void DenseMatrix<T>::_matvec_blas(T alpha, T beta,
DenseVector<T>& dest,
const DenseVector<T>& arg,
bool trans) const
{
// Ensure that dest and arg sizes are compatible
if (!trans)
{
// dest ~ A * arg
// (mx1) (mxn) * (nx1)
if ((dest.size() != this->m()) || (arg.size() != this->n()))
{
libMesh::out << "Improper input argument sizes!" << std::endl;
libmesh_error();
}
}
else // trans == true
{
// Ensure that dest and arg are proper size
// dest ~ A^T * arg
// (nx1) (nxm) * (mx1)
if ((dest.size() != this->n()) || (arg.size() != this->m()))
{
libMesh::out << "Improper input argument sizes!" << std::endl;
libmesh_error();
}
}
// Calling sequence for dgemv:
//
// dgemv(TRANS,M,N,ALPHA,A,LDA,X,INCX,BETA,Y,INCY)
// TRANS - CHARACTER*1, 't' for transpose, 'n' for non-transpose multiply
// We store everything in row-major order, so pass the transpose flag for
// non-transposed matvecs and the 'n' flag for transposed matvecs
char TRANS[] = "t";
if (trans)
TRANS[0] = 'n';
// M - INTEGER.
// On entry, M specifies the number of rows of the matrix A.
// In C/C++, pass the number of *cols* of A
int M = this->n();
// N - INTEGER.
// On entry, N specifies the number of columns of the matrix A.
// In C/C++, pass the number of *rows* of A
int N = this->m();
// ALPHA - DOUBLE PRECISION.
// The scalar constant passed to this function
// A - DOUBLE PRECISION array of DIMENSION ( LDA, n ).
// Before entry, the leading m by n part of the array A must
// contain the matrix of coefficients.
// The matrix, *this. Note that _matvec_blas is called from
// a const function, vector_mult(), and so we have made this function const
// as well. Since BLAS knows nothing about const, we have to cast it away
// now.
DenseMatrix<T>& a_ref = const_cast< DenseMatrix<T>& > ( *this );
std::vector<T>& a = a_ref.get_values();
// LDA - INTEGER.
// On entry, LDA specifies the first dimension of A as declared
// in the calling (sub) program. LDA must be at least
// max( 1, m ).
int LDA = M;
// X - DOUBLE PRECISION array of DIMENSION at least
// ( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n'
// and at least
// ( 1 + ( m - 1 )*abs( INCX ) ) otherwise.
// Before entry, the incremented array X must contain the
// vector x.
// Here, we must cast away the const-ness of "arg" since BLAS knows
// nothing about const
DenseVector<T>& x_ref = const_cast< DenseVector<T>& > ( arg );
std::vector<T>& x = x_ref.get_values();
// INCX - INTEGER.
// On entry, INCX specifies the increment for the elements of
// X. INCX must not be zero.
int INCX = 1;
// BETA - DOUBLE PRECISION.
// On entry, BETA specifies the scalar beta. When BETA is
// supplied as zero then Y need not be set on input.
// The second scalar constant passed to this function
// Y - DOUBLE PRECISION array of DIMENSION at least
// ( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n'
// and at least
// ( 1 + ( n - 1 )*abs( INCY ) ) otherwise.
// Before entry with BETA non-zero, the incremented array Y
// must contain the vector y. On exit, Y is overwritten by the
// updated vector y.
// The input vector "dest"
std::vector<T>& y = dest.get_values();
// INCY - INTEGER.
// On entry, INCY specifies the increment for the elements of
// Y. INCY must not be zero.
int INCY = 1;
// Finally, ready to call the BLAS function
BLASgemv_(TRANS, &M, &N, &alpha, &(a[0]), &LDA, &(x[0]), &INCX, &beta, &(y[0]), &INCY);
}
void DenseMatrix<T>::_evd_lapack (DenseVector<T> & lambda_real,
DenseVector<T> & lambda_imag)
{
// The calling sequence for dgeev is:
// DGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI, VL, LDVL, VR,
// LDVR, ILO, IHI, SCALE, ABNRM, RCONDE, RCONDV, WORK, LWORK, IWORK, INFO )
// BALANC (input) CHARACTER*1
// Indicates how the input matrix should be diagonally scaled
// and/or permuted to improve the conditioning of its
// eigenvalues.
