Vector *BPXPrecond(int NoLevels, QMatrix *A, Vector *y, Vector *c,
Matrix *R, Matrix *P, int Level,
IterProcType SmoothProc, int Nu,
PrecondProcType PrecondProc, double Omega,
IterProcType SmoothProcC, int NuC,
PrecondProcType PrecondProcC, double OmegaC)
/* BPX preconditioner (recursively defined) */
{
if (Level == 0) {
/* smoothing on the coarsest grid - NuC iterations */
V_SetAllCmp(&y[Level], 0.0);
(*SmoothProcC)(&A[Level], &y[Level], &c[Level], NuC, PrecondProcC, OmegaC);
} else {
/* smoothing - Nu iterations */
V_SetAllCmp(&y[Level], 0.0);
(*SmoothProc)(&A[Level], &y[Level], &c[Level], Nu, PrecondProc, Omega);
/* restiction of the residual to the coarser grid */
Asgn_VV(&c[Level - 1], Mul_MV(&R[Level - 1], &c[Level]));
/* smoothing on the coarser grid */
BPXPrecond(NoLevels, A, y, c, R, P, Level - 1,
SmoothProc, Nu, PrecondProc, Omega, SmoothProcC, NuC, PrecondProcC, OmegaC);
/* interpolation of the solution from coarser grid */
if (P != NULL)
AddAsgn_VV(&y[Level], Mul_MV(&P[Level], &y[Level - 1]));
else
AddAsgn_VV(&y[Level], Mul_MV(Transp_M(&R[Level - 1]), &y[Level - 1]));
}
return(&y[Level]);
}
NumericVector<T> &
LaspackVector<T>::operator = (const T s)
{
libmesh_assert (this->initialized());
libmesh_assert (this->closed());
V_SetAllCmp (&_vec, s);
return *this;
}
Vector *MGStep(int NoLevels, QMatrix *A, Vector *x, Vector *b,
Matrix *R, Matrix *P, int Level, int Gamma,
IterProcType SmoothProc, int Nu1, int Nu2,
PrecondProcType PrecondProc, double Omega,
IterProcType SolvProc, int NuC,
PrecondProcType PrecondProcC, double OmegaC)
/* one multigrid iteration */
{
int CoarseMGIter; /* multi grid iteration counter for coarser grid */
if (Level == 0) {
/* solving of system of equations for the residual on the coarsest grid */
(*SolvProc)(&A[Level], &x[Level], &b[Level], NuC, PrecondProcC, OmegaC);
} else {
/* pre-smoothing - Nu1 iterations */
(*SmoothProc)(&A[Level], &x[Level], &b[Level], Nu1, PrecondProc, Omega);
/* restiction of the residual to the coarser grid */
Asgn_VV(&b[Level - 1], Mul_MV(&R[Level - 1],
Sub_VV(&b[Level], Mul_QV(&A[Level], &x[Level]))));
/* initialisation of vector of unknowns on the coarser grid */
V_SetAllCmp(&x[Level - 1], 0.0);
/* solving of system of equations for the residual on the coarser grid */
for (CoarseMGIter = 1; CoarseMGIter <= Gamma; CoarseMGIter++)
MGStep(NoLevels, A, x, b, R, P, Level - 1, Gamma,
SmoothProc, Nu1, Nu2, PrecondProc, Omega,
SolvProc, NuC, PrecondProcC, OmegaC);
/* interpolation of the solution from the coarser grid */
if (P != NULL)
AddAsgn_VV(&x[Level], Mul_MV(&P[Level], &x[Level - 1]));
else
AddAsgn_VV(&x[Level], Mul_MV(Transp_M(&R[Level - 1]), &x[Level - 1]));
/* post-smoothing - Nu2 iterations */
(*SmoothProc)(&A[Level], &x[Level], &b[Level], Nu2, PrecondProc, Omega);
}
return(&x[Level]);
}
Vector *NestedMGIter(int NoLevels, QMatrix *A, Vector *x, Vector *b,
Matrix *R, Matrix *P, int Gamma,
