Vector *MGIter(int NoLevels, QMatrix *A, Vector *x, Vector *b,
Matrix *R, Matrix *P, int MaxIter, int Gamma,
IterProcType SmoothProc, int Nu1, int Nu2,
PrecondProcType PrecondProc, double Omega,
IterProcType SolvProc, int NuC,
PrecondProcType PrecondProcC, double OmegaC)
/* multigrid method with residual termination control */
{
int Iter;
double bNorm;
size_t Dim;
Vector r;
Dim = Q_GetDim(&A[NoLevels - 1]);
V_Constr(&r, "r", Dim, Normal, True);
if (LASResult() == LASOK) {
bNorm = l2Norm_V(&b[NoLevels - 1]);
Iter = 0;
/* r = b - A x(i) at NoLevels - 1 */
Asgn_VV(&r, Sub_VV(&b[NoLevels - 1], Mul_QV(&A[NoLevels - 1], &x[NoLevels - 1])));
while (!RTCResult(Iter, l2Norm_V(&r), bNorm, MGIterId)
&& Iter < MaxIter) {
Iter++;
/* one multigrid step */
MGStep(NoLevels, A, x, b, R, P, NoLevels - 1, Gamma,
SmoothProc, Nu1, Nu2, PrecondProc, Omega,
SolvProc, NuC, PrecondProcC, OmegaC);
/* r = b - A x(i) at NoLevels - 1 */
Asgn_VV(&r, Sub_VV(&b[NoLevels - 1], Mul_QV(&A[NoLevels - 1], &x[NoLevels - 1])));
}
}
V_Destr(&r);
return(&x[NoLevels - 1]);
}
void LaspackVector<T>::add_vector (const NumericVector<T> & vec_in,
const SparseMatrix<T> & mat_in)
{
// Make sure the data passed in are really in Laspack types
const LaspackVector<T> * vec = cast_ptr<const LaspackVector<T> *>(&vec_in);
const LaspackMatrix<T> * mat = cast_ptr<const LaspackMatrix<T> *>(&mat_in);
libmesh_assert(vec);
libmesh_assert(mat);
// += mat*vec
AddAsgn_VV (&_vec, Mul_QV(const_cast<QMatrix*>(&mat->_QMat),
const_cast<QVector*>(&vec->_vec)));
}
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 *BPXPCGIter(int NoLevels, QMatrix *A, Vector *z, Vector *r,
Matrix *R, Matrix *P, int MaxIter,
IterProcType SmoothProc, int Nu,
PrecondProcType PrecondProc, double Omega,
IterProcType SmoothProcC, int NuC,
PrecondProcType PrecondProcC, double OmegaC)
/* BPX preconditioned CG method */
{
int Iter;
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, BPXPCGIterId)
&& Iter < MaxIter) {
Iter++;
/* BPX preconditioner */
BPXPrecond(NoLevels, A, z, r, R, P, NoLevels - 1,
SmoothProc, Nu, PrecondProc, Omega, SmoothProcC, 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]);
}
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);
}