x = zeros(%b);
resNorm = 0;
resVec = resNorm;
else
x = x0;
// r = %b - %A*x;
if (matrixType ==1),
r = %b - %A*x;
r2 = r;
else
r = %b - %A(x,Aargs(:));
r2 = r;
end
resNorm = norm(r) / bnrm2;
resVec = resNorm;
end
if (verbose==1) then
printf(gettext(" Type of preconditionning #1 : %d\n"),precondType);
printf(gettext(" Type of preconditionning #2 : %d\n"),precondBis);
end
// Begin iteration
// Distinguish the number of iterations processed from the currentiter index
iter = 0
for currentiter = 1:maxIter
if (resNorm <= tol) then
if (verbose==1) then
printf(gettext(" New residual = %s < tol = %s => break\n"),string(resNorm),string(tol));
end
break;
end
iter = iter + 1
if (verbose==1) then
printf(gettext(" Iteration #%s/%s residual : %s\n"),string(currentiter),string(maxIter),string(resNorm));
printf(" x=\n");
disp(x);
end
rho = r2'*r;
if (rho == 0) then
break;
end
if (currentiter > 1) then
bet = (rho/rho_old)*(alp/ome);
p = r + bet*(p-ome*v);
else
p = r;
end
// Solve M' M2' P = p
if %M == [] & %M2 == [] then
P = p;
elseif %M2 == [] then
// Compute r so that M' P = p
if (precondType == 1) then
P = %M \ p;
elseif (precondType == 2) then
P = %M(p,Margs(:));
else
P = p;
end
else
// Compute r so that M' M2' P = p
if (precondBis == 1) then
P = %M \ p;
P = %M2 \ p;
elseif (precondBis == 2) then
P = %M(p,Margs(:));
P = %M2(p,M2args(:));
else
P = p;
end
end
// v = %A*P;
if (matrixType == 1),
v = %A*P;
else
v = %A(P);
end
alp = rho / (r2'*v);
s = r - alp*v;
// Check for convergence
if (norm(s) <= tol) then
x = x+alp*P;
resNorm = norm(s) / bnrm2;
if (verbose==1) then
printf(gettext(" New residual = %s < tol = %s => break\n"),string(resNorm),string(tol));
end
resVec = [resVec;resNorm];
break;
end
// Solve M M2 S = s
if %M == [] & %M2 == [] then
S = s;
elseif %M2 == [] then
// Compute r so that M S = s
if (precondType == 1) then
S = %M \ s;
elseif (precondType == 2) then
S = %M(s,Margs(:));
else
S = s;
end
else
// Compute r so that M M2 S = s
if (precondBis == 1) then
S = %M \ s;
S = %M2 \ s;
elseif (precondBis == 2) then
S = %M(s,Margs(:));
S = %M2(s,M2args(:));
else
S = s;
end
end
// t = %A*S;
if (matrixType == 1),
t = %A*S;
else
t = %A(S);
end
ome = (t'*s)/(t'*t);
x = x + alp*P+ome*S;
r = s - ome*t;
resNorm = norm(r) / bnrm2;
// Caution : transform the scalar resVec into vector resVec !
resVec = [resVec;resNorm];
rho_old = rho;
end
// Test for convergence
if (resNorm > tol) then
if (verbose==1) then
printf(gettext("Final residual = %s > tol =%s\n"),string(resNorm),string(tol));
printf(gettext("Algorithm fails\n"));
end
flag = 1;
if (lhs < 2) then
warning(msprintf(gettext("%s: Convergence error.\n"),"bicgstab"));
end
else
flag = 0;
if (verbose==1) then
printf(gettext("Algorithm pass\n"));
end
end
endfunction
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