/// \brief Verify the result of the "thin" QR factorization \f$A = QR\f$.
///
/// This method returns a list of three magnitudes:
/// - \f$\| A - QR \|_F\f$
/// - \f$\|I - Q^* Q\|_F\f$
/// - \f$\|A\|_F\f$
///
/// The notation $\f\| X \|\f$ denotes the Frobenius norm
/// (square root of sum of squares) of a matrix \f$X\f$.
/// Returning the Frobenius norm of \f$A\f$ allows you to scale
/// or not scale the residual \f$\|A - QR\|\f$ as you prefer.
virtual std::vector< magnitude_type >
verify (const multivector_type& A,
const multivector_type& Q,
const Teuchos::SerialDenseMatrix< local_ordinal_type, scalar_type >& R)
{
using Teuchos::ArrayRCP;
local_ordinal_type nrowsLocal_A, ncols_A, LDA;
local_ordinal_type nrowsLocal_Q, ncols_Q, LDQ;
fetchDims (A, nrowsLocal_A, ncols_A, LDA);
fetchDims (Q, nrowsLocal_Q, ncols_Q, LDQ);
if (nrowsLocal_A != nrowsLocal_Q)
throw std::runtime_error ("A and Q must have same number of rows");
else if (ncols_A != ncols_Q)
throw std::runtime_error ("A and Q must have same number of columns");
else if (ncols_A != R.numCols())
throw std::runtime_error ("A and R must have same number of columns");
else if (R.numRows() < R.numCols())
throw std::runtime_error ("R must have no fewer rows than columns");
// Const views suffice for verification
ArrayRCP< const scalar_type > A_ptr = fetchConstView (A);
ArrayRCP< const scalar_type > Q_ptr = fetchConstView (Q);
return global_verify (nrowsLocal_A, ncols_A, A_ptr.get(), LDA,
Q_ptr.get(), LDQ, R.values(), R.stride(),
pScalarMessenger_.get());
}
void assembleIRKState(
const int stageIndex,
const Teuchos::SerialDenseMatrix<int,Scalar> &A_in,
const Scalar dt,
const Thyra::VectorBase<Scalar> &x_base,
const Thyra::ProductVectorBase<Scalar> &x_stage_bar,
Teuchos::Ptr<Thyra::VectorBase<Scalar> > x_out_ptr
)
{
typedef ScalarTraits<Scalar> ST;
const int numStages_in = A_in.numRows();
TEUCHOS_ASSERT_IN_RANGE_UPPER_EXCLUSIVE( stageIndex, 0, numStages_in );
TEUCHOS_ASSERT_EQUALITY( A_in.numRows(), numStages_in );
TEUCHOS_ASSERT_EQUALITY( A_in.numCols(), numStages_in );
TEUCHOS_ASSERT_EQUALITY( x_stage_bar.productSpace()->numBlocks(), numStages_in );
Thyra::VectorBase<Scalar>& x_out = *x_out_ptr;
V_V( outArg(x_out), x_base );
for ( int j = 0; j < numStages_in; ++j ) {
Vp_StV( outArg(x_out), dt * A_in(stageIndex,j), *x_stage_bar.getVectorBlock(j) );
}
}
void
Stokhos::SmolyakPseudoSpectralOperator<ordinal_type,value_type,point_compare_type>::
transformPCE2QP_smolyak(
const value_type& alpha,
const Teuchos::SerialDenseMatrix<ordinal_type,value_type>& input,
Teuchos::SerialDenseMatrix<ordinal_type,value_type>& result,
const value_type& beta,
bool trans) const {
Teuchos::SerialDenseMatrix<ordinal_type,value_type> op_input, op_result;
result.scale(beta);
for (ordinal_type i=0; i<operators.size(); i++) {
Teuchos::RCP<operator_type> op = operators[i];
if (trans) {
op_input.reshape(input.numRows(), op->coeff_size());
op_result.reshape(result.numRows(), op->point_size());
}
else {
op_input.reshape(op->coeff_size(), input.numCols());
op_result.reshape(op->point_size(), result.numCols());
}
gather(scatter_maps[i], input, trans, op_input);
op->transformPCE2QP(smolyak_coeffs[i], op_input, op_result, 0.0, trans);
scatter(gather_maps[i], op_result, trans, result);
}
}
virtual ordinal_type ApplyInverse(
const Teuchos::SerialDenseMatrix<ordinal_type, value_type>& Input,
Teuchos::SerialDenseMatrix<ordinal_type, value_type>& Result,
ordinal_type m) const {
ordinal_type n=Input.numRows();
Teuchos::SerialDenseMatrix<ordinal_type, value_type> G(A);
Teuchos::SerialDenseMatrix<ordinal_type, value_type> z(n,1);
for (ordinal_type j=0; j<m; j++){
if (j==0){ // Compute z=D-1r
for (ordinal_type i=0; i<n; i++)
z(i,0)=Input(i,0)/A(i,i);
}
else {
//Compute G=invD(-L-U)=I-inv(D)A
for (ordinal_type i=0; i<n; i++){
for (ordinal_type j=0; j<n; j++){
if (j==i)
G(i,j)=0;
else
G(i,j)=-A(i,j)/A(i,i);
}
}
Result.assign(z);
//z=Gz+inv(D)r
Result.multiply(Teuchos::NO_TRANS,Teuchos::NO_TRANS,1.0, G, z, 1.0);
}
}
return 0;
}
void EpetraOpMultiVec::MvTimesMatAddMv ( double alpha, const MultiVec<double>& A,
const Teuchos::SerialDenseMatrix<int,double>& B, double beta )
{
Epetra_LocalMap LocalMap(B.numRows(), 0, Epetra_MV->Map().Comm());
Epetra_MultiVector B_Pvec(Epetra_DataAccess::View, LocalMap, B.values(), B.stride(), B.numCols());
EpetraOpMultiVec *A_vec = dynamic_cast<EpetraOpMultiVec *>(&const_cast<MultiVec<double> &>(A));
TEUCHOS_TEST_FOR_EXCEPTION( A_vec==NULL, std::invalid_argument, "Anasazi::EpetraOpMultiVec::SetBlocks() cast of MultiVec<double> to EpetraOpMultiVec failed.");
TEUCHOS_TEST_FOR_EXCEPTION(
Epetra_MV->Multiply( 'N', 'N', alpha, *(A_vec->GetEpetraMultiVector()), B_Pvec, beta ) != 0,
EpetraSpecializedMultiVecFailure, "Anasazi::EpetraOpMultiVec::MvTimesMatAddMv() call to Epetra_MultiVec::Multiply() returned a nonzero value.");
}
// Update *this with alpha * A * B + beta * (*this).
void MvTimesMatAddMv (ScalarType alpha, const Anasazi::MultiVec<ScalarType> &A,
const Teuchos::SerialDenseMatrix<int, ScalarType> &B,
ScalarType beta)
{
assert (Length_ == A.GetVecLength());
assert (B.numRows() == A.GetNumberVecs());
assert (B.numCols() <= NumberVecs_);
MyMultiVec* MyA;
MyA = dynamic_cast<MyMultiVec*>(&const_cast<Anasazi::MultiVec<ScalarType> &>(A));
assert(MyA!=NULL);
if ((*this)[0] == (*MyA)[0]) {
// If this == A, then need additional storage ...
// This situation is a bit peculiar but it may be required by
// certain algorithms.
