void showQMatrix( QMatrix *M ) {
int i, j;
printf( "Matrix: %s\n", Q_GetName( M ) );
for ( i = 0; i < Q_GetDim( M ); i++ ) {
printf( "Row %d: ", i );
for ( j = 0; j < Q_GetDim( M ); j++ )
if ( Equal( Q_GetEl( M, i+1, j+1 ), 0.0, 1e-5 ))
continue;
else
printf( "(%d) %.3lf ", j, Q_GetEl( M, i+1, j+1 ) );
printf( "\n" );
} /* for i */
} /* showQMatrix */
double GetMaxEigenval(QMatrix *A, PrecondProcType PrecondProc, double OmegaPrecond)
/* returns estimate for maximum eigenvalue of the matrix A */
{
double MaxEigenval;
EigenvalInfoType *EigenvalInfo;
Q_Lock(A);
if (LASResult() == LASOK) {
EigenvalInfo = (EigenvalInfoType *)*(Q_EigenvalInfo(A));
/* if eigenvalues not estimated yet, ... */
if (EigenvalInfo == NULL) {
EigenvalInfo = (EigenvalInfoType *)malloc(sizeof(EigenvalInfoType));
if (EigenvalInfo != NULL) {
*(Q_EigenvalInfo(A)) = (void *)EigenvalInfo;
EstimEigenvals(A, PrecondProc, OmegaPrecond);
} else {
LASError(LASMemAllocErr, "GetMaxEigenval", Q_GetName(A), NULL, NULL);
}
}
/* if eigenvalues estimated with an other preconditioner, ... */
if (EigenvalInfo->PrecondProcUsed != PrecondProc
|| EigenvalInfo->OmegaPrecondUsed != OmegaPrecond) {
EstimEigenvals(A, PrecondProc, OmegaPrecond);
}
if (LASResult() == LASOK)
MaxEigenval = EigenvalInfo->MaxEigenval;
else
MaxEigenval = 1.0;
} else {
MaxEigenval = 1.0;
}
return(MaxEigenval);
}
Vector *Test4_QV(QMatrix *Q, Vector *V)
/* VRes = Q * V ... implementation using local variables and pointers,
descended counting of matrix elements */
{
Vector *VRes;
char *VResName;
size_t Dim, Row, Clm, Len, ElCount;
size_t *QLen;
ElType **QEl, *PtrEl;
Real Sum, Cmp;
Real *VCmp, *VResCmp;
if (LASResult() == LASOK) {
if (Q->Dim == V->Dim) {
Dim = Q->Dim;
VRes = (Vector *)malloc(sizeof(Vector));
VResName = (char *)malloc((strlen(Q_GetName(Q)) + strlen(V_GetName(V)) + 10)
* sizeof(char));
if (VRes != NULL && VResName != NULL) {
sprintf(VResName, "(%s) * (%s)", Q_GetName(Q), V_GetName(V));
V_Constr(VRes, VResName, Dim, Tempor, True);
if (LASResult() == LASOK) {
/* initialisation of vector VRes */
if (Q->Symmetry || Q->ElOrder == Clmws)
V_SetAllCmp(VRes, 0.0);
/* analysis of multipliers of matrix Q and vector V
is not implemented yet */
/* multiplication of matrix elements by vector
components */
VCmp = V->Cmp;
VResCmp = VRes->Cmp;
QLen = Q->Len;
QEl = Q->El;
if (!Q->Symmetry) {
if (Q->ElOrder == Rowws) {
for (Row = Dim; Row > 0; Row--) {
Len = QLen[Row];
PtrEl = QEl[Row] + Len - 1;
Sum = 0.0;
for (ElCount = Len; ElCount > 0; ElCount--) {
Sum += (*PtrEl).Val * VCmp[(*PtrEl).Pos];
PtrEl--;
}
VResCmp[Row] = Sum;
}
}
if (Q->ElOrder == Clmws) {
for (Clm = Dim; Clm > 0; Clm--) {
Len = QLen[Clm];
PtrEl = QEl[Clm] + Len - 1;
Cmp = VCmp[Clm];
for (ElCount = Len; ElCount > 0; ElCount--) {
VResCmp[(*PtrEl).Pos] += (*PtrEl).Val * Cmp;
PtrEl--;;
}
}
}
} else {
/* multiplication by symmetric matrix is not
implemented yet */
V_SetAllCmp(VRes, 0.0);
}
}
} else {
LASError(LASMemAllocErr, "Mul_QV", Q_GetName(Q), V_GetName(V), NULL);
if (VRes != NULL)
free(VRes);
if (VResName != NULL)
free(VResName);
}
} else {
LASError(LASDimErr, "Mul_QV", Q_GetName(Q), V_GetName(V), NULL);
VRes = NULL;
}
} else {
VRes = NULL;
}
if (Q != NULL) {
