| [n,b,x,nelt,ia,ja,a,isym,matvec,msolve,itol,tol,itmax,iter,err,ierr,iunit,r,r0,p,q,u,v1,v2,rwork,iwork]=scgs(n,b,x,nelt,ia,ja,a,isym,matvec,msolve,itol,tol,itmax,iter,err,ierr,iunit,r,r0,p,q,u,v1,v2,rwork,iwork); |
function [n,b,x,nelt,ia,ja,a,isym,matvec,msolve,itol,tol,itmax,iter,err,ierr,iunit,r,r0,p,q,u,v1,v2,rwork,iwork]=scgs(n,b,x,nelt,ia,ja,a,isym,matvec,msolve,itol,tol,itmax,iter,err,ierr,iunit,r,r0,p,q,u,v1,v2,rwork,iwork);
%***BEGIN PROLOGUE SCGS
%***PURPOSE Preconditioned BiConjugate Gradient Squared Ax=b Solver.
% Routine to solve a Non-Symmetric linear system Ax = b
% using the Preconditioned BiConjugate Gradient Squared
% method.
%***LIBRARY SLATEC (SLAP)
%***CATEGORY D2A4, D2B4
%***TYPE SINGLE PRECISION (SCGS-S, DCGS-D)
%***KEYWORDS BICONJUGATE GRADIENT, ITERATIVE PRECONDITION,
% NON-SYMMETRIC LINEAR SYSTEM, SLAP, SPARSE
%***AUTHOR Greenbaum, Anne, (Courant Institute)
% Seager, Mark K., (LLNL)
% Lawrence Livermore National Laboratory
% PO BOX 808, L-60
% Livermore, CA 94550 (510) 423-3141
% seager@llnl.gov
%***DESCRIPTION
%
% *Usage:
% INTEGER N, NELT, IA(NELT), JA(NELT), ISYM, ITOL, ITMAX
% INTEGER ITER, IERR, IUNIT, IWORK(USER DEFINED)
% REAL B(N), X(N), A(NELT), TOL, ERR, R(N), R0(N), P(N)
% REAL Q(N), U(N), V1(N), V2(N), RWORK(USER DEFINED)
% EXTERNAL MATVEC, MSOLVE
%
% CALL SCGS(N, B, X, NELT, IA, JA, A, ISYM, MATVEC,
% $ MSOLVE, ITOL, TOL, ITMAX, ITER, ERR, IERR, IUNIT,
% $ R, R0, P, Q, U, V1, V2, RWORK, IWORK)
%
% *Arguments:
% N :IN Integer
% Order of the Matrix.
% B :IN Real B(N).
% Right-hand side vector.
% X :INOUT Real X(N).
% On input X is your initial guess for solution vector.
% On output X is the final approximate solution.
% NELT :IN Integer.
% Number of Non-Zeros stored in A.
% IA :IN Integer IA(NELT).
% JA :IN Integer JA(NELT).
% A :IN Real A(NELT).
% These arrays contain the matrix data structure for A.
% It could take any form. See 'Description', below,
% for more details.
% ISYM :IN Integer.
% Flag to indicate symmetric storage format.
% If ISYM=0, all non-zero entries of the matrix are stored.
% If ISYM=1, the matrix is symmetric, and only the upper
% or lower triangle of the matrix is stored.
% MATVEC :EXT External.
% Name of a routine which performs the matrix vector multiply
% operation Y = A*X given A and X. The name of the MATVEC
% routine must be declared external in the calling program.
% The calling sequence of MATVEC is:
% CALL MATVEC( N, X, Y, NELT, IA, JA, A, ISYM )
% Where N is the number of unknowns, Y is the product A*X upon
% return, X is an input vector. NELT, IA, JA, A and ISYM
% define the SLAP matrix data structure: see Description,below.
% MSOLVE :EXT External.