std::vector<ScalarType> tmp(NumberVecs_);
for (int i = 0 ; i < Length_ ; ++i) {
for (int v = 0; v < A.GetNumberVecs() ; ++v) {
tmp[v] = (*MyA)(i, v);
}
for (int v = 0 ; v < B.numCols() ; ++v) {
(*this)(i, v) *= beta;
ScalarType res = Teuchos::ScalarTraits<ScalarType>::zero();
for (int j = 0 ; j < A.GetNumberVecs() ; ++j) {
res += tmp[j] * B(j, v);
}
(*this)(i, v) += alpha * res;
}
}
}
else {
for (int i = 0 ; i < Length_ ; ++i) {
for (int v = 0 ; v < B.numCols() ; ++v) {
(*this)(i, v) *= beta;
ScalarType res = 0.0;
for (int j = 0 ; j < A.GetNumberVecs() ; ++j) {
res += (*MyA)(i, j) * B(j, v);
}
(*this)(i, v) += alpha * res;
}
}
}
}
void EpetraMultiVec::MvTransMv ( const double alpha, const MultiVec<double>& A,
Teuchos::SerialDenseMatrix<int,double>& B) const
{
EpetraMultiVec *A_vec = dynamic_cast<EpetraMultiVec *>(&const_cast<MultiVec<double> &>(A));
if (A_vec) {
Epetra_LocalMap LocalMap(B.numRows(), 0, Map().Comm());
Epetra_MultiVector B_Pvec(View, LocalMap, B.values(), B.stride(), B.numCols());
int info = B_Pvec.Multiply( 'T', 'N', alpha, *A_vec, *this, 0.0 );
TEST_FOR_EXCEPTION(info!=0, EpetraMultiVecFailure,
"Belos::EpetraMultiVec::MvTransMv call to Multiply() returned a nonzero value.");
}
}
// Generic BLAS level 3 matrix multiplication
// \f$\text{this}\leftarrow \alpha A B+\beta\text{this}\f$
void gemm(const Real alpha,
const MV& A,
const Teuchos::SerialDenseMatrix<int,Real> &B,
const Real beta) {
// Scale this by beta
this->scale(beta);
for(int i=0;i<B.numRows();++i) {
for(int j=0;j<B.numCols();++j) {
mvec_[j]->axpy(alpha*B(i,j),*A.getVector(i));
}
}
}
void EpetraMultiVec::MvTimesMatAddMv ( const double alpha, const MultiVec<double>& A,
const Teuchos::SerialDenseMatrix<int,double>& B, const double beta )
{
Epetra_LocalMap LocalMap(B.numRows(), 0, Map().Comm());
Epetra_MultiVector B_Pvec(View, LocalMap, B.values(), B.stride(), B.numCols());
EpetraMultiVec *A_vec = dynamic_cast<EpetraMultiVec *>(&const_cast<MultiVec<double> &>(A));
TEST_FOR_EXCEPTION(A_vec==NULL, EpetraMultiVecFailure,
"Belos::EpetraMultiVec::MvTimesMatAddMv cast from Belos::MultiVec<> to Belos::EpetraMultiVec failed.");
int info = Multiply( 'N', 'N', alpha, *A_vec, B_Pvec, beta );
TEST_FOR_EXCEPTION(info!=0, EpetraMultiVecFailure,
"Belos::EpetraMultiVec::MvTimesMatAddMv call to Multiply() returned a nonzero value.");
}
LocalOrdinal
revealRank (Kokkos::MultiVector<Scalar, NodeType>& Q,
Teuchos::SerialDenseMatrix<LocalOrdinal, Scalar>& R,
const magnitude_type& tol,
const bool contiguousCacheBlocks = false) const
{
typedef Kokkos::MultiVector<Scalar, NodeType> KMV;
const LocalOrdinal nrows = static_cast<LocalOrdinal> (Q.getNumRows());
const LocalOrdinal ncols = static_cast<LocalOrdinal> (Q.getNumCols());
const LocalOrdinal ldq = static_cast<LocalOrdinal> (Q.getStride());
Teuchos::ArrayRCP<Scalar> Q_ptr = Q.getValuesNonConst();
// Take the easy exit if available.
if (ncols == 0)
return 0;
//
// FIXME (mfh 16 Jul 2010) We _should_ compute the SVD of R (as
// the copy B) on Proc 0 only. This would ensure that all
// processors get the same SVD and rank (esp. in a heterogeneous
// computing environment). For now, we just do this computation
// redundantly, and hope that all the returned rank values are
// the same.
//
matrix_type U (ncols, ncols, STS::zero());
const ordinal_type rank =
reveal_R_rank (ncols, R.values(), R.stride(),
U.get(), U.lda(), tol);
if (rank < ncols)
{
// cerr << ">>> Rank of R: " << rank << " < ncols=" << ncols << endl;
// cerr << ">>> Resulting U:" << endl;
// print_local_matrix (cerr, ncols, ncols, R, ldr);
// cerr << endl;
// If R is not full rank: reveal_R_rank() already computed
// the SVD \f$R = U \Sigma V^*\f$ of (the input) R, and
// overwrote R with \f$\Sigma V^*\f$. Now, we compute \f$Q
// := Q \cdot U\f$, respecting cache blocks of Q.
Q_times_B (nrows, ncols, Q_ptr.getRawPtr(), ldq,
U.get(), U.lda(), contiguousCacheBlocks);
}
return rank;
}
void EpetraMultiVec::MvTransMv ( double alpha, const MultiVec<double>& A,
Teuchos::SerialDenseMatrix<int,double>& B
#ifdef HAVE_ANASAZI_EXPERIMENTAL
, ConjType conj
#endif
) const
{
EpetraMultiVec *A_vec = dynamic_cast<EpetraMultiVec *>(&const_cast<MultiVec<double> &>(A));
if (A_vec) {
Epetra_LocalMap LocalMap(B.numRows(), 0, Map().Comm());
Epetra_MultiVector B_Pvec(View, LocalMap, B.values(), B.stride(), B.numCols());
TEUCHOS_TEST_FOR_EXCEPTION(
B_Pvec.Multiply( 'T', 'N', alpha, *A_vec, *this, 0.0 ) != 0,
EpetraMultiVecFailure, "Anasazi::EpetraMultiVec::MvTransMv() call to Epetra_MultiVec::Multiply() returned a nonzero value.");
}
}
void
Stokhos::SmolyakPseudoSpectralOperator<ordinal_type,value_type,point_compare_type>::
scatter(
const Teuchos::Array<ordinal_type>& map,
const Teuchos::SerialDenseMatrix<ordinal_type,value_type>& input,
bool trans,
Teuchos::SerialDenseMatrix<ordinal_type,value_type>& result) const {
if (trans) {
for (ordinal_type j=0; j<map.size(); j++)
for (ordinal_type i=0; i<input.numRows(); i++)
result(i,map[j]) += input(i,j);
}
else {
for (ordinal_type j=0; j<input.numCols(); j++)
for (ordinal_type i=0; i<map.size(); i++)
result(map[i],j) += input(i,j);
}
}
// Compute a dense matrix B through the matrix-matrix multiply alpha * A^H * (*this).
void MvTransMv (ScalarType alpha, const Anasazi::MultiVec<ScalarType>& A,
Teuchos::SerialDenseMatrix< int, ScalarType >& B
#ifdef HAVE_ANASAZI_EXPERIMENTAL
, Anasazi::ConjType conj
#endif
) const
{
MyMultiVec* MyA;
MyA = dynamic_cast<MyMultiVec*>(&const_cast<Anasazi::MultiVec<ScalarType> &>(A));
assert (MyA != 0);
assert (A.GetVecLength() == Length_);
assert (NumberVecs_ <= B.numCols());
assert (A.GetNumberVecs() <= B.numRows());
#ifdef HAVE_ANASAZI_EXPERIMENTAL
if (conj == Anasazi::CONJ) {
#endif
for (int v = 0 ; v < A.GetNumberVecs() ; ++v) {
for (int w = 0 ; w < NumberVecs_ ; ++w) {
ScalarType value = 0.0;
for (int i = 0 ; i < Length_ ; ++i) {
value += Teuchos::ScalarTraits<ScalarType>::conjugate((*MyA)(i, v)) * (*this)(i, w);
}
B(v, w) = alpha * value;
}
}
#ifdef HAVE_ANASAZI_EXPERIMENTAL
} else {
for (int v = 0 ; v < A.GetNumberVecs() ; ++v) {
for (int w = 0 ; w < NumberVecs_ ; ++w) {
ScalarType value = 0.0;
for (int i = 0 ; i < Length_ ; ++i) {
value += (*MyA)(i, v) * (*this)(i, w);
}
B(v, w) = alpha * value;
}
}
}
#endif
}
// Compute a dense matrix B through the matrix-matrix multiply alpha * A^H * (*this).
void MvTransMv (const ScalarType alpha, const Belos::MultiVec<ScalarType>& A,
Teuchos::SerialDenseMatrix< int, ScalarType >& B) const
{
MyMultiVec* MyA;
MyA = dynamic_cast<MyMultiVec*>(&const_cast<Belos::MultiVec<ScalarType> &>(A));
TEUCHOS_ASSERT(MyA != NULL);
assert (A.GetGlobalLength() == Length_);
assert (NumberVecs_ <= B.numCols());
assert (A.GetNumberVecs() <= B.numRows());
for (int v = 0 ; v < A.GetNumberVecs() ; ++v) {
for (int w = 0 ; w < NumberVecs_ ; ++w) {
ScalarType value = 0.0;
for (int i = 0 ; i < Length_ ; ++i) {
value += Teuchos::ScalarTraits<ScalarType>::conjugate((*MyA)(i, v)) * (*this)(i, w);
}
B(v, w) = alpha * value;
}
}
}
void EpetraOpMultiVec::MvTransMv ( double alpha, const MultiVec<double>& A,
Teuchos::SerialDenseMatrix<int,double>& B
#ifdef HAVE_ANASAZI_EXPERIMENTAL
, ConjType conj
#endif
) const
{
EpetraOpMultiVec *A_vec = dynamic_cast<EpetraOpMultiVec *>(&const_cast<MultiVec<double> &>(A));
if (A_vec) {
Epetra_LocalMap LocalMap(B.numRows(), 0, Epetra_MV->Map().Comm());
Epetra_MultiVector B_Pvec(Epetra_DataAccess::View, LocalMap, B.values(), B.stride(), B.numCols());
int info = Epetra_OP->Apply( *Epetra_MV, *Epetra_MV_Temp );
TEUCHOS_TEST_FOR_EXCEPTION( info != 0, EpetraSpecializedMultiVecFailure,
"Anasazi::EpetraOpMultiVec::MvTransMv(): Error returned from Epetra_Operator::Apply()" );
TEUCHOS_TEST_FOR_EXCEPTION(
B_Pvec.Multiply( 'T', 'N', alpha, *(A_vec->GetEpetraMultiVector()), *Epetra_MV_Temp, 0.0 ) != 0,
EpetraSpecializedMultiVecFailure, "Anasazi::EpetraOpMultiVec::MvTransMv() call to Epetra_MultiVector::Multiply() returned a nonzero value.");
}
}
void updateGuess(Teuchos::SerialDenseVector<int, std::complex<double> >& myCurrentGuess,
Teuchos::SerialDenseVector<int, std::complex<double> >& myTargetsCalculated,
Teuchos::SerialDenseMatrix<int, std::complex<double> >& myJacobian,
Teuchos::LAPACK<int, std::complex<double> >& myLAPACK
)
{
//v = J(inverse) * (-F(x))
//new guess = v + old guess
myTargetsCalculated *= -1.0;
//Perform an LU factorization of this matrix.
int ipiv[NUMDIMENSIONS], info;
char TRANS = 'N';
myLAPACK.GETRF( NUMDIMENSIONS, NUMDIMENSIONS, myJacobian.values(), myJacobian.stride(), ipiv, &info );
// Solve the linear system.
myLAPACK.GETRS( TRANS, NUMDIMENSIONS, 1, myJacobian.values(), myJacobian.stride(),
ipiv, myTargetsCalculated.values(), myTargetsCalculated.stride(), &info );
//We have overwritten myTargetsCalculated with guess update values
//myBLAS.AXPY(NUMDIMENSIONS, 1.0, myGuessAdjustment.values(), 1, myCurrentGuess.values(), 1);
myCurrentGuess += myTargetsCalculated;
}
/*! \brief Update \c mv with \f$ \alpha A B + \beta mv \f$.