if (Q->Instance == Tempor) {
Q_Destr(Q);
free(Q);
}
}
if (V != NULL) {
if (V->Instance == Tempor) {
V_Destr(V);
free(V);
}
}
return(VRes);
}
Vector *Test1_QV(QMatrix *Q, Vector *V)
/* VRes = Q * V ... very simple implementation */
{
Vector *VRes;
char *VResName;
size_t Dim, Row, Clm, ElCount;
ElType **QEl;
if (LASResult() == LASOK) {
if (Q->Dim == V->Dim) {
Dim = Q->Dim;
QEl = Q->El;
VRes = (Vector *)malloc(sizeof(Vector));
VResName = (char *)malloc((strlen(Q_GetName(Q)) + strlen(V_GetName(V)) + 10)
* sizeof(char));
if (VRes != NULL && VResName != NULL) {
sprintf(VResName, "(%s) * (%s)", Q_GetName(Q), V_GetName(V));
V_Constr(VRes, VResName, Dim, Tempor, True);
if (LASResult() == LASOK) {
/* initialisation of vector VRes */
V_SetAllCmp(VRes, 0.0);
/* analysis of multipliers of matrix Q and vector V
is not implemented yet */
/* multiplication of matrix elements by vector
components */
if (!Q->Symmetry) {
if (Q->ElOrder == Rowws) {
for (Row = 1; Row <= Dim; Row++) {
for (ElCount = 0; ElCount < Q->Len[Row]; ElCount++) {
Clm = QEl[Row][ElCount].Pos;
VRes->Cmp[Row] += QEl[Row][ElCount].Val * V->Cmp[Clm];
}
}
}
if (Q->ElOrder == Clmws) {
for (Clm = 1; Clm <= Dim; Clm++) {
for (ElCount = 0; ElCount < Q->Len[Clm]; ElCount++) {
Row = QEl[Clm][ElCount].Pos;
VRes->Cmp[Row] += QEl[Clm][ElCount].Val * V->Cmp[Clm];
}
}
}
} else {
/* multiplication by symmetric matrix is not
implemented yet */
V_SetAllCmp(VRes, 0.0);
}
}
} else {
LASError(LASMemAllocErr, "Mul_QV", Q_GetName(Q), V_GetName(V), NULL);
if (VRes != NULL)
free(VRes);
if (VResName != NULL)
free(VResName);
}
} else {
LASError(LASDimErr, "Mul_QV", Q_GetName(Q), V_GetName(V), NULL);
VRes = NULL;
}
} else {
VRes = NULL;
}
if (Q != NULL) {
if (Q->Instance == Tempor) {
Q_Destr(Q);
free(Q);
}
}
if (V != NULL) {
if (V->Instance == Tempor) {
V_Destr(V);
free(V);
}
}
return(VRes);
}
static void EstimEigenvals(QMatrix *A, PrecondProcType PrecondProc, double OmegaPrecond)
/* estimates extremal eigenvalues of the matrix A by means of the Lanczos method */
{
/*
* for details to the Lanczos algorithm see
*
* G. H. Golub, Ch. F. van Loan:
* Matrix Computations;
* North Oxford Academic, Oxford, 1986
*
* (for modification for preconditioned matrices compare with sec. 10.3)
*
*/
double LambdaMin = 0.0, LambdaMax = 0.0;
double LambdaMinOld, LambdaMaxOld;
double GershBoundMin = 0.0, GershBoundMax = 0.0;
double *Alpha, *Beta;
size_t Dim, j;
Boolean Found;
Vector q, qOld, h, p;
Q_Lock(A);
Dim = Q_GetDim(A);
V_Constr(&q, "q", Dim, Normal, True);
V_Constr(&qOld, "qOld", Dim, Normal, True);
V_Constr(&h, "h", Dim, Normal, True);
if (PrecondProc != NULL)
V_Constr(&p, "p", Dim, Normal, True);
if (LASResult() == LASOK) {
Alpha = (double *)malloc((Dim + 1) * sizeof(double));
Beta = (double *)malloc((Dim + 1) * sizeof(double));
if (Alpha != NULL && Beta != NULL) {
j = 0;
V_SetAllCmp(&qOld, 0.0);
V_SetRndCmp(&q);
if (Q_KerDefined(A))
OrthoRightKer_VQ(&q, A);
if (Q_GetSymmetry(A) && PrecondProc != NULL) {
(*PrecondProc)(A, &p, &q, OmegaPrecond);
MulAsgn_VS(&q, 1.0 / sqrt(Mul_VV(&q, &p)));
} else {
MulAsgn_VS(&q, 1.0 / l2Norm_V(&q));
}
Beta[0] = 1.0;
do {
j++;
if (Q_GetSymmetry(A) && PrecondProc != NULL) {
/* p = M^(-1) q */
(*PrecondProc)(A, &p, &q, OmegaPrecond);
