/*
Routines related to orthogonalization.
See the SLEPc Technical Report STR-1 for a detailed explanation.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
SLEPc - Scalable Library for Eigenvalue Problem Computations
Copyright (c) 2002-2007, Universidad Politecnica de Valencia, Spain
This file is part of SLEPc. See the README file for conditions of use
and additional information.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
*/
#include "private/ipimpl.h" /*I "slepcip.h" I*/
#include "slepcblaslapack.h"
/*
IPOrthogonalizeMGS - Compute one step of Modified Gram-Schmidt
*/
#undef __FUNCT__
#define __FUNCT__ "IPOrthogonalizeMGS"
static PetscErrorCode IPOrthogonalizeMGS(IP ip,PetscInt n,PetscTruth *which,Vec *V,Vec v,PetscScalar *H)
{
PetscErrorCode ierr;
PetscInt j;
PetscFunctionBegin;
for (j=0; j<n; j++)
if (!which || which[j]) {
/* h_j = ( v, v_j ) */
ierr = IPInnerProduct(ip,v,V[j],&H[j]);CHKERRQ(ierr);
/* v <- v - h_j v_j */
ierr = VecAXPY(v,-H[j],V[j]);CHKERRQ(ierr);
}
PetscFunctionReturn(0);
}
/*
IPOrthogonalizeCGS - Compute |v'| (estimated), |v| and one step of CGS with only one global synchronization
*/
#undef __FUNCT__
#define __FUNCT__ "IPOrthogonalizeCGS"
PetscErrorCode IPOrthogonalizeCGS(IP ip,PetscInt n,PetscTruth *which,Vec *V,Vec v,PetscScalar *H,PetscReal *onorm,PetscReal *norm,Vec work)
{
PetscErrorCode ierr;
PetscInt j;
PetscScalar alpha;
PetscReal sum;
PetscFunctionBegin;
/* h = W^* v ; alpha = (v , v) */
if (which) {
/* use select array */
for (j=0; j<n; j++)
if (which[j]) { ierr = IPInnerProductBegin(ip,v,V[j],&H[j]);CHKERRQ(ierr); }
if (onorm || norm) { ierr = IPInnerProductBegin(ip,v,v,&alpha);CHKERRQ(ierr); }
for (j=0; j<n; j++)
if (which[j]) { ierr = IPInnerProductEnd(ip,v,V[j],&H[j]);CHKERRQ(ierr); }
if (onorm || norm) { ierr = IPInnerProductEnd(ip,v,v,&alpha);CHKERRQ(ierr); }
} else { /* merge comunications */
if (onorm || norm) {
ierr = IPMInnerProductBegin(ip,v,n,V,H);CHKERRQ(ierr);
ierr = IPInnerProductBegin(ip,v,v,&alpha);CHKERRQ(ierr);
ierr = IPMInnerProductEnd(ip,v,n,V,H);CHKERRQ(ierr);
ierr = IPInnerProductEnd(ip,v,v,&alpha);CHKERRQ(ierr);
} else { /* use simpler function */
ierr = IPMInnerProduct(ip,v,n,V,H);CHKERRQ(ierr);
}
}
/* q = v - V h */
ierr = VecSet(work,0.0);CHKERRQ(ierr);
if (which) {
for (j=0; j<n; j++)
if (which[j]) { ierr = VecAXPY(work,H[j],V[j]);CHKERRQ(ierr); }
} else {
ierr = VecMAXPY(work,n,H,V);CHKERRQ(ierr);
}
ierr = VecAXPY(v,-1.0,work);CHKERRQ(ierr);
/* compute |v| */
if (onorm) *onorm = sqrt(PetscRealPart(alpha));
/* compute |v'| */
if (norm) {
sum = 0.0;
for (j=0; j<n; j++)
if (!which || which[j])
sum += PetscRealPart(H[j] * PetscConj(H[j]));
*norm = PetscRealPart(alpha)-sum;
if (*norm <= 0.0) {
ierr = IPNorm(ip,v,norm);CHKERRQ(ierr);
} else *norm = sqrt(*norm);
}
PetscFunctionReturn(0);
}
/*
IPOrthogonalizeGS - Compute |v'|, |v| and one step of CGS or MGS
*/
#undef __FUNCT__
#define __FUNCT__ "IPOrthogonalizeGS"
static PetscErrorCode IPOrthogonalizeGS(IP ip,PetscInt n,PetscTruth *which,Vec *V,Vec v,PetscScalar *H,PetscReal *onorm,PetscReal *norm,Vec work)
{
PetscErrorCode ierr;
PetscFunctionBegin;
switch (ip->orthog_type) {
case IP_CGS_ORTH:
ierr = IPOrthogonalizeCGS(ip,n,which,V,v,H,onorm,norm,work);CHKERRQ(ierr);
break;
case IP_MGS_ORTH:
/* compute |v| */
if (onorm) { ierr = IPNorm(ip,v,onorm);CHKERRQ(ierr); }
ierr = IPOrthogonalizeMGS(ip,n,which,V,v,H);CHKERRQ(ierr);
/* compute |v'| */
if (norm) { ierr = IPNorm(ip,v,norm);CHKERRQ(ierr); }
break;
default:
SETERRQ(PETSC_ERR_ARG_WRONG,"Unknown orthogonalization type");
}
PetscFunctionReturn(0);
}
#undef __FUNCT__
#define __FUNCT__ "IPOrthogonalize"
/*@
IPOrthogonalize - Orthogonalize a vector with respect to a set of vectors.