*/
static void MvTimesMatAddMv( const double alpha, const _MV & A,
const Teuchos::SerialDenseMatrix<int,double>& B,
const double beta, _MV & mv )
{
// Out::os() << "MvTimesMatAddMv()" << endl;
int n = B.numCols();
// Out::os() << "B.numCols()=" << n << endl;
TEST_FOR_EXCEPT(mv.size() != n);
for (int j=0; j<mv.size(); j++)
{
Vector<double> tmp;
if (beta==one())
{
tmp = mv[j].copy();
}
else if (beta==zero())
{
tmp = mv[j].copy();
tmp.setToConstant(zero());
}
else
{
tmp = beta * mv[j];
}
if (alpha != zero())
{
for (int i=0; i<A.size(); i++)
{
tmp = tmp + alpha*B(i,j)*A[i];
}
}
mv[j].acceptCopyOf(tmp);
}
}
ordinal_type
Stokhos::CGDivisionExpansionStrategy<ordinal_type,value_type,node_type>::
CG(const Teuchos::SerialDenseMatrix<ordinal_type, value_type> & A,
Teuchos::SerialDenseMatrix<ordinal_type,value_type> & X,
const Teuchos::SerialDenseMatrix<ordinal_type,value_type> & B,
ordinal_type max_iter,
value_type tolerance,
ordinal_type prec_iter,
ordinal_type order ,
ordinal_type m,
ordinal_type PrecNum,
const Teuchos::SerialDenseMatrix<ordinal_type, value_type> & M,
ordinal_type diag)
{
ordinal_type n = A.numRows();
ordinal_type k=0;
value_type resid;
Teuchos::SerialDenseMatrix<ordinal_type, value_type> Ax(n,1);
Ax.multiply(Teuchos::NO_TRANS,Teuchos::NO_TRANS,1.0, A, X, 0.0);
Teuchos::SerialDenseMatrix<ordinal_type, value_type> r(Teuchos::Copy,B);
r-=Ax;
resid=r.normFrobenius();
Teuchos::SerialDenseMatrix<ordinal_type, value_type> p(r);
Teuchos::SerialDenseMatrix<ordinal_type, value_type> rho(1,1);
Teuchos::SerialDenseMatrix<ordinal_type, value_type> oldrho(1,1);
Teuchos::SerialDenseMatrix<ordinal_type, value_type> pAp(1,1);
Teuchos::SerialDenseMatrix<ordinal_type, value_type> Ap(n,1);
value_type b;
value_type a;
while (resid > tolerance && k < max_iter){
Teuchos::SerialDenseMatrix<ordinal_type, value_type> z(r);
//Solve Mz=r
if (PrecNum != 0){
if (PrecNum == 1){
Stokhos::DiagPreconditioner<ordinal_type, value_type> precond(M);
precond.ApplyInverse(r,z,prec_iter);
}
else if (PrecNum == 2){
Stokhos::JacobiPreconditioner<ordinal_type, value_type> precond(M);
precond.ApplyInverse(r,z,2);
}
else if (PrecNum == 3){
Stokhos::GSPreconditioner<ordinal_type, value_type> precond(M,0);
precond.ApplyInverse(r,z,1);
}
else if (PrecNum == 4){
Stokhos::SchurPreconditioner<ordinal_type, value_type> precond(M, order, m, diag);
precond.ApplyInverse(r,z,prec_iter);
}
}
rho.multiply(Teuchos::TRANS,Teuchos::NO_TRANS,1.0, r, z, 0.0);
if (k==0){
p.assign(z);
rho.multiply(Teuchos::TRANS, Teuchos::NO_TRANS, 1.0, r, z, 0.0);
}
else {
b=rho(0,0)/oldrho(0,0);
p.scale(b);
p+=z;
}
Ap.multiply(Teuchos::NO_TRANS,Teuchos::NO_TRANS,1.0, A, p, 0.0);
pAp.multiply(Teuchos::TRANS,Teuchos::NO_TRANS,1.0, p, Ap, 0.0);
a=rho(0,0)/pAp(0,0);
Teuchos::SerialDenseMatrix<ordinal_type, value_type> scalep(p);
scalep.scale(a);
X+=scalep;
Ap.scale(a);
r-=Ap;
oldrho.assign(rho);
resid=r.normFrobenius();
k++;
}
//std::cout << "iteration count " << k << std::endl;
return 0;
}
//Mean-Based Preconditioned GMRES
int pregmres(const Teuchos::SerialDenseMatrix<int, double> & A, const Teuchos::SerialDenseMatrix<int,double> & X,const Teuchos::SerialDenseMatrix<int,double> & B, int max_iter, double tolerance)
{
int n;
int k;
double resid;
k=1;
n=A.numRows();
std::cout << A << std::endl;
Teuchos::SerialDenseMatrix<int, double> D(n,1);
//Get diagonal entries of A
for (int i=0; i<n; i++){
D(i,0)=A(i,i);
}
Teuchos::SerialDenseMatrix<int, double> Ax(n,1);
Ax.multiply(Teuchos::NO_TRANS,Teuchos::NO_TRANS,1.0, A, X, 0.0);
Teuchos::SerialDenseMatrix<int, double> r0(B);
r0-=Ax;
resid=r0.normFrobenius();
//define vector v=r/norm(r) where r=b-Ax
Teuchos::SerialDenseMatrix<int, double> v(n,1);
r0.scale(1/resid);
Teuchos::SerialDenseMatrix<int, double> h(1,1);
//Matrix of orthog basis vectors V
Teuchos::SerialDenseMatrix<int, double> V(n,1);
//Set v=r0/norm(r0) to be 1st col of V
for (int i=0; i<n; i++){
V(i,0)=r0(i,0);
}
//right hand side
Teuchos::SerialDenseMatrix<int, double> bb(1,1);
bb(0,0)=resid;
Teuchos::SerialDenseMatrix<int, double> w(n,1);
Teuchos::SerialDenseMatrix<int, double> c;
Teuchos::SerialDenseMatrix<int, double> s;
while (resid > tolerance && k < max_iter){
std::cout << "k = " << k << std::endl;
h.reshape(k+1,k);
//Arnoldi iteration(Gram-Schmidt )
V.reshape(n,k+1);
//set vk to be kth col of V
Teuchos::SerialDenseMatrix<int, double> vk(Teuchos::Copy, V, n,1,0,k-1);
//Preconditioning step w=AMj(-1)vj
w.multiply(Teuchos::NO_TRANS, Teuchos::NO_TRANS, 1/D(k-1,0), A, vk, 0.0);
Teuchos::SerialDenseMatrix<int, double> vi(n,1);
Teuchos::SerialDenseMatrix<int, double> ip(1,1);
for (int i=0; i<k; i++){
//set vi to be ith col of V
Teuchos::SerialDenseMatrix<int, double> vi(Teuchos::Copy, V, n,1,0,i);
//Calculate inner product
ip.multiply(Teuchos::TRANS, Teuchos::NO_TRANS, 1.0, vi, w, 0.0);
h(i,k-1)= ip(0,0);
//scale vi by h(i,k-1)
vi.scale(ip(0,0));
w-=vi;
}
h(k,k-1)=w.normFrobenius();
w.scale(1.0/w.normFrobenius());
//add column vk+1=w to V
for (int i=0; i<n; i++){
V(i,k)=w(i,0);
}
//Solve upper hessenberg least squares problem via Givens rotations
//Compute previous Givens rotations
for (int i=0; i<k-1; i++){
h(i,k-1)=c(i,0)*h(i,k-1)+s(i,0)*h(i+1,k-1);
h(i+1,k-1)=-s(i,0)*h(i,k-1)+c(i,0)*h(i+1,k-1);
}
//Compute next Givens rotations
c.reshape(k,1);
s.reshape(k,1);
bb.reshape(k+1,1);
double l = sqrt(h(k-1,k-1)*h(k-1,k-1)+h(k,k-1)*h(k,k-1));
c(k-1,0)=h(k-1,k-1)/l;
s(k-1,0)=h(k,k-1)/l;
std::cout <<" h(k,k-1)= " << h(k,k-1) << std::endl;
// Givens rotation on h and bb
h(k-1,k-1)=l;
h(k,k-1)=0;
bb(k-1,0)=c(k-1,0)*bb(k-1,0);
bb(k,0)=-s(k-1,0)*bb(k-1,0);
//Determine residual
resid = fabs(bb(k,0));
std::cout << "resid = " << resid << std::endl;
k=k+1;
}
//Extract upper triangular square matrix
bb.reshape(h.numRows()-1 ,1);
//Solve linear system
int info;
Teuchos::LAPACK<int, double> lapack;
lapack.TRTRS('U', 'N', 'N', h.numRows()-1, 1, h.values(), h.stride(), bb.values(), bb.stride(),&info);
//Found y=Mx
for (int i=0; i<k-1; i++){
bb(i,0)=bb(i,0)/D(i,0);
}
V.reshape(n,k-1);
Teuchos::SerialDenseMatrix<int, double> ans(X);
ans.multiply(Teuchos::NO_TRANS, Teuchos::NO_TRANS, 1.0, V, bb, 1.0);
std::cout << "ans= " << ans << std::endl;
std::cout << "h= " << h << std::endl;
return 0;
}
void
Stokhos::SmolyakPseudoSpectralOperator<ordinal_type,value_type,point_compare_type>::
