//---------------------------------------------------------
void eig(const DMat& A, DVec& Re, DMat& VL, DMat& VR, bool bL, bool bR)
//---------------------------------------------------------
{
// Compute eigensystem of a real general matrix
// Currently NOT returning imaginary components
static DMat B;
if (!A.is_square()) { umERROR("eig(A)", "matrix is not square."); }
int N = A.num_rows();
int LDA=N, LDVL=N, LDVR=N, ldwork=10*N, info=0;
Re.resize(N); // store REAL components of eigenvalues in Re
VL.resize(N,N); // storage for LEFT eigenvectors
VR.resize(N,N); // storage for RIGHT eigenvectors
DVec Im(N); // NOT returning imaginary components
DVec work(ldwork, 0.0);
// Work on a copy of A
B = A;
char jobL = bL ? 'V' : 'N'; // calc LEFT eigenvectors?
char jobR = bR ? 'V' : 'N'; // calc RIGHT eigenvectors?
GEEV (jobL,jobR, N, B.data(), LDA, Re.data(), Im.data(),
VL.data(), LDVL, VR.data(), LDVR, work.data(), ldwork, info);
if (info < 0) {
umERROR("eig(A, Re,Im)", "Error in input argument (%d)\nNo solution computed.", -info);
} else if (info > 0) {
umLOG(1, "eig(A, Re,Im): ...\n"
"\nThe QR algorithm failed to compute all the"
"\neigenvalues, and no eigenvectors have been"
"\ncomputed; elements %d+1:N of WR and WI contain"
"\neigenvalues which have converged.\n", info);
}
#if (0)
// Return (Re,Imag) parts of eigenvalues as columns of Ev
Ev.resize(N,2);
Ev.set_col(1, Re);
Ev.set_col(2, Im);
#endif
#ifdef _DEBUG
//#####################################################
// Check for imaginary components in eigenvalues
//#####################################################
double im_max = Im.max_val_abs();
if (im_max > 1e-6) {
umERROR("eig(A)", "imaginary components in eigenvalues.");
}
//#####################################################
#endif
}
//---------------------------------------------------------
DMat& qr(DMat& A, bool in_place)
//---------------------------------------------------------
{
// Form orthogonal QR factorization of A(m,n).
// The result Q is represented as a product of
// min(m, n) elementary reflectors.
int M=A.num_rows(), N=A.num_cols(), LDA=A.num_rows();
int min_mn = A.min_mn(), info=0; DVec tau(min_mn);
if (in_place)
{
// factorize arg
GEQRF(M, N, A.data(), LDA, tau.data(), info);
if (info) { umERROR("qr(A)", "dgeqrf reports: info = %d", info); }
//A.set_qrtau(tau); // H(i) = I - tau * v * v'
A.set_factmode(FACT_QR); // indicate factored state
return A;
}
else
{
// factorize copy of arg
DMat* tmp = new DMat(A, OBJ_temp, "qr(A)");
GEQRF (M, N, tmp->data(), LDA, tau.data(), info);
if (info) { umERROR("qr(A)", "dgeqrf reports: info = %d", info); }
//tmp->set_qrtau(tau); // H(i) = I - tau * v * v'
tmp->set_factmode(FACT_QR); // indicate factored state
return (*tmp);
}
}
//---------------------------------------------------------
DMat& chol(DMat& A, bool in_place)
//---------------------------------------------------------
{
// Given symmetric positive-definite matrix A,
// return its Cholesky-factorization for use
// later in solving (multiple) linear systems.