// = 'N': Do not diagonally scale or permute;
char BALANC = 'N';
// JOBVL (input) CHARACTER*1
// = 'N': left eigenvectors of A are not computed;
// = 'V': left eigenvectors of A are computed.
char JOBVL = 'N';
// JOBVR (input) CHARACTER*1
// = 'N': right eigenvectors of A are not computed;
// = 'V': right eigenvectors of A are computed.
char JOBVR = 'N';
// SENSE (input) CHARACTER*1
// Determines which reciprocal condition numbers are computed.
// = 'N': None are computed;
// = 'E': Computed for eigenvalues only;
// = 'V': Computed for right eigenvectors only;
// = 'B': Computed for eigenvalues and right eigenvectors.
char SENSE = 'N';
// N (input) int *
// The number of rows/cols of the matrix A. N >= 0.
libmesh_assert( this->m() == this->n() );
int N = this->m();
// A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
// On entry, the N-by-N matrix A.
// On exit, A has been overwritten.
// Here, we pass &(_val[0]).
// LDA (input) int *
// The leading dimension of the array A. LDA >= max(1,N).
int LDA = N;
// WR (output) DOUBLE PRECISION array, dimension (N)
// WI (output) DOUBLE PRECISION array, dimension (N)
// WR and WI contain the real and imaginary parts,
// respectively, of the computed eigenvalues. Complex
// conjugate pairs of eigenvalues appear consecutively
// with the eigenvalue having the positive imaginary part
// first.
lambda_real.resize(N);
lambda_imag.resize(N);
// VL (output) DOUBLE PRECISION array, dimension (LDVL,N)
// If JOBVL = 'V', the left eigenvectors u(j) are stored one
// after another in the columns of VL, in the same order
// as their eigenvalues.
// If JOBVL = 'N', VL is not referenced.
// If the j-th eigenvalue is real, then u(j) = VL(:,j),
// the j-th column of VL.
// If the j-th and (j+1)-st eigenvalues form a complex
// conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and
// u(j+1) = VL(:,j) - i*VL(:,j+1).
// Just set to NULL here.
// LDVL (input) INTEGER
// The leading dimension of the array VL. LDVL >= 1; if
// JOBVL = 'V', LDVL >= N.
int LDVL = 1;
// VR (output) DOUBLE PRECISION array, dimension (LDVR,N)
// If JOBVR = 'V', the right eigenvectors v(j) are stored one
// after another in the columns of VR, in the same order
// as their eigenvalues.
// If JOBVR = 'N', VR is not referenced.
// If the j-th eigenvalue is real, then v(j) = VR(:,j),
// the j-th column of VR.
// If the j-th and (j+1)-st eigenvalues form a complex
// conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and
// v(j+1) = VR(:,j) - i*VR(:,j+1).
// Just set to NULL here.
// LDVR (input) INTEGER
// The leading dimension of the array VR. LDVR >= 1; if
// JOBVR = 'V', LDVR >= N.
int LDVR = 1;
// Outputs (unused)
int ILO = 0;
int IHI = 0;
std::vector<T> SCALE(N);
T ABNRM;
std::vector<T> RCONDE(N);
std::vector<T> RCONDV(N);
// WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
// On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
//
// LWORK (input) INTEGER
// The dimension of the array WORK.
int LWORK = 3*N;
std::vector<T> WORK( LWORK );
// IWORK (workspace) INTEGER array, dimension (2*N-2)
// If SENSE = 'N' or 'E', not referenced.
// Just set to NULL
// INFO (output) INTEGER
// = 0: successful exit
// < 0: if INFO = -i, the i-th argument had an illegal value.
// > 0: if INFO = i, the QR algorithm failed to compute all the
// eigenvalues, and no eigenvectors or condition numbers
// have been computed; elements 1:ILO-1 and i+1:N of WR
// and WI contain eigenvalues which have converged.
int INFO = 0;
// Get references to raw data
std::vector<T> & lambda_real_val = lambda_real.get_values();
std::vector<T> & lambda_imag_val = lambda_imag.get_values();
// Ready to call the actual factorization routine through SLEPc's interface
LAPACKgeevx_( &BALANC, &JOBVL, &JOBVR, &SENSE, &N, &(_val[0]), &LDA, &lambda_real_val[0],
&lambda_imag_val[0], libmesh_nullptr, &LDVL, libmesh_nullptr, &LDVR, &ILO, &IHI, &SCALE[0], &ABNRM,
&RCONDE[0], &RCONDV[0], &WORK[0], &LWORK, libmesh_nullptr, &INFO );
// Check return value for errors
if (INFO != 0)
libmesh_error_msg("INFO=" << INFO << ", Error during Lapack eigenvalue calculation!");
}
void DenseMatrix<T>::_lu_back_substitute_lapack (const DenseVector<T>& b,
DenseVector<T>& x)
{
// The calling sequence for getrs is:
// dgetrs(TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
// trans (input) char*
// 'n' for no tranpose, 't' for transpose
char TRANS[] = "t";
// N (input) int*
// The order of the matrix A. N >= 0.