apply_direct(
const Teuchos::SerialDenseMatrix<ordinal_type,value_type>& A,
const value_type& alpha,
const Teuchos::SerialDenseMatrix<ordinal_type,value_type>& input,
Teuchos::SerialDenseMatrix<ordinal_type,value_type>& result,
const value_type& beta,
bool trans) const {
if (trans) {
TEUCHOS_ASSERT(input.numCols() <= A.numCols());
TEUCHOS_ASSERT(result.numCols() == A.numRows());
TEUCHOS_ASSERT(result.numRows() == input.numRows());
blas.GEMM(Teuchos::NO_TRANS, Teuchos::TRANS, input.numRows(),
A.numRows(), input.numCols(), alpha, input.values(),
input.stride(), A.values(), A.stride(), beta,
result.values(), result.stride());
}
else {
TEUCHOS_ASSERT(input.numRows() <= A.numCols());
TEUCHOS_ASSERT(result.numRows() == A.numRows());
TEUCHOS_ASSERT(result.numCols() == input.numCols());
blas.GEMM(Teuchos::NO_TRANS, Teuchos::NO_TRANS, A.numRows(),
input.numCols(), input.numRows(), alpha, A.values(),
A.stride(), input.values(), input.stride(), beta,
result.values(), result.stride());
}
}
ordinal_type
Stokhos::MonomialGramSchmidtPCEBasis<ordinal_type, value_type>::
buildReducedBasis(
ordinal_type max_p,
value_type threshold,
const Teuchos::SerialDenseMatrix<ordinal_type,value_type>& A,
const Teuchos::SerialDenseMatrix<ordinal_type,value_type>& F,
const Teuchos::Array<value_type>& weights,
Teuchos::Array< Stokhos::MultiIndex<ordinal_type> >& terms_,
Teuchos::Array<ordinal_type>& num_terms_,
Teuchos::SerialDenseMatrix<ordinal_type,value_type>& Qp_,
Teuchos::SerialDenseMatrix<ordinal_type,value_type>& Q_)
{
// Compute basis terms -- 2-D array giving powers for each linear index
ordinal_type max_sz;
CPBUtils::compute_terms(max_p, this->d, max_sz, terms_, num_terms_);
// Compute B matrix -- monomials in F
// for i=0,...,nqp-1
// for j=0,...,sz-1
// B(i,j) = F(i,1)^terms_[j][1] * ... * F(i,d)^terms_[j][d]
// where sz is the total size of a basis up to order p and terms_[j]
// is an array of powers for each term in the total-order basis
ordinal_type nqp = weights.size();
SDM B(nqp, max_sz);
for (ordinal_type i=0; i<nqp; i++) {
for (ordinal_type j=0; j<max_sz; j++) {
B(i,j) = 1.0;
for (ordinal_type k=0; k<this->d; k++)
B(i,j) *= std::pow(F(i,k), terms_[j][k]);
}
}
// Rescale columns of B to have unit norm
for (ordinal_type j=0; j<max_sz; j++) {
value_type nrm = 0.0;
for (ordinal_type i=0; i<nqp; i++)
nrm += B(i,j)*B(i,j)*weights[i];
nrm = std::sqrt(nrm);
for (ordinal_type i=0; i<nqp; i++)
B(i,j) /= nrm;
}
// Compute our new basis -- each column of Q is the new basis evaluated
// at the original quadrature points. Constraint pivoting so first d+1
// columns and included in Q.
SDM R;
Teuchos::Array<ordinal_type> piv(max_sz);
for (int i=0; i<this->d+1; i++)
piv[i] = 1;
typedef Stokhos::OrthogonalizationFactory<ordinal_type,value_type> SOF;
ordinal_type sz_ = SOF::createOrthogonalBasis(
this->orthogonalization_method, threshold, this->verbose, B, weights,
Q_, R, piv);
// Compute Qp = A^T*W*Q
SDM tmp(nqp, sz_);
Qp_.reshape(this->pce_sz, sz_);
for (ordinal_type i=0; i<nqp; i++)
for (ordinal_type j=0; j<sz_; j++)
tmp(i,j) = Q_(i,j)*weights[i];
ordinal_type ret =
Qp_.multiply(Teuchos::TRANS, Teuchos::NO_TRANS, 1.0, A, tmp, 0.0);
TEUCHOS_ASSERT(ret == 0);
// It isn't clear that Qp is orthogonal, but if you derive the projection
// matrix from the original space to the reduced, you end up with
// Q^T*W*A = Qp^T
return sz_;
}
//GMRES
int gmres(const Teuchos::SerialDenseMatrix<int, double> & A, Teuchos::SerialDenseMatrix<int,double> X,const Teuchos::SerialDenseMatrix<int,double> & B, int max_iter, double tolerance)
{
int n;
int k;
double resid;
k=1;
n=A.numRows();
std::cout << "A= " << A << std::endl;
std::cout << "B= " << B << std::endl;
//Teuchos::SerialDenseMatrix<int, double> Ax(n,1);
//Ax.multiply(Teuchos::NO_TRANS,Teuchos::NO_TRANS,1.0, A, X, 0.0);
Teuchos::SerialDenseMatrix<int, double> r0(B);
//r0-=Ax;
resid=r0.normFrobenius();
std::cout << "resid= " << resid << std::endl;
//define vector v=r/norm(r) where r=b-Ax
r0.scale(1/resid);
Teuchos::SerialDenseMatrix<int, double> h(1,1);
//Matrix of orthog basis vectors V
Teuchos::SerialDenseMatrix<int, double> V(n,1);
//Set v=r0/norm(r0) to be 1st col of V
for (int i=0; i<n; i++){
V(i,0)=r0(i,0);
}
//right hand side
Teuchos::SerialDenseMatrix<int, double> bb(1,1);
bb(0,0)=resid;
Teuchos::SerialDenseMatrix<int, double> w(n,1);
Teuchos::SerialDenseMatrix<int, double> c;
Teuchos::SerialDenseMatrix<int, double> s;
while (resid > tolerance && k < max_iter){
std::cout << "k = " << k << std::endl;
h.reshape(k+1,k);
//Arnoldi iteration(Gram-Schmidt )
V.reshape(n,k+1);
//set vk to be kth col of V
Teuchos::SerialDenseMatrix<int, double> vk(Teuchos::Copy, V, n,1,0,k-1);
w.multiply(Teuchos::NO_TRANS, Teuchos::NO_TRANS, 1.0, A, vk, 0.0);
Teuchos::SerialDenseMatrix<int, double> vi(n,1);
Teuchos::SerialDenseMatrix<int, double> ip(1,1);
for (int i=0; i<k; i++){
//set vi to be ith col of V
Teuchos::SerialDenseMatrix<int, double> vi(Teuchos::Copy, V, n,1,0,i);
//Calculate inner product
ip.multiply(Teuchos::TRANS, Teuchos::NO_TRANS, 1.0, vi, w, 0.0);
h(i,k-1)= ip(0,0);
//scale vi by h(i,k-1)
vi.scale(ip(0,0));
w-=vi;
}
h(k,k-1)=w.normFrobenius();
w.scale(1.0/w.normFrobenius());
//add column vk+1=w to V
for (int i=0; i<n; i++){
V(i,k)=w(i,0);
}
//Solve upper hessenberg least squares problem via Givens rotations
//Compute previous Givens rotations
for (int i=0; i<k-1; i++){
// double hi=h(i,k-1);
// double hi1=h(i+1,k-1);
// h(i,k-1)=c(i,0)*h(i,k-1)+s(i,0)*h(i+1,k-1);
// h(i+1,k-1)=-1*s(i,0)*h(i,k-1)+c(i,0)*h(i+1,k-1);
// h(i,k-1)=c(i,0)*hi+s(i,0)*hi1;
// h(i+1,k-1)=-1*s(i,0)*hi+c(i,0)*hi1;
double q=c(i,0)*h(i,k-1)+s(i,0)*h(i+1,k-1);
h(i+1,k-1)=-1*s(i,0)*h(i,k-1)+c(i,0)*h(i+1,k-1);
h(i,k-1)=q;
}
//Compute next Givens rotations
c.reshape(k,1);
s.reshape(k,1);
bb.reshape(k+1,1);
double l = sqrt(h(k-1,k-1)*h(k-1,k-1)+h(k,k-1)*h(k,k-1));
c(k-1,0)=h(k-1,k-1)/l;
s(k-1,0)=h(k,k-1)/l;
std::cout << "c " << c(k-1,0)<<std::endl;
std::cout << "s " << s(k-1,0)<<std::endl;
// Givens rotation on h and bb
// h(k-1,k-1)=l;
// h(k,k-1)=0;
double hk=h(k,k-1);
double hk1=h(k-1,k-1);
h(k-1,k-1)=c(k-1,0)*hk1+s(k-1,0)*hk;
h(k,k-1)=-1*s(k-1,0)*hk1+c(k-1,0)*hk;
std::cout << "l = " << l <<std::endl;
std::cout << "h(k-1,k-1) = should be l " << h(k-1,k-1) <<std::endl;