% Name of a routine which solves a linear system MZ = R for Z
% given R with the preconditioning matrix M (M is supplied via
% RWORK and IWORK arrays). The name of the MSOLVE routine
% must be declared external in the calling program. The
% calling sequence of MSOLVE is:
% CALL MSOLVE(N, R, Z, NELT, IA, JA, A, ISYM, RWORK, IWORK)
% Where N is the number of unknowns, R is the right-hand side
% vector, and Z is the solution upon return. NELT, IA, JA, A
% and ISYM define the SLAP matrix data structure: see
% Description, below. RWORK is a real array that can be used
% to pass necessary preconditioning information and/or
% workspace to MSOLVE. IWORK is an integer work array for the
% same purpose as RWORK.
% ITOL :IN Integer.
% Flag to indicate type of convergence criterion.
% If ITOL=1, iteration stops when the 2-norm of the residual
% divided by the 2-norm of the right-hand side is less than TOL.
% This routine must calculate the residual from R = A*X - B.
% This is unnatural and hence expensive for this type of iter-
% ative method. ITOL=2 is *STRONGLY* recommended.
% If ITOL=2, iteration stops when the 2-norm of M-inv times the
% residual divided by the 2-norm of M-inv times the right hand
% side is less than TOL, where M-inv time a vector is the pre-
% conditioning step. This is the *NATURAL* stopping for this
% iterative method and is *STRONGLY* recommended.
% ITOL=11 is often useful for checking and comparing different
% routines. For this case, the user must supply the 'exact'
% solution or a very accurate approximation (one with an error
% much less than TOL) through a common block,
% COMMON /SSLBLK/ SOLN( )
% If ITOL=11, iteration stops when the 2-norm of the difference
% between the iterative approximation and the user-supplied
% solution divided by the 2-norm of the user-supplied solution
% is less than TOL.
% TOL :INOUT Real.
% Convergence criterion, as described above. (Reset if IERR=4.)
% ITMAX :IN Integer.
% Maximum number of iterations.
% ITER :OUT Integer.
% Number of iterations required to reach convergence, or
% ITMAX+1 if convergence criterion could not be achieved in
% ITMAX iterations.
% ERR :OUT Real.
% Error estimate of error in final approximate solution, as
% defined by ITOL.
% IERR :OUT Integer.
% Return error flag.
% IERR = 0 => All went well.
% IERR = 1 => Insufficient space allocated for WORK or IWORK.
% IERR = 2 => Method failed to converge in ITMAX steps.
% IERR = 3 => Error in user input.
% Check input values of N, ITOL.
% IERR = 4 => User error tolerance set too tight.
% Reset to 500*R1MACH(3). Iteration proceeded.
% IERR = 5 => Breakdown of the method detected.
% (r0,r) approximately 0.
% IERR = 6 => Stagnation of the method detected.
% (r0,v) approximately 0.
% IUNIT :IN Integer.
% Unit number on which to write the error at each iteration,
% if this is desired for monitoring convergence. If unit
% number is 0, no writing will occur.
% R :WORK Real R(N).
% R0 :WORK Real R0(N).
% P :WORK Real P(N).
% Q :WORK Real Q(N).
% U :WORK Real U(N).
% V1 :WORK Real V1(N).
% V2 :WORK Real V2(N).
% Real arrays used for workspace.
% RWORK :WORK Real RWORK(USER DEFINED).
% Real array that can be used for workspace in MSOLVE.
% IWORK :WORK Integer IWORK(USER DEFINED).
% Integer array that can be used for workspace in MSOLVE.
%
% *Description
% This routine does not care what matrix data structure is
% used for A and M. It simply calls the MATVEC and MSOLVE
% routines, with the arguments as described above. The user
% could write any type of structure and the appropriate MATVEC
% and MSOLVE routines. It is assumed that A is stored in the
% IA, JA, A arrays in some fashion and that M (or INV(M)) is
% stored in IWORK and RWORK in some fashion. The SLAP
% routines SSDBCG and SSLUCS are examples of this procedure.