Collective on IP
Input Parameters:
+ ip - the inner product (IP) context
. n - number of columns of V
. which - logical array indicating columns of V to be used
. V - set of vectors
- work - workspace vector
Input/Output Parameter:
. v - (input) vector to be orthogonalized and (output) result of
orthogonalization
Output Parameter:
+ H - coefficients computed during orthogonalization
. norm - norm of the vector after being orthogonalized
- lindep - flag indicating that refinement did not improve the quality
of orthogonalization
Notes:
This function applies an orthogonal projector to project vector v onto the
orthogonal complement of the span of the columns of V.
On exit, v0 = [V v]*H, where v0 is the original vector v.
This routine does not normalize the resulting vector.
Level: developer
.seealso: IPSetOrthogonalization(), IPBiOrthogonalize()
@*/
PetscErrorCode IPOrthogonalize(IP ip,PetscInt n,PetscTruth *which,Vec *V,Vec v,PetscScalar *H,PetscReal *norm,PetscTruth *lindep,Vec work)
{
PetscErrorCode ierr;
PetscScalar lh[100],*h,lc[100],*c;
PetscTruth allocatedh = PETSC_FALSE,allocatedc = PETSC_FALSE,allocatedw = PETSC_FALSE;
PetscReal onrm,nrm;
PetscInt j,k;
PetscFunctionBegin;
if (n==0) {
if (norm) { ierr = IPNorm(ip,v,norm);CHKERRQ(ierr); }
if (lindep) *lindep = PETSC_FALSE;
PetscFunctionReturn(0);
}
ierr = PetscLogEventBegin(IP_Orthogonalize,ip,0,0,0);CHKERRQ(ierr);
/* allocate H, c and work if needed */
if (!H) {
if (n<=100) h = lh;
else {
ierr = PetscMalloc(n*sizeof(PetscScalar),&h);CHKERRQ(ierr);
allocatedh = PETSC_TRUE;
}
} else h = H;
if (ip->orthog_ref != IP_ORTH_REFINE_NEVER) {
if (n<=100) c = lc;
else {
ierr = PetscMalloc(n*sizeof(PetscScalar),&c);CHKERRQ(ierr);
allocatedc = PETSC_TRUE;
}
}
if (!work && ip->orthog_type == IP_CGS_ORTH) {
ierr = VecDuplicate(v,&work);CHKERRQ(ierr);
allocatedw = PETSC_TRUE;
}
/* orthogonalize and compute onorm */
switch (ip->orthog_ref) {
case IP_ORTH_REFINE_NEVER:
ierr = IPOrthogonalizeGS(ip,n,which,V,v,h,PETSC_NULL,PETSC_NULL,work);CHKERRQ(ierr);
/* compute |v| */
if (norm) { ierr = IPNorm(ip,v,norm);CHKERRQ(ierr); }
/* linear dependence check does not work without refinement */
if (lindep) *lindep = PETSC_FALSE;
break;
case IP_ORTH_REFINE_ALWAYS:
ierr = IPOrthogonalizeGS(ip,n,which,V,v,h,PETSC_NULL,PETSC_NULL,work);CHKERRQ(ierr);
if (lindep) {
ierr = IPOrthogonalizeGS(ip,n,which,V,v,c,&onrm,&nrm,work);CHKERRQ(ierr);
if (norm) *norm = nrm;
if (nrm < ip->orthog_eta * onrm) *lindep = PETSC_TRUE;
else *lindep = PETSC_FALSE;
} else {
ierr = IPOrthogonalizeGS(ip,n,which,V,v,c,PETSC_NULL,norm,work);CHKERRQ(ierr);
}
for (j=0;j<n;j++)
if (!which || which[j]) h[j] += c[j];
break;
case IP_ORTH_REFINE_IFNEEDED:
ierr = IPOrthogonalizeGS(ip,n,which,V,v,h,&onrm,&nrm,work);CHKERRQ(ierr);