std::cout << "h(k,k-1) = should be 0 " << h(k,k-1) <<std::endl;
bb(k,0)=-1*s(k-1,0)*bb(k-1,0);
bb(k-1,0)=c(k-1,0)*bb(k-1,0);
//Determine residual
resid =fabs(bb(k,0));
std::cout << "resid = " << resid <<std::endl;
k++;
}
//Extract upper triangular square matrix
bb.reshape(h.numRows()-1 ,1);
//Solve linear system
int info;
std::cout << "bb pre solve = " << bb << std::endl;
std::cout << "h= " << h << std::endl;
Teuchos::LAPACK<int, double> lapack;
lapack.TRTRS('U', 'N', 'N', h.numRows()-1, 1, h.values(), h.stride(), bb.values(), bb.stride(),&info);
V.reshape(n,k-1);
std::cout << "V= " << V << std::endl;
std::cout << "y= " << bb << std::endl;
X.multiply(Teuchos::NO_TRANS, Teuchos::NO_TRANS, 1.0, V, bb, 1.0);
std::cout << "X= " << X << std::endl;
//Check V is orthogoanl
// Teuchos::SerialDenseMatrix<int, double> vtv(V);
// vtv.multiply(Teuchos::TRANS, Teuchos::NO_TRANS, 1.0, V, V, 0.0);
// std::cout << "Vtv" << vtv << std::endl;
return 0;
}
// CG
int CG(const Teuchos::SerialDenseMatrix<int, double> & A, Teuchos::SerialDenseMatrix<int,double> X,const Teuchos::SerialDenseMatrix<int,double> & B, int max_iter, double tolerance, Stokhos::DiagPreconditioner<int,double> prec)
{
int n;
int k=0;
double resid;
n=A.numRows();
std::cout << "A= " << A << std::endl;
std::cout << "B= " << B << std::endl;
Teuchos::SerialDenseMatrix<int, double> Ax(n,1);
Ax.multiply(Teuchos::NO_TRANS,Teuchos::NO_TRANS,1.0, A, X, 0.0);
Teuchos::SerialDenseMatrix<int, double> r(B);
r-=Ax;
resid=r.normFrobenius();
Teuchos::SerialDenseMatrix<int, double> rho(1,1);
Teuchos::SerialDenseMatrix<int, double> oldrho(1,1);
Teuchos::SerialDenseMatrix<int, double> pAp(1,1);
Teuchos::SerialDenseMatrix<int, double> Ap(n,1);
double b;
double a;
Teuchos::SerialDenseMatrix<int, double> p(r);
while (resid > tolerance && k < max_iter){
Teuchos::SerialDenseMatrix<int, double> z(r);
//z=M-1r
// prec.ApplyInverse(r,z);
rho.multiply(Teuchos::TRANS,Teuchos::NO_TRANS,1.0, r, z, 0.0);
if (k==0){
p.assign(z);
rho.multiply(Teuchos::TRANS, Teuchos::NO_TRANS, 1.0, r, z, 0.0);
}
else {
b=rho(0,0)/oldrho(0,0);
p.scale(b);
p+=z;
}
Ap.multiply(Teuchos::NO_TRANS,Teuchos::NO_TRANS,1.0, A, p, 0.0);
pAp.multiply(Teuchos::TRANS,Teuchos::NO_TRANS,1.0, p, Ap, 0.0);
a=rho(0,0)/pAp(0,0);
Teuchos::SerialDenseMatrix<int, double> scalep(p);
scalep.scale(a);
X+=scalep;
Ap.scale(a);
r-=Ap;
oldrho.assign(rho);
resid=r.normFrobenius();
k++;
}
std::cout << "X= " << X << std::endl;
return 0;
}
void DislocationDensity<EvalT, Traits>::
evaluateFields(typename Traits::EvalData workset)
{
Teuchos::SerialDenseMatrix<int, double> A;
Teuchos::SerialDenseMatrix<int, double> X;
Teuchos::SerialDenseMatrix<int, double> B;
Teuchos::SerialDenseSolver<int, double> solver;
A.shape(numNodes,numNodes);
X.shape(numNodes,numNodes);
B.shape(numNodes,numNodes);
// construct Identity for RHS
for (int i = 0; i < numNodes; ++i)
B(i,i) = 1.0;
for (int i=0; i < G.size() ; i++) G[i] = 0.0;
// construct the node --> point operator
for (std::size_t cell=0; cell < workset.numCells; ++cell)
{
for (std::size_t node=0; node < numNodes; ++node)
for (std::size_t qp=0; qp < numQPs; ++qp)
A(qp,node) = BF(cell,node,qp);
X = 0.0;
solver.setMatrix( Teuchos::rcp( &A, false) );
solver.setVectors( Teuchos::rcp( &X, false ), Teuchos::rcp( &B, false ) );
// Solve the system A X = B to find A_inverse
int status = 0;
status = solver.factor();
status = solver.solve();
// compute nodal Fp
nodalFp.initialize(0.0);
for (std::size_t node=0; node < numNodes; ++node)
for (std::size_t qp=0; qp < numQPs; ++qp)
for (std::size_t i=0; i < numDims; ++i)
for (std::size_t j=0; j < numDims; ++j)
nodalFp(node,i,j) += X(node,qp) * Fp(cell,qp,i,j);
// compute the curl using nodalFp
curlFp.initialize(0.0);
for (std::size_t node=0; node < numNodes; ++node)
{
for (std::size_t qp=0; qp < numQPs; ++qp)
{
curlFp(qp,0,0) += nodalFp(node,0,2) * GradBF(cell,node,qp,1) - nodalFp(node,0,1) * GradBF(cell,node,qp,2);
curlFp(qp,0,1) += nodalFp(node,1,2) * GradBF(cell,node,qp,1) - nodalFp(node,1,1) * GradBF(cell,node,qp,2);
curlFp(qp,0,2) += nodalFp(node,2,2) * GradBF(cell,node,qp,1) - nodalFp(node,2,1) * GradBF(cell,node,qp,2);
curlFp(qp,1,0) += nodalFp(node,0,0) * GradBF(cell,node,qp,2) - nodalFp(node,0,2) * GradBF(cell,node,qp,0);
curlFp(qp,1,1) += nodalFp(node,1,0) * GradBF(cell,node,qp,2) - nodalFp(node,1,2) * GradBF(cell,node,qp,0);
curlFp(qp,1,2) += nodalFp(node,2,0) * GradBF(cell,node,qp,2) - nodalFp(node,2,2) * GradBF(cell,node,qp,0);
curlFp(qp,2,0) += nodalFp(node,0,1) * GradBF(cell,node,qp,0) - nodalFp(node,0,0) * GradBF(cell,node,qp,1);
curlFp(qp,2,1) += nodalFp(node,1,1) * GradBF(cell,node,qp,0) - nodalFp(node,1,0) * GradBF(cell,node,qp,1);
curlFp(qp,2,2) += nodalFp(node,2,1) * GradBF(cell,node,qp,0) - nodalFp(node,2,0) * GradBF(cell,node,qp,1);
}
}
for (std::size_t qp=0; qp < numQPs; ++qp)
for (std::size_t i=0; i < numDims; ++i)
for (std::size_t j=0; j < numDims; ++j)
for (std::size_t k=0; k < numDims; ++k)
G(cell,qp,i,j) += Fp(cell,qp,i,k) * curlFp(qp,k,j);
}
}
int
Stokhos::GMRESDivisionExpansionStrategy<ordinal_type,value_type,node_type>::
GMRES(const Teuchos::SerialDenseMatrix<int, double> & A, Teuchos::SerialDenseMatrix<int,double> & X, const Teuchos::SerialDenseMatrix<int,double> & B, int max_iter, double tolerance, int prec_iter, int order, int dim, int PrecNum, const Teuchos::SerialDenseMatrix<int, double> & M, int diag)
{
int n = A.numRows();
int k = 1;
double resid;
Teuchos::SerialDenseMatrix<int, double> P(n,n);
Teuchos::SerialDenseMatrix<int, double> Ax(n,1);
Ax.multiply(Teuchos::NO_TRANS,Teuchos::NO_TRANS,1.0, A, X, 0.0);
Teuchos::SerialDenseMatrix<int, double> r0(B);
r0-=Ax;
resid=r0.normFrobenius();
//define vector v=r/norm(r) where r=b-Ax
Teuchos::SerialDenseMatrix<int, double> v(n,1);
r0.scale(1/resid);
Teuchos::SerialDenseMatrix<int, double> h(1,1);
//Matrix of orthog basis vectors V
Teuchos::SerialDenseMatrix<int, double> V(n,1);
//Set v=r0/norm(r0) to be 1st col of V
for (int i=0; i<n; i++) {
V(i,0)=r0(i,0);
}
//right hand side
Teuchos::SerialDenseMatrix<int, double> bb(1,1);
bb(0,0)=resid;
Teuchos::SerialDenseMatrix<int, double> w(n,1);
Teuchos::SerialDenseMatrix<int, double> c;
Teuchos::SerialDenseMatrix<int, double> s;
while (resid > tolerance && k < max_iter) {
h.reshape(k+1,k);
//Arnoldi iteration(Gram-Schmidt )