IterProcType SmoothProc, int Nu1, int Nu2,
PrecondProcType PrecondProc, double Omega,
IterProcType SolvProc, int NuC,
PrecondProcType PrecondProcC, double OmegaC)
/* nested multigrid method */
{
int Level;
/* solution of system of equations on coarsest grid */
V_SetAllCmp(&x[0], 0.0);
MGStep(NoLevels, A, x, b, R, P, 0, Gamma,
SmoothProc, Nu1, Nu2, PrecondProc, Omega,
SolvProc, NuC, PrecondProcC, OmegaC);
for (Level = 1; Level < NoLevels; Level++) {
/* prolongation of solution to finer grid */
if (P != NULL)
Asgn_VV(&x[Level], Mul_MV(&P[Level], &x[Level - 1]));
else
Asgn_VV(&x[Level], Mul_MV(Transp_M(&R[Level - 1]), &x[Level - 1]));
/* solution of system of equations on finer grid with
multigrid method */
MGStep(NoLevels, A, x, b, R, P, Level, Gamma,
SmoothProc, Nu1, Nu2, PrecondProc, Omega,
SolvProc, NuC, PrecondProcC, OmegaC);
}
/* submission of reached accuracy to RTC */
RTCResult(1, l2Norm_V(Sub_VV(&b[NoLevels - 1],
Mul_QV(&A[NoLevels - 1], &x[NoLevels - 1]))),
l2Norm_V(&b[NoLevels - 1]), NestedMGIterId);
return(&x[NoLevels - 1]);
}
Vector *MGPCGIter(int NoLevels, QMatrix *A, Vector *z, Vector *r,
Matrix *R, Matrix *P, int MaxIter, int NoMGIter, int Gamma,
IterProcType SmoothProc, int Nu1, int Nu2,
PrecondProcType PrecondProc, double Omega,
IterProcType SolvProc, int NuC,
PrecondProcType PrecondProcC, double OmegaC)/* multigrid preconditioned CG method */
{
int Iter, MGIter;
double Alpha, Beta, Rho, RhoOld = 0.0;
double bNorm;
size_t Dim;
Vector x, p, q, b;
Dim = Q_GetDim(&A[NoLevels - 1]);
V_Constr(&x, "x", Dim, Normal, True);
V_Constr(&p, "p", Dim, Normal, True);
V_Constr(&q, "q", Dim, Normal, True);
V_Constr(&b, "b", Dim, Normal, True);
if (LASResult() == LASOK) {
/* copy solution and right hand side stored in parameters z and r */
Asgn_VV(&x, &z[NoLevels - 1]);
Asgn_VV(&b, &r[NoLevels - 1]);
bNorm = l2Norm_V(&b);
Iter = 0;
Asgn_VV(&r[NoLevels - 1], Sub_VV(&b, Mul_QV(&A[NoLevels - 1], &x)));
while (!RTCResult(Iter, l2Norm_V(&r[NoLevels - 1]), bNorm, MGPCGIterId)
&& Iter < MaxIter) {
Iter++;
/* multigrid preconditioner */
V_SetAllCmp(&z[NoLevels - 1], 0.0);
for (MGIter = 1; MGIter <= NoMGIter; MGIter++)
MGStep(NoLevels, A, z, r, R, P, NoLevels - 1, Gamma,
SmoothProc, Nu1, Nu2, PrecondProc, Omega,
SolvProc, NuC, PrecondProcC, OmegaC);
Rho = Mul_VV(&r[NoLevels - 1], &z[NoLevels - 1]);
if (Iter == 1) {
Asgn_VV(&p, &z[NoLevels - 1]);
} else {
Beta = Rho / RhoOld;
Asgn_VV(&p, Add_VV(&z[NoLevels - 1], Mul_SV(Beta, &p)));
}
Asgn_VV(&q, Mul_QV(&A[NoLevels - 1], &p));
Alpha = Rho / Mul_VV(&p, &q);
AddAsgn_VV(&x, Mul_SV(Alpha, &p));
SubAsgn_VV(&r[NoLevels - 1], Mul_SV(Alpha, &q));
RhoOld = Rho;
}
/* put solution and right hand side vectors back */
Asgn_VV(&z[NoLevels - 1], &x);
Asgn_VV(&r[NoLevels - 1], &b);
}
V_Destr(&x);
V_Destr(&p);
V_Destr(&q);
V_Destr(&b);
return(&z[NoLevels - 1]);
}
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);
}