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/**************************************************************************
**
** Copyright (C) 1993 David E. Steward & Zbigniew Leyk, all rights reserved.
**
** Meschach Library
**
** This Meschach Library is provided "as is" without any express
** or implied warranty of any kind with respect to this software.
** In particular the authors shall not be liable for any direct,
** indirect, special, incidental or consequential damages arising
** in any way from use of the software.
**
** Everyone is granted permission to copy, modify and redistribute this
** Meschach Library, provided:
** 1. All copies contain this copyright notice.
** 2. All modified copies shall carry a notice stating who
** made the last modification and the date of such modification.
** 3. No charge is made for this software or works derived from it.
** This clause shall not be construed as constraining other software
** distributed on the same medium as this software, nor is a
** distribution fee considered a charge.
**
***************************************************************************/
/*
Conjugate gradient routines file
Uses sparse matrix input & sparse Cholesky factorisation in pccg().
All the following routines use routines to define a matrix
rather than use any explicit representation
(with the exeception of the pccg() pre-conditioner)
The matrix A is defined by
VEC *(*A)(void *params, VEC *x, VEC *y)
where y = A.x on exit, and y is returned. The params argument is
intended to make it easier to re-use & modify such routines.
If we have a sparse matrix data structure
SPMAT *A_mat;
then these can be used by passing sp_mv_mlt as the function, and
A_mat as the param.
*/
#include <stdio.h>
#include <math.h>
#include "matrix.h"
#include "sparse.h"
static char rcsid[] = "$Id: conjgrad.c,v 1.4 1994/01/13 05:36:45 des Exp $";
/* #define MAX_ITER 10000 */
static int max_iter = 10000;
int cg_num_iters;
/* matrix-as-routine type definition */
/* #ifdef ANSI_C */
/* typedef VEC *(*MTX_FN)(void *params, VEC *x, VEC *out); */
/* #else */
typedef VEC *(*MTX_FN)();
/* #endif */
#ifdef ANSI_C
VEC *spCHsolve(SPMAT *,VEC *,VEC *);
#else
VEC *spCHsolve();
#endif
/* cg_set_maxiter -- sets maximum number of iterations if numiter > 1
-- just returns current max_iter otherwise
-- returns old maximum */
int cg_set_maxiter(numiter)
int numiter;
{
int temp;
if ( numiter < 2 )
return max_iter;
temp = max_iter;
max_iter = numiter;
return temp;
}
/* pccg -- solves A.x = b using pre-conditioner M
(assumed factored a la spCHfctr())
-- results are stored in x (if x != NULL), which is returned */
VEC *pccg(A,A_params,M_inv,M_params,b,eps,x)
MTX_FN A, M_inv;
VEC *b, *x;
double eps;
void *A_params, *M_params;
{
VEC *r = VNULL, *p = VNULL, *q = VNULL, *z = VNULL;
int k;
Real alpha, beta, ip, old_ip, norm_b;
if ( ! A || ! b )
error(E_NULL,"pccg");
if ( x == b )
error(E_INSITU,"pccg");
x = v_resize(x,b->dim);
if ( eps <= 0.0 )
eps = MACHEPS;
r = v_get(b->dim);
p = v_get(b->dim);
q = v_get(b->dim);
z = v_get(b->dim);
norm_b = v_norm2(b);
v_zero(x);
r = v_copy(b,r);
old_ip = 0.0;
for ( k = 0; ; k++ )
{
if ( v_norm2(r) < eps*norm_b )
break;
if ( k > max_iter )
error(E_ITER,"pccg");
if ( M_inv )
(*M_inv)(M_params,r,z);
else
v_copy(r,z); /* M == identity */
ip = in_prod(z,r);
if ( k ) /* if ( k > 0 ) ... */
{
beta = ip/old_ip;
p = v_mltadd(z,p,beta,p);
}
else /* if ( k == 0 ) ... */
{
beta = 0.0;
p = v_copy(z,p);
old_ip = 0.0;
}
q = (*A)(A_params,p,q);
alpha = ip/in_prod(p,q);
x = v_mltadd(x,p,alpha,x);
r = v_mltadd(r,q,-alpha,r);
old_ip = ip;
}
cg_num_iters = k;
V_FREE(p);
V_FREE(q);
V_FREE(r);
V_FREE(z);
return x;
}
/* sp_pccg -- a simple interface to pccg() which uses sparse matrix
data structures
-- assumes that LLT contains the Cholesky factorisation of the
actual pre-conditioner */
VEC *sp_pccg(A,LLT,b,eps,x)
SPMAT *A, *LLT;
VEC *b, *x;
double eps;