int N = this->m();
// NRHS (input) int*
// The number of right hand sides, i.e., the number of columns
// of the matrix B. NRHS >= 0.
int NRHS = 1;
// A (input) DOUBLE PRECISION array, dimension (LDA,N)
// The factors L and U from the factorization A = P*L*U
// as computed by dgetrf.
// Here, we pass &(_val[0])
// LDA (input) int*
// The leading dimension of the array A. LDA >= max(1,N).
int LDA = N;
// ipiv (input) int array, dimension (N)
// The pivot indices from DGETRF; for 1<=i<=N, row i of the
// matrix was interchanged with row IPIV(i).
// Here, we pass &(_pivots[0]) which was computed in _lu_decompose_lapack
// B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
// On entry, the right hand side matrix B.
// On exit, the solution matrix X.
// Here, we pass a copy of the rhs vector's data array in x, so that the
// passed right-hand side b is unmodified. I don't see a way around this
// copy if we want to maintain an unmodified rhs in LibMesh.
x = b;
std::vector<T>& x_vec = x.get_values();
// We can avoid the copy if we don't care about overwriting the RHS: just
// pass b to the Lapack routine and then swap with x before exiting
// std::vector<T>& x_vec = b.get_values();
// LDB (input) int*
// The leading dimension of the array B. LDB >= max(1,N).
int LDB = N;
// INFO (output) int*
// = 0: successful exit
// < 0: if INFO = -i, the i-th argument had an illegal value
int INFO = 0;
// Finally, ready to call the Lapack getrs function
LAPACKgetrs_(TRANS, &N, &NRHS, &(_val[0]), &LDA, &(_pivots[0]), &(x_vec[0]), &LDB, &INFO);
// Check return value for errors
if (INFO != 0)
{
libMesh::out << "INFO="
<< INFO
<< ", Error during Lapack LU solve!" << std::endl;
libmesh_error();
}
// Don't do this if you already made a copy of b above
// Swap b and x. The solution will then be in x, and whatever was originally
// in x, maybe garbage, maybe nothing, will be in b.
// FIXME: Rewrite the LU and Cholesky solves to just take one input, and overwrite
// the input. This *should* make user code simpler, as they don't have to create
// an extra vector just to pass it in to the solve function!
// b.swap(x);
}
void DenseMatrix<T>::_svd_solve_lapack(const DenseVector<T> & rhs,
DenseVector<T> & x,
Real rcond) const
{
// Since BLAS is expecting column-major storage, we first need to
// make a transposed copy of *this, then pass it to the gelss
// routine instead of the original. This extra copy is kind of a
// bummer, it might be better if we could use the full SVD to
// compute the least-squares solution instead... Note that it isn't
// completely terrible either, since A_trans gets overwritten by
// Lapack, and we usually would end up making a copy of A outside
// the function call anyway.
DenseMatrix<T> A_trans;
this->get_transpose(A_trans);
// M is INTEGER
// The number of rows of the input matrix. M >= 0.
// This is actually the number of *columns* of A_trans.
int M = A_trans.n();
// N is INTEGER
// The number of columns of the matrix A. N >= 0.
// This is actually the number of *rows* of A_trans.
int N = A_trans.m();
// We'll use the min and max of (M,N) several times below.
int max_MN = std::max(M,N);
int min_MN = std::min(M,N);
// NRHS is INTEGER
// The number of right hand sides, i.e., the number of columns
// of the matrices B and X. NRHS >= 0.
// This could later be generalized to solve for multiple right-hand
// sides...
int NRHS = 1;
// A is DOUBLE PRECISION array, dimension (LDA,N)
// On entry, the M-by-N matrix A.
// On exit, the first min(m,n) rows of A are overwritten with
// its right singular vectors, stored rowwise.
//
// The data vector that will be passed to Lapack.
std::vector<T> & A_trans_vals = A_trans.get_values();
// LDA is INTEGER
// The leading dimension of the array A. LDA >= max(1,M).
int LDA = M;
// B is DOUBLE PRECISION array, dimension (LDB,NRHS)
// On entry, the M-by-NRHS right hand side matrix B.