%
% Two examples of matrix data structures are the: 1) SLAP
% Triad format and 2) SLAP Column format.
%
% =================== S L A P Triad format ===================
%
% In this format only the non-zeros are stored. They may
% appear in *ANY* order. The user supplies three arrays of
% length NELT, where NELT is the number of non-zeros in the
% matrix: (IA(NELT), JA(NELT), A(NELT)). For each non-zero
% the user puts the row and column index of that matrix
% element in the IA and JA arrays. The value of the non-zero
% matrix element is placed in the corresponding location of
% the A array. This is an extremely easy data structure to
% generate. On the other hand it is not too efficient on
% vector computers for the iterative solution of linear
% systems. Hence, SLAP changes this input data structure to
% the SLAP Column format for the iteration (but does not
% change it back).
%
% Here is an example of the SLAP Triad storage format for a
% 5x5 Matrix. Recall that the entries may appear in any order.
%
% 5x5 Matrix SLAP Triad format for 5x5 matrix on left.
% 1 2 3 4 5 6 7 8 9 10 11
% |11 12 0 0 15| A: 51 12 11 33 15 53 55 22 35 44 21
% |21 22 0 0 0| IA: 5 1 1 3 1 5 5 2 3 4 2
% | 0 0 33 0 35| JA: 1 2 1 3 5 3 5 2 5 4 1
% | 0 0 0 44 0|
% |51 0 53 0 55|
%
% =================== S L A P Column format ==================
%
% In this format the non-zeros are stored counting down
% columns (except for the diagonal entry, which must appear
% first in each 'column') and are stored in the real array A.
% In other words, for each column in the matrix put the
% diagonal entry in A. Then put in the other non-zero
% elements going down the column (except the diagonal) in
% order. The IA array holds the row index for each non-zero.
% The JA array holds the offsets into the IA, A arrays for the
% beginning of each column. That is, IA(JA(ICOL)),
% A(JA(ICOL)) points to the beginning of the ICOL-th column in
% IA and A. IA(JA(ICOL+1)-1), A(JA(ICOL+1)-1) points to the
% end of the ICOL-th column. Note that we always have JA(N+1)
% = NELT+1, where N is the number of columns in the matrix and
% NELT is the number of non-zeros in the matrix.
%
% Here is an example of the SLAP Column storage format for a
% 5x5 Matrix (in the A and IA arrays '|' denotes the end of a
% column):
%
% 5x5 Matrix SLAP Column format for 5x5 matrix on left.
% 1 2 3 4 5 6 7 8 9 10 11
% |11 12 0 0 15| A: 11 21 51 | 22 12 | 33 53 | 44 | 55 15 35
% |21 22 0 0 0| IA: 1 2 5 | 2 1 | 3 5 | 4 | 5 1 3
% | 0 0 33 0 35| JA: 1 4 6 8 9 12
% | 0 0 0 44 0|
% |51 0 53 0 55|
%
% *Cautions:
% This routine will attempt to write to the Fortran logical output
% unit IUNIT, if IUNIT ~= 0. Thus, the user must make sure that
% this logical unit is attached to a file or terminal before calling
% this routine with a non-zero value for IUNIT. This routine does
% not check for the validity of a non-zero IUNIT unit number.
%
%***SEE ALSO SSDCGS, SSLUCS
%***REFERENCES 1. P. Sonneveld, CGS, a fast Lanczos-type solver
% for nonsymmetric linear systems, Delft University
% of Technology Report 84-16, Department of Mathe-
% matics and Informatics, Delft, The Netherlands.
% 2. E. F. Kaasschieter, The solution of non-symmetric
% linear systems by biconjugate gradients or conjugate
% gradients squared, Delft University of Technology
% Report 86-21, Department of Mathematics and Informa-
% tics, Delft, The Netherlands.