/* h = A p */
Asgn_VV(&h, Mul_QV(A, &p));
if (Q_KerDefined(A))
OrthoRightKer_VQ(&h, A);
/* Alpha = p . h */
Alpha[j] = Mul_VV(&p, &h);
/* r = h - Alpha q - Beta qOld */
SubAsgn_VV(&h, Add_VV(Mul_SV(Alpha[j], &q), Mul_SV(Beta[j-1], &qOld)));
/* z = M^(-1) r */
(*PrecondProc)(A, &p, &h, OmegaPrecond);
/* Beta = sqrt(r . z) */
Beta[j] = sqrt(Mul_VV(&h, &p));
Asgn_VV(&qOld, &q);
/* q = r / Beta */
Asgn_VV(&q, Mul_SV(1.0 / Beta[j], &h));
} else {
/* h = A p */
if (Q_GetSymmetry(A)) {
Asgn_VV(&h, Mul_QV(A, &q));
} else {
if (PrecondProc != NULL) {
(*PrecondProc)(A, &h, Mul_QV(A, &q), OmegaPrecond);
(*PrecondProc)(Transp_Q(A), &h, &h, OmegaPrecond);
Asgn_VV(&h, Mul_QV(Transp_Q(A), &h));
} else {
Asgn_VV(&h, Mul_QV(Transp_Q(A), Mul_QV(A, &q)));
}
}
if (Q_KerDefined(A))
OrthoRightKer_VQ(&h, A);
/* Alpha = q . h */
Alpha[j] = Mul_VV(&q, &h);
/* r = h - Alpha q - Beta qOld */
SubAsgn_VV(&h, Add_VV(Mul_SV(Alpha[j], &q), Mul_SV(Beta[j-1], &qOld)));
/* Beta = || r || */
Beta[j] = l2Norm_V(&h);
Asgn_VV(&qOld, &q);
/* q = r / Beta */
Asgn_VV(&q, Mul_SV(1.0 / Beta[j], &h));
}
LambdaMaxOld = LambdaMax;
LambdaMinOld = LambdaMin;
/* determination of extremal eigenvalues of the tridiagonal matrix
(Beta[i-1] Alpha[i] Beta[i]) (where 1 <= i <= j)
by means of the method of bisection; bounds for eigenvalues
are determined after Gershgorin circle theorem */
if (j == 1) {
GershBoundMin = Alpha[1] - fabs(Beta[1]);
GershBoundMax = Alpha[1] + fabs(Beta[1]);
LambdaMin = Alpha[1];
LambdaMax = Alpha[1];
} else {
GershBoundMin = min(Alpha[j] - fabs(Beta[j]) - fabs(Beta[j - 1]),
GershBoundMin);
GershBoundMax = max(Alpha[j] + fabs(Beta[j]) + fabs(Beta[j - 1]),
GershBoundMax);
SearchEigenval(j, Alpha, Beta, 1, GershBoundMin, LambdaMin,
&Found, &LambdaMin);
if (!Found)
SearchEigenval(j, Alpha, Beta, 1, GershBoundMin, GershBoundMax,
&Found, &LambdaMin);
SearchEigenval(j, Alpha, Beta, j, LambdaMax, GershBoundMax,
&Found, &LambdaMax);
if (!Found)
SearchEigenval(j, Alpha, Beta, j, GershBoundMin, GershBoundMax,
&Found, &LambdaMax);
}
} while (!IsZero(Beta[j]) && j < Dim
&& (fabs(LambdaMin - LambdaMinOld) > EigenvalEps * LambdaMin
|| fabs(LambdaMax - LambdaMaxOld) > EigenvalEps * LambdaMax)
&& LASResult() == LASOK);
if (Q_GetSymmetry(A)) {
LambdaMin = (1.0 - j * EigenvalEps) * LambdaMin;
} else {
LambdaMin = (1.0 - sqrt(j) * EigenvalEps) * sqrt(LambdaMin);
}
if (Alpha != NULL)
free(Alpha);
if (Beta != NULL)
free(Beta);
} else {
LASError(LASMemAllocErr, "EstimEigenvals", Q_GetName(A), NULL, NULL);
}
}
V_Destr(&q);
V_Destr(&qOld);
V_Destr(&h);
if (PrecondProc != NULL)
V_Destr(&p);
if (LASResult() == LASOK) {
((EigenvalInfoType *)*(Q_EigenvalInfo(A)))->MinEigenval = LambdaMin;
((EigenvalInfoType *)*(Q_EigenvalInfo(A)))->MaxEigenval = LambdaMax;
((EigenvalInfoType *)*(Q_EigenvalInfo(A)))->PrecondProcUsed = PrecondProc;
((EigenvalInfoType *)*(Q_EigenvalInfo(A)))->OmegaPrecondUsed = OmegaPrecond;
} else {
((EigenvalInfoType *)*(Q_EigenvalInfo(A)))->MinEigenval = 1.0;
((EigenvalInfoType *)*(Q_EigenvalInfo(A)))->MaxEigenval = 1.0;
((EigenvalInfoType *)*(Q_EigenvalInfo(A)))->PrecondProcUsed = NULL;
((EigenvalInfoType *)*(Q_EigenvalInfo(A)))->OmegaPrecondUsed = 1.0;
}
Q_Unlock(A);
}