/* ||q|| < eta ||h|| */
k = 1;
while (k<3 && nrm < ip->orthog_eta * onrm) {
k++;
switch (ip->orthog_type) {
case IP_CGS_ORTH:
ierr = IPOrthogonalizeGS(ip,n,which,V,v,c,&onrm,&nrm,work);CHKERRQ(ierr);
break;
case IP_MGS_ORTH:
onrm = nrm;
ierr = IPOrthogonalizeGS(ip,n,which,V,v,c,PETSC_NULL,&nrm,work);CHKERRQ(ierr);
break;
default:
SETERRQ(PETSC_ERR_ARG_WRONG,"Unknown orthogonalization type");
}
for (j=0;j<n;j++)
if (!which || which[j]) h[j] += c[j];
}
if (norm) *norm = nrm;
if (lindep) {
if (nrm < ip->orthog_eta * onrm) *lindep = PETSC_TRUE;
else *lindep = PETSC_FALSE;
}
break;
default:
SETERRQ(PETSC_ERR_ARG_WRONG,"Unknown orthogonalization refinement");
}
/* free work space */
if (allocatedc) { ierr = PetscFree(c);CHKERRQ(ierr); }
if (allocatedh) { ierr = PetscFree(h);CHKERRQ(ierr); }
if (allocatedw) { ierr = VecDestroy(work);CHKERRQ(ierr); }
ierr = PetscLogEventEnd(IP_Orthogonalize,ip,0,0,0);CHKERRQ(ierr);
PetscFunctionReturn(0);
}
#undef __FUNCT__
#define __FUNCT__ "IPQRDecomposition"
/*@
IPQRDecomposition - Compute the QR factorization of a set of vectors.
Collective on IP
Input Parameters:
+ ip - the eigenproblem solver context
. V - set of vectors
. m - starting column
. n - ending column
. ldr - leading dimension of R
- work - workspace vector
Output Parameter:
. R - triangular matrix of QR factorization
Notes:
This routine orthonormalizes the columns of V so that V'*V=I up to
working precision. It assumes that the first m columns (from 0 to m-1)
are already orthonormal. The coefficients of the orthogonalization are
stored in R so that V*R is equal to the original V.
In some cases, this routine makes V B-orthonormal, that is, V'*B*V=I.
If one of the vectors is linearly dependent wrt the rest (at working
precision) then it is replaced by a random vector.
Level: developer
.seealso: IPOrthogonalize(), IPNorm(), IPInnerProduct().
@*/
PetscErrorCode IPQRDecomposition(IP ip,Vec *V,PetscInt m,PetscInt n,PetscScalar *R,PetscInt ldr,Vec work)
{
PetscErrorCode ierr;
PetscInt k;
PetscReal norm;
PetscTruth lindep;
PetscFunctionBegin;
for (k=m; k<n; k++) {
/* orthogonalize v_k with respect to v_0, ..., v_{k-1} */
if (R) { ierr = IPOrthogonalize(ip,k,PETSC_NULL,V,V[k],&R[0+ldr*k],&norm,&lindep,work);CHKERRQ(ierr); }
else { ierr = IPOrthogonalize(ip,k,PETSC_NULL,V,V[k],PETSC_NULL,&norm,&lindep,work);CHKERRQ(ierr); }
/* normalize v_k: r_{k,k} = ||v_k||_2; v_k = v_k/r_{k,k} */
if (norm==0.0 || lindep) {
PetscInfo(ip,"Linearly dependent vector found, generating a new random vector\n");
ierr = SlepcVecSetRandom(V[k]);CHKERRQ(ierr);
ierr = IPNorm(ip,V[k],&norm);CHKERRQ(ierr);
}
ierr = VecScale(V[k],1.0/norm);CHKERRQ(ierr);
if (R) R[k+ldr*k] = norm;
}
PetscFunctionReturn(0);
}
/*
Biorthogonalization routine using classical Gram-Schmidt with refinement.