V.reshape(n,k+1);
//set vk to be kth col of V
Teuchos::SerialDenseMatrix<int, double> vk(Teuchos::Copy, V, n,1,0,k-1);
//Preconditioning step: solve Mz=vk
Teuchos::SerialDenseMatrix<int, double> z(vk);
if (PrecNum == 1) {
Stokhos::DiagPreconditioner precond(M);
precond.ApplyInverse(vk,z,prec_iter);
}
else if (PrecNum == 2) {
Stokhos::JacobiPreconditioner precond(M);
precond.ApplyInverse(vk,z,2);
}
else if (PrecNum == 3) {
Stokhos::GSPreconditioner precond(M,1);
precond.ApplyInverse(vk,z,1);
}
else if (PrecNum == 4) {
Stokhos::SchurPreconditioner precond(M, order, dim, diag);
precond.ApplyInverse(vk,z,prec_iter);
}
w.multiply(Teuchos::NO_TRANS, Teuchos::NO_TRANS, 1, A, z, 0.0);
Teuchos::SerialDenseMatrix<int, double> vi(n,1);
Teuchos::SerialDenseMatrix<int, double> ip(1,1);
for (int i=0; i<k; i++) {
//set vi to be ith col of V
Teuchos::SerialDenseMatrix<int, double> vi(Teuchos::Copy, V, n,1,0,i);
//Calculate inner product
ip.multiply(Teuchos::TRANS, Teuchos::NO_TRANS, 1.0, vi, w, 0.0);
h(i,k-1)= ip(0,0);
//scale vi by h(i,k-1)
vi.scale(ip(0,0));
w-=vi;
}
h(k,k-1)=w.normFrobenius();
w.scale(1.0/h(k,k-1));
//add column vk+1=w to V
for (int i=0; i<n; i++) {
V(i,k)=w(i,0);
}
//Solve upper hessenberg least squares problem via Givens rotations
//Compute previous Givens rotations
for (int i=0; i<k-1; i++) {
double q=c(i,0)*h(i,k-1)+s(i,0)*h(i+1,k-1);
h(i+1,k-1)=-1*s(i,0)*h(i,k-1)+c(i,0)*h(i+1,k-1);
h(i,k-1)=q;
}
//Compute next Givens rotations
c.reshape(k,1);
s.reshape(k,1);
bb.reshape(k+1,1);
double l = sqrt(h(k-1,k-1)*h(k-1,k-1)+h(k,k-1)*h(k,k-1));
c(k-1,0)=h(k-1,k-1)/l;
s(k-1,0)=h(k,k-1)/l;
// Givens rotation on h and bb
h(k-1,k-1)=l;
h(k,k-1)=0;
bb(k,0)=-s(k-1,0)*bb(k-1,0);
bb(k-1,0)=c(k-1,0)*bb(k-1,0);
//Determine residual
resid = fabs(bb(k,0));
k++;
}
//Extract upper triangular square matrix
bb.reshape(h.numRows()-1 ,1);
//Solve linear system
int info;
Teuchos::LAPACK<int, double> lapack;
lapack.TRTRS('U', 'N', 'N', h.numRows()-1, 1, h.values(), h.stride(), bb.values(), bb.stride(),&info);
Teuchos::SerialDenseMatrix<int, double> ans(X);
V.reshape(n,k-1);
ans.multiply(Teuchos::NO_TRANS, Teuchos::NO_TRANS, 1.0, V, bb, 0.0);
if (PrecNum == 1) {
Stokhos::DiagPreconditioner precond(M);
precond.ApplyInverse(ans,ans,prec_iter);
}
else if (PrecNum == 2) {
Stokhos::JacobiPreconditioner precond(M);
precond.ApplyInverse(ans,ans,2);
}
else if (PrecNum == 3) {
Stokhos::GSPreconditioner precond(M,1);
precond.ApplyInverse(ans,ans,1);
}
else if (PrecNum == 4) {
Stokhos::SchurPreconditioner precond(M, order, dim, diag);
precond.ApplyInverse(ans,ans,prec_iter);
}
X+=ans;
std::cout << "iteration count= " << k-1 << std::endl;
return 0;
}
ordinal_type
Stokhos::MonomialProjGramSchmidtPCEBasis<ordinal_type, value_type>::
buildReducedBasis(
ordinal_type max_p,
value_type threshold,
const Teuchos::SerialDenseMatrix<ordinal_type,value_type>& A,
const Teuchos::SerialDenseMatrix<ordinal_type,value_type>& F,
const Teuchos::Array<value_type>& weights,
Teuchos::Array< Stokhos::MultiIndex<ordinal_type> >& terms_,
Teuchos::Array<ordinal_type>& num_terms_,
Teuchos::SerialDenseMatrix<ordinal_type,value_type>& Qp_,
Teuchos::SerialDenseMatrix<ordinal_type,value_type>& Q_)
{
// Compute basis terms -- 2-D array giving powers for each linear index
ordinal_type max_sz;
CPBUtils::compute_terms(max_p, this->d, max_sz, terms_, num_terms_);
// Compute B matrix -- monomials in F
// for i=0,...,nqp-1
// for j=0,...,sz-1
// B(i,j) = F(i,1)^terms_[j][1] * ... * F(i,d)^terms_[j][d]
// where sz is the total size of a basis up to order p and terms_[j]
// is an array of powers for each term in the total-order basis
ordinal_type nqp = weights.size();
SDM B(nqp, max_sz);
for (ordinal_type i=0; i<nqp; i++) {
for (ordinal_type j=0; j<max_sz; j++) {
B(i,j) = 1.0;
for (ordinal_type k=0; k<this->d; k++)
B(i,j) *= std::pow(F(i,k), terms_[j][k]);
}
}
// Project B into original basis -- should use SPAM for this
SDM Bp(this->pce_sz, max_sz);
const Teuchos::Array<value_type>& basis_norms =
this->pce_basis->norm_squared();
for (ordinal_type i=0; i<this->pce_sz; i++) {
for (ordinal_type j=0; j<max_sz; j++) {
Bp(i,j) = 0.0;
for (ordinal_type k=0; k<nqp; k++)
Bp(i,j) += weights[k]*B(k,j)*A(k,i);
Bp(i,j) /= basis_norms[i];
}
}
// Rescale columns of Bp to have unit norm
for (ordinal_type j=0; j<max_sz; j++) {
value_type nrm = 0.0;
for (ordinal_type i=0; i<this->pce_sz; i++)
nrm += Bp(i,j)*Bp(i,j)*basis_norms[i];
nrm = std::sqrt(nrm);
for (ordinal_type i=0; i<this->pce_sz; i++)
Bp(i,j) /= nrm;
}
// Compute our new basis -- each column of Qp is the coefficients of the
// new basis in the original basis. Constraint pivoting so first d+1
// columns and included in Qp.
Teuchos::Array<value_type> w(this->pce_sz, 1.0);
SDM R;
Teuchos::Array<ordinal_type> piv(max_sz);
for (int i=0; i<this->d+1; i++)
piv[i] = 1;
typedef Stokhos::OrthogonalizationFactory<ordinal_type,value_type> SOF;
ordinal_type sz_ = SOF::createOrthogonalBasis(
this->orthogonalization_method, threshold, this->verbose, Bp, w,
Qp_, R, piv);
// Evaluate new basis at original quadrature points
Q_.reshape(nqp, sz_);
ordinal_type ret =
Q_.multiply(Teuchos::NO_TRANS, Teuchos::NO_TRANS, 1.0, A, Qp_, 0.0);
TEUCHOS_ASSERT(ret == 0);
return sz_;
}
int
main (int argc, char *argv[])
{
using namespace Anasazi;
using Teuchos::RCP;
using Teuchos::rcp;
using std::endl;
#ifdef HAVE_MPI
// Initialize MPI
MPI_Init (&argc, &argv);
#endif // HAVE_MPI
// Create an Epetra communicator
#ifdef HAVE_MPI
Epetra_MpiComm Comm (MPI_COMM_WORLD);
#else
Epetra_SerialComm Comm;
#endif // HAVE_MPI
// Create an Anasazi output manager
BasicOutputManager<double> printer;
printer.stream(Errors) << Anasazi_Version() << std::endl << std::endl;
// Get the sorting std::string from the command line
std::string which ("LM");
Teuchos::CommandLineProcessor cmdp (false, true);
cmdp.setOption("sort", &which, "Targetted eigenvalues (SM or LM).");
if (cmdp.parse (argc, argv) != Teuchos::CommandLineProcessor::PARSE_SUCCESSFUL) {
#ifdef HAVE_MPI
MPI_Finalize ();
#endif // HAVE_MPI
return -1;
}
// Dimension of the matrix
//
// Discretization points in any one direction.
const int nx = 10;
// Size of matrix nx*nx
const int NumGlobalElements = nx*nx;
// Construct a Map that puts approximately the same number of
// equations on each process.