// On exit, B is overwritten by the N-by-NRHS solution
// matrix X. If m >= n and RANK = n, the residual
// sum-of-squares for the solution in the i-th column is given
// by the sum of squares of elements n+1:m in that column.
//
// Since we don't want the user's rhs vector to be overwritten by
// the solution, we copy the rhs values into the solution vector "x"
// now. x needs to be long enough to hold both the (Nx1) solution
// vector or the (Mx1) rhs, so size it to the max of those.
x.resize(max_MN);
for (unsigned i=0; i<rhs.size(); ++i)
x(i) = rhs(i);
// Make the syntax below simpler by grabbing a reference to this array.
std::vector<T> & B = x.get_values();
// LDB is INTEGER
// The leading dimension of the array B. LDB >= max(1,max(M,N)).
int LDB = x.size();
// S is DOUBLE PRECISION array, dimension (min(M,N))
// The singular values of A in decreasing order.
// The condition number of A in the 2-norm = S(1)/S(min(m,n)).
std::vector<T> S(min_MN);
// RCOND is DOUBLE PRECISION
// RCOND is used to determine the effective rank of A.
// Singular values S(i) <= RCOND*S(1) are treated as zero.
// If RCOND < 0, machine precision is used instead.
Real RCOND = rcond;
// RANK is INTEGER
// The effective rank of A, i.e., the number of singular values
// which are greater than RCOND*S(1).
int RANK = 0;
// LWORK is INTEGER
// The dimension of the array WORK. LWORK >= 1, and also:
// LWORK >= 3*min(M,N) + max( 2*min(M,N), max(M,N), NRHS )
// For good performance, LWORK should generally be larger.
//
// If LWORK = -1, then a workspace query is assumed; the routine
// only calculates the optimal size of the WORK array, returns
// this value as the first entry of the WORK array, and no error
// message related to LWORK is issued by XERBLA.
//
// The factor of 1.5 is arbitrary and is used to satisfy the "should
// generally be larger" clause.
int LWORK = 1.5 * (3*min_MN + std::max(2*min_MN, std::max(max_MN, NRHS)));
// WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
// On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
std::vector<T> WORK(LWORK);
// INFO is INTEGER
// = 0: successful exit
// < 0: if INFO = -i, the i-th argument had an illegal value.
// > 0: the algorithm for computing the SVD failed to converge;
// if INFO = i, i off-diagonal elements of an intermediate
// bidiagonal form did not converge to zero.
int INFO = 0;
// LAPACKgelss_(const PetscBLASInt *, // M
// const PetscBLASInt *, // N
// const PetscBLASInt *, // NRHS
// PetscScalar *, // A
// const PetscBLASInt *, // LDA
// PetscScalar *, // B
// const PetscBLASInt *, // LDB
// PetscReal *, // S(out) = singular values of A in increasing order
// const PetscReal *, // RCOND = tolerance for singular values
// PetscBLASInt *, // RANK(out) = number of "non-zero" singular values
// PetscScalar *, // WORK
// const PetscBLASInt *, // LWORK
// PetscBLASInt *); // INFO
LAPACKgelss_(&M, &N, &NRHS, &A_trans_vals[0], &LDA, &B[0], &LDB, &S[0], &RCOND, &RANK, &WORK[0], &LWORK, &INFO);
// Check for errors in the Lapack call
if (INFO < 0)
libmesh_error_msg("Error, argument " << -INFO << " to LAPACKgelss_ had an illegal value.");
if (INFO > 0)
libmesh_error_msg("The algorithm for computing the SVD failed to converge!");
// Debugging: print singular values and information about condition number:
// libMesh::err << "RCOND=" << RCOND << std::endl;
// libMesh::err << "Singular values: " << std::endl;
// for (unsigned i=0; i<S.size(); ++i)
// libMesh::err << S[i] << std::endl;
// libMesh::err << "The condition number of A is approximately: " << S[0]/S.back() << std::endl;
// Lapack has already written the solution into B, but it will be
// the wrong size for non-square problems, so we need to resize it
// correctly. The size of the solution vector should be the number
// of columns of the original A matrix. Unfortunately, resizing a
// DenseVector currently also zeros it out (unlike a std::vector) so
// we'll resize the underlying storage directly (the size is not
// stored independently elsewhere).
x.get_values().resize(this->n());
}