*/
#undef __FUNCT__
#define __FUNCT__ "IPCGSBiOrthogonalization"
static PetscErrorCode IPCGSBiOrthogonalization(IP ip,PetscInt n_,Vec *V,Vec *W,Vec v,PetscScalar *H,PetscReal *hnorm,PetscReal *norm)
{
#if defined(SLEPC_MISSING_LAPACK_GELQF) || defined(SLEPC_MISSING_LAPACK_ORMLQ)
PetscFunctionBegin;
SETERRQ(PETSC_ERR_SUP,"xGELQF - Lapack routine is unavailable.");
#else
PetscErrorCode ierr;
PetscBLASInt j,ione=1,lwork,info,n=n_;
PetscScalar shh[100],*lhh,*vw,*tau,one=1.0,*work;
Vec w;
PetscFunctionBegin;
/* Don't allocate small arrays */
if (n<=100) lhh = shh;
else { ierr = PetscMalloc(n*sizeof(PetscScalar),&lhh);CHKERRQ(ierr); }
ierr = PetscMalloc(n*n*sizeof(PetscScalar),&vw);CHKERRQ(ierr);
ierr = VecDuplicate(v,&w);CHKERRQ(ierr);
for (j=0;j<n;j++) {
ierr = IPMInnerProduct(ip,V[j],n,W,vw+j*n);CHKERRQ(ierr);
}
lwork = n;
ierr = PetscMalloc(n*sizeof(PetscScalar),&tau);CHKERRQ(ierr);
ierr = PetscMalloc(lwork*sizeof(PetscScalar),&work);CHKERRQ(ierr);
LAPACKgelqf_(&n,&n,vw,&n,tau,work,&lwork,&info);
if (info) SETERRQ1(PETSC_ERR_LIB,"Error in Lapack xGELQF %i",info);
/*** First orthogonalization ***/
/* h = W^* v */
/* q = v - V h */
ierr = IPMInnerProduct(ip,v,n,W,H);CHKERRQ(ierr);
BLAStrsm_("L","L","N","N",&n,&ione,&one,vw,&n,H,&n,1,1,1,1);
LAPACKormlq_("L","N",&n,&ione,&n,vw,&n,tau,H,&n,work,&lwork,&info,1,1);
if (info) SETERRQ1(PETSC_ERR_LIB,"Error in Lapack xORMLQ %i",info);
ierr = VecSet(w,0.0);CHKERRQ(ierr);
ierr = VecMAXPY(w,n,H,V);CHKERRQ(ierr);
ierr = VecAXPY(v,-1.0,w);CHKERRQ(ierr);
/* compute norm of v */
if (norm) { ierr = IPNorm(ip,v,norm);CHKERRQ(ierr); }
if (n>100) { ierr = PetscFree(lhh);CHKERRQ(ierr); }
ierr = PetscFree(vw);CHKERRQ(ierr);
ierr = PetscFree(tau);CHKERRQ(ierr);
ierr = PetscFree(work);CHKERRQ(ierr);
ierr = VecDestroy(w);CHKERRQ(ierr);
PetscFunctionReturn(0);
#endif
}
#undef __FUNCT__
#define __FUNCT__ "IPBiOrthogonalize"
/*@
IPBiOrthogonalize - Bi-orthogonalize a vector with respect to a set of vectors.
Collective on IP
Input Parameters:
+ ip - the inner product context
. n - number of columns of V
. V - set of vectors
- W - set of vectors
Input/Output Parameter:
. v - vector to be orthogonalized
Output Parameter:
+ H - coefficients computed during orthogonalization
- norm - norm of the vector after being orthogonalized
Notes:
This function applies an oblique projector to project vector v onto the
span of the columns of V along the orthogonal complement of the column
space of W.
On exit, v0 = [V v]*H, where v0 is the original vector v.
This routine does not normalize the resulting vector.
Level: developer
.seealso: IPSetOrthogonalization(), IPOrthogonalize()
@*/
PetscErrorCode IPBiOrthogonalize(IP ip,PetscInt n,Vec *V,Vec *W,Vec v,PetscScalar *H,PetscReal *norm)
{
PetscErrorCode ierr;
PetscScalar lh[100],*h;
PetscTruth allocated = PETSC_FALSE;
PetscReal lhnrm,*hnrm,lnrm,*nrm;
PetscFunctionBegin;
if (n==0) {
if (norm) { ierr = IPNorm(ip,v,norm);CHKERRQ(ierr); }
} else {
ierr = PetscLogEventBegin(IP_Orthogonalize,ip,0,0,0);CHKERRQ(ierr);
/* allocate H if needed */
if (!H) {
if (n<=100) h = lh;
else {
ierr = PetscMalloc(n*sizeof(PetscScalar),&h);CHKERRQ(ierr);
allocated = PETSC_TRUE;
}
} else h = H;
/* retrieve hnrm and nrm for linear dependence check or conditional refinement */
if (ip->orthog_ref == IP_ORTH_REFINE_IFNEEDED) {
hnrm = &lhnrm;
if (norm) nrm = norm;
else nrm = &lnrm;
} else {
hnrm = PETSC_NULL;
nrm = norm;
}
switch (ip->orthog_type) {
case IP_CGS_ORTH:
ierr = IPCGSBiOrthogonalization(ip,n,V,W,v,h,hnrm,nrm);CHKERRQ(ierr);
break;
default:
SETERRQ(PETSC_ERR_ARG_WRONG,"Unknown orthogonalization type");
}
if (allocated) { ierr = PetscFree(h);CHKERRQ(ierr); }
ierr = PetscLogEventEnd(IP_Orthogonalize,ip,0,0,0);CHKERRQ(ierr);
}
PetscFunctionReturn(0);
}