{ return pccg(sp_mv_mlt,A,spCHsolve,LLT,b,eps,x); }
/*
Routines for performing the CGS (Conjugate Gradient Squared)
algorithm of P. Sonneveld:
"CGS, a fast Lanczos-type solver for nonsymmetric linear
systems", SIAM J. Sci. & Stat. Comp. v. 10, pp. 36--52
*/
/* cgs -- uses CGS to compute a solution x to A.x=b
-- the matrix A is not passed explicitly, rather a routine
A is passed where A(x,Ax,params) computes
Ax = A.x
-- the computed solution is passed */
VEC *cgs(A,A_params,b,r0,tol,x)
MTX_FN A;
VEC *x, *b;
VEC *r0; /* tilde r0 parameter -- should be random??? */
double tol; /* error tolerance used */
void *A_params;
{
VEC *p, *q, *r, *u, *v, *tmp1, *tmp2;
Real alpha, beta, norm_b, rho, old_rho, sigma;
int iter;
if ( ! A || ! x || ! b || ! r0 )
error(E_NULL,"cgs");
if ( x->dim != b->dim || r0->dim != x->dim )
error(E_SIZES,"cgs");
if ( tol <= 0.0 )
tol = MACHEPS;
p = v_get(x->dim);
q = v_get(x->dim);
r = v_get(x->dim);
u = v_get(x->dim);
v = v_get(x->dim);
tmp1 = v_get(x->dim);
tmp2 = v_get(x->dim);
norm_b = v_norm2(b);
(*A)(A_params,x,tmp1);
v_sub(b,tmp1,r);
v_zero(p); v_zero(q);
old_rho = 1.0;
iter = 0;
while ( v_norm2(r) > tol*norm_b )
{
if ( ++iter > max_iter ) break;
/* error(E_ITER,"cgs"); */
rho = in_prod(r0,r);
if ( old_rho == 0.0 )
error(E_SING,"cgs");
beta = rho/old_rho;
v_mltadd(r,q,beta,u);
v_mltadd(q,p,beta,tmp1);
v_mltadd(u,tmp1,beta,p);
(*A)(A_params,p,v);
sigma = in_prod(r0,v);
if ( sigma == 0.0 )
error(E_SING,"cgs");
alpha = rho/sigma;
v_mltadd(u,v,-alpha,q);
v_add(u,q,tmp1);
(*A)(A_params,tmp1,tmp2);
v_mltadd(r,tmp2,-alpha,r);
v_mltadd(x,tmp1,alpha,x);
old_rho = rho;
}
cg_num_iters = iter;
V_FREE(p); V_FREE(q); V_FREE(r);
V_FREE(u); V_FREE(v);
V_FREE(tmp1); V_FREE(tmp2);
return x;
}
/* sp_cgs -- simple interface for SPMAT data structures */
VEC *sp_cgs(A,b,r0,tol,x)
SPMAT *A;
VEC *b, *r0, *x;
double tol;
{ return cgs(sp_mv_mlt,A,b,r0,tol,x); }
/*
Routine for performing LSQR -- the least squares QR algorithm
of Paige and Saunders:
"LSQR: an algorithm for sparse linear equations and
sparse least squares", ACM Trans. Math. Soft., v. 8
pp. 43--71 (1982)
*/
/* lsqr -- sparse CG-like least squares routine:
-- finds min_x ||A.x-b||_2 using A defined through A & AT
-- returns x (if x != NULL) */
VEC *lsqr(A,AT,A_params,b,tol,x)
MTX_FN A, AT; /* AT is A transposed */
VEC *x, *b;
double tol; /* error tolerance used */
void *A_params;
{
VEC *u, *v, *w, *tmp;
Real alpha, beta, norm_b, phi, phi_bar,
rho, rho_bar, rho_max, theta;
Real s, c; /* for Givens' rotations */
int iter, m, n;
if ( ! b || ! x )
error(E_NULL,"lsqr");
if ( tol <= 0.0 )
tol = MACHEPS;
m = b->dim; n = x->dim;
u = v_get((u_int)m);
v = v_get((u_int)n);
w = v_get((u_int)n);
tmp = v_get((u_int)n);
norm_b = v_norm2(b);
v_zero(x);
beta = v_norm2(b);
if ( beta == 0.0 )
return x;
sv_mlt(1.0/beta,b,u);
tracecatch((*AT)(A_params,u,v),"lsqr");
alpha = v_norm2(v);
if ( alpha == 0.0 )
return x;
sv_mlt(1.0/alpha,v,v);
v_copy(v,w);
phi_bar = beta; rho_bar = alpha;
rho_max = 1.0;
iter = 0;
do {
if ( ++iter > max_iter )
error(E_ITER,"lsqr");
tmp = v_resize(tmp,m);
tracecatch((*A) (A_params,v,tmp),"lsqr");
v_mltadd(tmp,u,-alpha,u);
beta = v_norm2(u); sv_mlt(1.0/beta,u,u);
tmp = v_resize(tmp,n);
tracecatch((*AT)(A_params,u,tmp),"lsqr");
v_mltadd(tmp,v,-beta,v);
alpha = v_norm2(v); sv_mlt(1.0/alpha,v,v);
rho = sqrt(rho_bar*rho_bar+beta*beta);
if ( rho > rho_max )
rho_max = rho;
c = rho_bar/rho;
s = beta/rho;
theta = s*alpha;
rho_bar = -c*alpha;
phi = c*phi_bar;
phi_bar = s*phi_bar;
/* update x & w */
if ( rho == 0.0 )
error(E_SING,"lsqr");
v_mltadd(x,w,phi/rho,x);
v_mltadd(v,w,-theta/rho,w);
} while ( fabs(phi_bar*alpha*c) > tol*norm_b/rho_max );
cg_num_iters = iter;
V_FREE(tmp); V_FREE(u); V_FREE(v); V_FREE(w);
return x;
}
/* sp_lsqr -- simple interface for SPMAT data structures */
VEC *sp_lsqr(A,b,tol,x)
SPMAT *A;
VEC *b, *x;
double tol;
{ return lsqr(sp_mv_mlt,sp_vm_mlt,A,b,tol,x); }
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