% 3. Mark K. Seager, A SLAP for the Masses, in
% G. F. Carey, Ed., Parallel Supercomputing: Methods,
% Algorithms and Applications, Wiley, 1989, pp.135-155.
%***ROUTINES CALLED ISSCGS, R1MACH, SAXPY, SDOT
%***REVISION HISTORY (YYMMDD)
% 871119 DATE WRITTEN
% 881213 Previous REVISION DATE
% 890915 Made changes requested at July 1989 CML Meeting. (MKS)
% 890921 Removed TeX from comments. (FNF)
% 890922 Numerous changes to prologue to make closer to SLATEC
% standard. (FNF)
% 890929 Numerous changes to reduce SP/DP differences. (FNF)
% 891004 Added new reference.
% 910411 Prologue converted to Version 4.0 format. (BAB)
% 910502 Removed MATVEC and MSOLVE from ROUTINES CALLED list. (FNF)
% 920407 COMMON BLOCK renamed SSLBLK. (WRB)
% 920511 Added complete declaration section. (WRB)
% 920929 Corrected format of references. (FNF)
% 921019 Changed 500.0 to 500 to reduce SP/DP differences. (FNF)
% 921113 Corrected C***CATEGORY line. (FNF)
%***end PROLOGUE SCGS
% .. Scalar Arguments ..
% .. Array Arguments ..
persistent ak akm bk bnrm fuzz i k rhon rhonm1 sigma solnrm tolmin ;
rwork_shape=size(rwork);rwork=reshape(rwork,1,[]);
iwork_shape=size(iwork);iwork=reshape(iwork,1,[]);
% .. subroutine Arguments ..
% .. Local Scalars ..
if isempty(ak), ak=0; end;
if isempty(akm), akm=0; end;
if isempty(bk), bk=0; end;
if isempty(bnrm), bnrm=0; end;
if isempty(fuzz), fuzz=0; end;
if isempty(rhon), rhon=0; end;
if isempty(rhonm1), rhonm1=0; end;
if isempty(sigma), sigma=0; end;
if isempty(solnrm), solnrm=0; end;
if isempty(tolmin), tolmin=0; end;
if isempty(i), i=0; end;
if isempty(k), k=0; end;
% .. External Functions ..
% .. External Subroutines ..
% .. Intrinsic Functions ..
%***FIRST EXECUTABLE STATEMENT SCGS
%
% Check some of the input data.
%
iter = 0;
ierr = 0;
if( n<1 )
ierr = 3;
else;
tolmin = 500.*r1mach(3);
if( tol<tolmin )
tol = tolmin;
ierr = 4;
end;
%
% Calculate initial residual and pseudo-residual, and check
% stopping criterion.
[n,x,r,nelt,ia,ja,a,isym]=matvec(n,x,r,nelt,ia,ja,a,isym);
for i = 1 : n;
v1(i) = r(i) - b(i);
end; i = fix(n+1);
[n,v1,r,nelt,ia,ja,a,isym,rwork,iwork]=msolve(n,v1,r,nelt,ia,ja,a,isym,rwork,iwork);
%
if( isscgs(n,b,x,nelt,ia,ja,a,isym,matvec,msolve,itol,tol,itmax,iter,err,ierr,iunit,r,r0,p,q,u,v1,v2,rwork,iwork,ak,bk,bnrm,solnrm)==0 )
if( ierr==0 )
%
% Set initial values.
%
fuzz = r1mach(3).^2;
for i = 1 : n;
r0(i) = r(i);
end; i = fix(n+1);
rhonm1 = 1;
%
% ***** ITERATION LOOP *****
%
for k = 1 : itmax;
iter = fix(k);
%
% Calculate coefficient BK and direction vectors U, V and P.