Epetra_Map Map (NumGlobalElements, 0, Comm);
// Get update list and number of local equations from newly created Map.
int NumMyElements = Map.NumMyElements ();
std::vector<int> MyGlobalElements (NumMyElements);
Map.MyGlobalElements (&MyGlobalElements[0]);
// Create an integer vector NumNz that is used to build the Petra
// matrix. NumNz[i] is the number of OFF-DIAGONAL terms for the
// i-th global equation on this process.
std::vector<int> NumNz (NumMyElements);
/* We are building a matrix of block structure:
| T -I |
|-I T -I |
| -I T |
| ... -I|
| -I T|
where each block is dimension nx by nx and the matrix is on the order of
nx*nx. The block T is a tridiagonal matrix.
*/
for (int i=0; i<NumMyElements; ++i) {
if (MyGlobalElements[i] == 0 || MyGlobalElements[i] == NumGlobalElements-1 ||
MyGlobalElements[i] == nx-1 || MyGlobalElements[i] == nx*(nx-1) ) {
NumNz[i] = 3;
}
else if (MyGlobalElements[i] < nx || MyGlobalElements[i] > nx*(nx-1) ||
MyGlobalElements[i]%nx == 0 || (MyGlobalElements[i]+1)%nx == 0) {
NumNz[i] = 4;
}
else {
NumNz[i] = 5;
}
}
// Create an Epetra_Matrix
RCP<Epetra_CrsMatrix> A = rcp (new Epetra_CrsMatrix (Epetra_DataAccess::Copy, Map, &NumNz[0]));
// Compute coefficients for discrete convection-diffution operator
const double one = 1.0;
std::vector<double> Values(4);
std::vector<int> Indices(4);
double rho = 0.0;
double h = one /(nx+1);
double h2 = h*h;
double c = 5.0e-01*rho/ h;
Values[0] = -one/h2 - c; Values[1] = -one/h2 + c; Values[2] = -one/h2; Values[3]= -one/h2;
double diag = 4.0 / h2;
int NumEntries;
for (int i=0; i<NumMyElements; ++i) {
if (MyGlobalElements[i]==0) {
Indices[0] = 1;
Indices[1] = nx;
NumEntries = 2;
int info = A->InsertGlobalValues(MyGlobalElements[i], NumEntries, &Values[1], &Indices[0]);
TEUCHOS_TEST_FOR_EXCEPTION
(info != 0, std::runtime_error, "InsertGlobalValues returned info = "
<< info << " != 0." );
}
else if (MyGlobalElements[i] == nx*(nx-1)) {
Indices[0] = nx*(nx-1)+1;
Indices[1] = nx*(nx-2);
NumEntries = 2;
int info = A->InsertGlobalValues(MyGlobalElements[i], NumEntries, &Values[1], &Indices[0]);
TEUCHOS_TEST_FOR_EXCEPTION
(info != 0, std::runtime_error, "InsertGlobalValues returned info = "
<< info << " != 0." );
}
else if (MyGlobalElements[i] == nx-1) {
Indices[0] = nx-2;
NumEntries = 1;
int info = A->InsertGlobalValues(MyGlobalElements[i], NumEntries, &Values[0], &Indices[0]);
TEUCHOS_TEST_FOR_EXCEPTION
(info != 0, std::runtime_error, "InsertGlobalValues returned info = "
<< info << " != 0." );
Indices[0] = 2*nx-1;
info = A->InsertGlobalValues(MyGlobalElements[i], NumEntries, &Values[2], &Indices[0]);
TEUCHOS_TEST_FOR_EXCEPTION
(info != 0, std::runtime_error, "InsertGlobalValues returned info = "
<< info << " != 0." );
}
else if (MyGlobalElements[i] == NumGlobalElements-1) {
Indices[0] = NumGlobalElements-2;
NumEntries = 1;
int info = A->InsertGlobalValues(MyGlobalElements[i], NumEntries, &Values[0], &Indices[0]);
TEUCHOS_TEST_FOR_EXCEPTION
(info != 0, std::runtime_error, "InsertGlobalValues returned info = "
<< info << " != 0." );
Indices[0] = nx*(nx-1)-1;
info = A->InsertGlobalValues(MyGlobalElements[i], NumEntries, &Values[2], &Indices[0]);
TEUCHOS_TEST_FOR_EXCEPTION
(info != 0, std::runtime_error, "InsertGlobalValues returned info = "
<< info << " != 0." );
}
else if (MyGlobalElements[i] < nx) {
Indices[0] = MyGlobalElements[i]-1;
Indices[1] = MyGlobalElements[i]+1;
Indices[2] = MyGlobalElements[i]+nx;
NumEntries = 3;
int info = A->InsertGlobalValues(MyGlobalElements[i], NumEntries, &Values[0], &Indices[0]);
TEUCHOS_TEST_FOR_EXCEPTION
(info != 0, std::runtime_error, "InsertGlobalValues returned info = "
<< info << " != 0." );
}
else if (MyGlobalElements[i] > nx*(nx-1)) {
Indices[0] = MyGlobalElements[i]-1;
Indices[1] = MyGlobalElements[i]+1;
Indices[2] = MyGlobalElements[i]-nx;
NumEntries = 3;
int info = A->InsertGlobalValues(MyGlobalElements[i], NumEntries, &Values[0], &Indices[0]);
TEUCHOS_TEST_FOR_EXCEPTION
(info != 0, std::runtime_error, "InsertGlobalValues returned info = "
<< info << " != 0." );
}
else if (MyGlobalElements[i]%nx == 0) {
Indices[0] = MyGlobalElements[i]+1;
Indices[1] = MyGlobalElements[i]-nx;
Indices[2] = MyGlobalElements[i]+nx;
NumEntries = 3;
int info = A->InsertGlobalValues(MyGlobalElements[i], NumEntries, &Values[1], &Indices[0]);
TEUCHOS_TEST_FOR_EXCEPTION
(info != 0, std::runtime_error, "InsertGlobalValues returned info = "
<< info << " != 0." );
}
else if ((MyGlobalElements[i]+1)%nx == 0) {
Indices[0] = MyGlobalElements[i]-nx;
Indices[1] = MyGlobalElements[i]+nx;
NumEntries = 2;
int info = A->InsertGlobalValues(MyGlobalElements[i], NumEntries, &Values[2], &Indices[0]);
TEUCHOS_TEST_FOR_EXCEPTION
(info != 0, std::runtime_error, "InsertGlobalValues returned info = "
<< info << " != 0." );
Indices[0] = MyGlobalElements[i]-1;
NumEntries = 1;
info = A->InsertGlobalValues(MyGlobalElements[i], NumEntries, &Values[0], &Indices[0]);
TEUCHOS_TEST_FOR_EXCEPTION
(info != 0, std::runtime_error, "InsertGlobalValues returned info = "
<< info << " != 0." );
}
else {
Indices[0] = MyGlobalElements[i]-1;
Indices[1] = MyGlobalElements[i]+1;
Indices[2] = MyGlobalElements[i]-nx;
Indices[3] = MyGlobalElements[i]+nx;
NumEntries = 4;
int info = A->InsertGlobalValues(MyGlobalElements[i], NumEntries, &Values[0], &Indices[0]);
TEUCHOS_TEST_FOR_EXCEPTION
(info != 0, std::runtime_error, "InsertGlobalValues returned info = "
<< info << " != 0." );
}
// Put in the diagonal entry
int info = A->InsertGlobalValues(MyGlobalElements[i], 1, &diag, &MyGlobalElements[i]);
TEUCHOS_TEST_FOR_EXCEPTION
(info != 0, std::runtime_error, "InsertGlobalValues returned info = "
<< info << " != 0." );
}
// Finish up
int info = A->FillComplete ();
TEUCHOS_TEST_FOR_EXCEPTION
(info != 0, std::runtime_error, "A->FillComplete() returned info = "
<< info << " != 0." );
A->SetTracebackMode (1); // Shutdown Epetra Warning tracebacks
// Create a identity matrix for the temporary mass matrix
RCP<Epetra_CrsMatrix> M = rcp (new Epetra_CrsMatrix (Epetra_DataAccess::Copy, Map, 1));
for (int i=0; i<NumMyElements; i++) {
Values[0] = one;
Indices[0] = i;
NumEntries = 1;
info = M->InsertGlobalValues(MyGlobalElements[i], NumEntries, &Values[0], &Indices[0]);
TEUCHOS_TEST_FOR_EXCEPTION
(info != 0, std::runtime_error, "M->InsertGlobalValues() returned info = "
<< info << " != 0." );
}
// Finish up
info = M->FillComplete ();
TEUCHOS_TEST_FOR_EXCEPTION
(info != 0, std::runtime_error, "M->FillComplete() returned info = "
<< info << " != 0." );
M->SetTracebackMode (1); // Shutdown Epetra Warning tracebacks
//************************************
// Call the LOBPCG solver manager
//***********************************
//
// Variables used for the LOBPCG Method
const int nev = 10;
const int blockSize = 5;
const int maxIters = 500;
const double tol = 1.0e-8;
typedef Epetra_MultiVector MV;
typedef Epetra_Operator OP;
typedef MultiVecTraits<double, Epetra_MultiVector> MVT;
// Create an Epetra_MultiVector for an initial vector to start the
// solver. Note: This needs to have the same number of columns as
// the blocksize.