[rhon ,n,r0,dumvar4,r]=sdot(n,r0,1,r,1);
if( abs(rhonm1)<fuzz )
%
% Breakdown of method detected.
ierr = 5;
rwork_shape=zeros(rwork_shape);rwork_shape(:)=rwork(1:numel(rwork_shape));rwork=rwork_shape;
iwork_shape=zeros(iwork_shape);iwork_shape(:)=iwork(1:numel(iwork_shape));iwork=iwork_shape;
return;
else;
bk = rhon./rhonm1;
if( iter==1 )
for i = 1 : n;
u(i) = r(i);
p(i) = r(i);
end; i = fix(n+1);
else;
for i = 1 : n;
u(i) = r(i) + bk.*q(i);
v1(i) = q(i) + bk.*p(i);
end; i = fix(n+1);
for i = 1 : n;
p(i) = u(i) + bk.*v1(i);
end; i = fix(n+1);
end;
%
% Calculate coefficient AK, new iterate X, Q
[n,p,v2,nelt,ia,ja,a,isym]=matvec(n,p,v2,nelt,ia,ja,a,isym);
[n,v2,v1,nelt,ia,ja,a,isym,rwork,iwork]=msolve(n,v2,v1,nelt,ia,ja,a,isym,rwork,iwork);
[sigma ,n,r0,dumvar4,v1]=sdot(n,r0,1,v1,1);
if( abs(sigma)<fuzz )
%
% Stagnation of method detected.
ierr = 6;
rwork_shape=zeros(rwork_shape);rwork_shape(:)=rwork(1:numel(rwork_shape));rwork=rwork_shape;
iwork_shape=zeros(iwork_shape);iwork_shape(:)=iwork(1:numel(iwork_shape));iwork=iwork_shape;
return;
else;
ak = rhon./sigma;
akm = -ak;
for i = 1 : n;
q(i) = u(i) + akm.*v1(i);
end; i = fix(n+1);
for i = 1 : n;
v1(i) = u(i) + q(i);
end; i = fix(n+1);
% X = X - ak*V1.
[n,akm,v1,dumvar4,x]=saxpy(n,akm,v1,1,x,1);
% -1
% R = R - ak*M *A*V1
[n,v1,v2,nelt,ia,ja,a,isym]=matvec(n,v1,v2,nelt,ia,ja,a,isym);
[n,v2,v1,nelt,ia,ja,a,isym,rwork,iwork]=msolve(n,v2,v1,nelt,ia,ja,a,isym,rwork,iwork);
[n,akm,v1,dumvar4,r]=saxpy(n,akm,v1,1,r,1);
%
% check stopping criterion.
if( isscgs(n,b,x,nelt,ia,ja,a,isym,matvec,msolve,itol,tol,itmax,iter,err,ierr,iunit,r,r0,p,q,u,v1,v2,rwork,iwork,ak,bk,bnrm,solnrm)~=0 )
rwork_shape=zeros(rwork_shape);rwork_shape(:)=rwork(1:numel(rwork_shape));rwork=rwork_shape;
iwork_shape=zeros(iwork_shape);iwork_shape(:)=iwork(1:numel(iwork_shape));iwork=iwork_shape;
return;
end;
%
% Update RHO.
rhonm1 = rhon;
end;
end;
end; k = fix(itmax+1);
%
% ***** end of loop *****
% Stopping criterion not satisfied.
iter = fix(itmax + 1);
ierr = 2;
rwork_shape=zeros(rwork_shape);rwork_shape(:)=rwork(1:numel(rwork_shape));rwork=rwork_shape;
iwork_shape=zeros(iwork_shape);iwork_shape(:)=iwork(1:numel(iwork_shape));iwork=iwork_shape;
return;
end;
end;
end;
%------------- LAST LINE OF SCGS FOLLOWS ----------------------------
rwork_shape=zeros(rwork_shape);rwork_shape(:)=rwork(1:numel(rwork_shape));rwork=rwork_shape;
iwork_shape=zeros(iwork_shape);iwork_shape(:)=iwork(1:numel(iwork_shape));iwork=iwork_shape;
end %subroutine scgs
%DECK SCHDC
|
|