RCP<Epetra_MultiVector> ivec = rcp (new Epetra_MultiVector (Map, blockSize));
ivec->Random (); // fill the initial vector with random values
// Create the eigenproblem.
RCP<BasicEigenproblem<double, MV, OP> > MyProblem =
rcp (new BasicEigenproblem<double, MV, OP> (A, ivec));
// Inform the eigenproblem that the operator A is symmetric
MyProblem->setHermitian (true);
// Set the number of eigenvalues requested
MyProblem->setNEV (nev);
// Tell the eigenproblem that you are finishing passing it information.
const bool success = MyProblem->setProblem ();
if (! success) {
printer.print (Errors, "Anasazi::BasicEigenproblem::setProblem() reported an error.\n");
#ifdef HAVE_MPI
MPI_Finalize ();
#endif // HAVE_MPI
return -1;
}
// Create parameter list to pass into the solver manager
Teuchos::ParameterList MyPL;
MyPL.set ("Which", which);
MyPL.set ("Block Size", blockSize);
MyPL.set ("Maximum Iterations", maxIters);
MyPL.set ("Convergence Tolerance", tol);
MyPL.set ("Full Ortho", true);
MyPL.set ("Use Locking", true);
// Create the solver manager
LOBPCGSolMgr<double, MV, OP> MySolverMan (MyProblem, MyPL);
// Solve the problem
ReturnType returnCode = MySolverMan.solve ();
// Get the eigenvalues and eigenvectors from the eigenproblem
Eigensolution<double,MV> sol = MyProblem->getSolution ();
std::vector<Value<double> > evals = sol.Evals;
RCP<MV> evecs = sol.Evecs;
// Compute residuals.
std::vector<double> normR (sol.numVecs);
if (sol.numVecs > 0) {
Teuchos::SerialDenseMatrix<int,double> T (sol.numVecs, sol.numVecs);
Epetra_MultiVector tempAevec (Map, sol.numVecs );
T.putScalar (0.0);
for (int i = 0; i < sol.numVecs; ++i) {
T(i,i) = evals[i].realpart;
}
A->Apply (*evecs, tempAevec);
MVT::MvTimesMatAddMv (-1.0, *evecs, T, 1.0, tempAevec);
MVT::MvNorm (tempAevec, normR);
}
// Print the results
std::ostringstream os;
os.setf (std::ios_base::right, std::ios_base::adjustfield);
os << "Solver manager returned "
<< (returnCode == Converged ? "converged." : "unconverged.") << endl;
os << endl;
os << "------------------------------------------------------" << endl;
os << std::setw(16) << "Eigenvalue"
<< std::setw(18) << "Direct Residual"
<< endl;
os << "------------------------------------------------------" << endl;
for (int i = 0; i < sol.numVecs; ++i) {
os << std::setw(16) << evals[i].realpart
<< std::setw(18) << normR[i] / evals[i].realpart
<< endl;
}
os << "------------------------------------------------------" << endl;
printer.print (Errors, os.str ());
#ifdef HAVE_MPI
MPI_Finalize ();
#endif // HAVE_MPI
return 0;
}
void
factorExplicit (Kokkos::MultiVector<Scalar, NodeType>& A,
Kokkos::MultiVector<Scalar, NodeType>& Q,
Teuchos::SerialDenseMatrix<LocalOrdinal, Scalar>& R,
const bool contiguousCacheBlocks,
const bool forceNonnegativeDiagonal=false)
{
using Teuchos::asSafe;
typedef Kokkos::MultiVector<Scalar, NodeType> KMV;
// Tsqr currently likes LocalOrdinal ordinals, but
// Kokkos::MultiVector has size_t ordinals. Do conversions
// here.
//
// Teuchos::asSafe() can do safe conversion (e.g., checking for
// overflow when casting to a narrower integer type), if a
// custom specialization is defined for
// Teuchos::ValueTypeConversionTraits<size_t, LocalOrdinal>.
// Otherwise, this has the same (potentially) unsafe effect as
// static_cast<LocalOrdinal>(...) would have.
const LocalOrdinal A_numRows = asSafe<LocalOrdinal> (A.getNumRows());
const LocalOrdinal A_numCols = asSafe<LocalOrdinal> (A.getNumCols());
const LocalOrdinal A_stride = asSafe<LocalOrdinal> (A.getStride());
const LocalOrdinal Q_numRows = asSafe<LocalOrdinal> (Q.getNumRows());
const LocalOrdinal Q_numCols = asSafe<LocalOrdinal> (Q.getNumCols());
const LocalOrdinal Q_stride = asSafe<LocalOrdinal> (Q.getStride());
// Sanity checks for matrix dimensions
if (A_numRows < A_numCols) {
std::ostringstream os;
os << "In Tsqr::factorExplicit: input matrix A has " << A_numRows
<< " local rows, and " << A_numCols << " columns. The input "
"matrix must have at least as many rows on each processor as "
"there are columns.";
throw std::invalid_argument(os.str());
} else if (A_numRows != Q_numRows) {
std::ostringstream os;
os << "In Tsqr::factorExplicit: input matrix A and output matrix Q "
"must have the same number of rows. A has " << A_numRows << " rows"
" and Q has " << Q_numRows << " rows.";
throw std::invalid_argument(os.str());
} else if (R.numRows() < R.numCols()) {
std::ostringstream os;
os << "In Tsqr::factorExplicit: output matrix R must have at least "
"as many rows as columns. R has " << R.numRows() << " rows and "
<< R.numCols() << " columns.";
throw std::invalid_argument(os.str());
} else if (A_numCols != R.numCols()) {
std::ostringstream os;
os << "In Tsqr::factorExplicit: input matrix A and output matrix R "
"must have the same number of columns. A has " << A_numCols
<< " columns and R has " << R.numCols() << " columns.";
throw std::invalid_argument(os.str());
}
// Check for quick exit, based on matrix dimensions
if (Q_numCols == 0)
return;
// Hold on to nonconst views of A and Q. This will make TSQR
// correct (if perhaps inefficient) for all possible Kokkos Node
// types, even GPU nodes.
Teuchos::ArrayRCP<scalar_type> A_ptr = A.getValuesNonConst();
Teuchos::ArrayRCP<scalar_type> Q_ptr = Q.getValuesNonConst();
R.putScalar (STS::zero());
NodeOutput nodeResults =
nodeTsqr_->factor (A_numRows, A_numCols, A_ptr.getRawPtr(), A_stride,
R.values(), R.stride(), contiguousCacheBlocks);
// FIXME (mfh 19 Oct 2010) Replace actions on raw pointer with
// actions on the Kokkos::MultiVector or at least the ArrayRCP.
nodeTsqr_->fill_with_zeros (Q_numRows, Q_numCols,
Q_ptr.getRawPtr(), Q_stride,
contiguousCacheBlocks);
matview_type Q_rawView (Q_numRows, Q_numCols,
Q_ptr.getRawPtr(), Q_stride);
matview_type Q_top_block =
nodeTsqr_->top_block (Q_rawView, contiguousCacheBlocks);
if (Q_top_block.nrows() < R.numCols()) {
std::ostringstream os;
os << "The top block of Q has too few rows. This means that the "
<< "the intranode TSQR implementation has a bug in its top_block"
<< "() method. The top block should have at least " << R.numCols()
<< " rows, but instead has only " << Q_top_block.ncols()
<< " rows.";
throw std::logic_error (os.str());
}
{
matview_type Q_top (R.numCols(), Q_numCols, Q_top_block.get(),
Q_top_block.lda());
matview_type R_view (R.numRows(), R.numCols(), R.values(), R.stride());
distTsqr_->factorExplicit (R_view, Q_top, forceNonnegativeDiagonal);
}
nodeTsqr_->apply (ApplyType::NoTranspose,
A_numRows, A_numCols, A_ptr.getRawPtr(), A_stride,
nodeResults, Q_numCols, Q_ptr.getRawPtr(), Q_stride,
contiguousCacheBlocks);
// If necessary, force the R factor to have a nonnegative diagonal.
if (forceNonnegativeDiagonal &&
! QR_produces_R_factor_with_nonnegative_diagonal()) {
details::NonnegDiagForcer<LocalOrdinal, Scalar, STS::isComplex> forcer;
matview_type Q_mine (Q_numRows, Q_numCols, Q_ptr.getRawPtr(), Q_stride);
matview_type R_mine (R.numRows(), R.numCols(), R.values(), R.stride());
forcer.force (Q_mine, R_mine);
}
// "Commit" the changes to the multivector.
A_ptr = Teuchos::null;
Q_ptr = Teuchos::null;
}