int M=A.num_rows(), LDA=A.num_rows(), info=0;
char uplo = 'U';
if (in_place)
{
// factorize arg
POTRF (uplo, M, A.data(), LDA, info);
if (info) { umERROR("chol(A)", "dpotrf reports: info = %d", info); }
A.zero_below_diag();
A.set_factmode(FACT_CHOL); // indicate factored state
return A;
}
else
{
// factorize copy of arg
DMat* tmp = new DMat(A, OBJ_temp, "chol(A)");
POTRF (uplo, M, tmp->data(), LDA, info);
if (info) { umERROR("chol(A)", "dpotrf reports: info = %d", info); }
tmp->zero_below_diag();
tmp->set_factmode(FACT_CHOL); // indicate factored state
#if (0)
// compare with Matlab
tmp->print(g_MSGFile, "chol", "lf", 4, 8);
#endif
return (*tmp);
}
}
//---------------------------------------------------------
DMat& lu(DMat& A, bool in_place)
//---------------------------------------------------------
{
// Given square matrix A, return its lu-factorization
// for use later in solving (multiple) linear systems.
if (!A.is_square()) { umERROR("lu(A)", "matrix is not square."); }
int rows=A.num_rows(); int N=rows, LDA=rows, info=0;
int* ipiv = umIVector(rows);
if (in_place)
{
// factorize arg
GETRF(N, N, A.data(), LDA, ipiv, info);
if (info) { umERROR("lu(A)", "dgetrf reports: info = %d", info); }
A.set_pivots(ipiv); // store pivots
A.set_factmode(FACT_LUP); // indicate factored state
return A;
}
else
{
// factorize copy of arg
DMat* tmp = new DMat(A, OBJ_temp, "lu(A)");
GETRF(N, N, tmp->data(), LDA, ipiv, info);
if (info) { umERROR("lu(A)", "dgetrf reports: info = %d", info); }
tmp->set_pivots(ipiv); // store pivots
tmp->set_factmode(FACT_LUP); // indicate factored state
return (*tmp);
}
}
// compute eigensystem of a real symmetric matrix
//---------------------------------------------------------
void eig_sym(const DMat& A, DVec& ev, DMat& Q, bool bDoEVecs)
//---------------------------------------------------------
{
if (!A.is_square()) { umERROR("eig_sym(A)", "matrix is not square."); }
int N = A.num_rows();
int LDA=N, LDVL=N, LDVR=N, ldwork=10*N, info=0;
DVec work(ldwork, 0.0, OBJ_temp, "work_TMP");
Q = A; // Calculate eigenvectors in Q (optional)
ev.resize(N); // Calculate eigenvalues in ev
char jobV = bDoEVecs ? 'V' : 'N';
SYEV (jobV,'U', N, Q.data(), LDA, ev.data(), work.data(), ldwork, info);
if (info < 0) {
umERROR("eig_sym(A, Re,Im)", "Error in input argument (%d)\nNo solution computed.", -info);
} else if (info > 0) {
umLOG(1, "eig_sym(A, W): ...\n"
"\nthe algorithm failed to converge;"
"\n%d off-diagonal elements of an intermediate"
"\ntridiagonal form did not converge to zero.\n", info);
}
}
// DPOSV uses Cholesky factorization A=U^T*U, A=L*L^T
// to compute the solution to a real system of linear
// equations A*X=B, where A is a square, (N,N) symmetric
// positive definite matrix and X and B are (N,NRHS).
//
// If the system is over or under-determined,
// (i.e. A is not square), then pass the problem
// to the Least-squares solver (DGELSS) below.
//---------------------------------------------------------
void umSOLVE_CH(const DMat& mat, const DMat& B, DMat& X)
//---------------------------------------------------------
{
if (!mat.ok()) {umWARNING("umSOLVE_CH()", "system is empty"); return;}
if (!mat.is_square()) {
umSOLVE_LS(mat, B, X); // return a least-squares solution.
return;
}
DMat A(mat); // Work with a copy of input array.
X = B; // initialize solution with rhs
int rows=A.num_rows(), LDA=A.num_rows(), cols=A.num_cols();
int LDB=X.num_rows(), NRHS=X.num_cols(), info=0;
assert(LDB >= rows); // enough space for solutions?
// Solve the system.
POSV('U', rows, NRHS, A.data(), LDA, X.data(), LDB, info);
if (info < 0) {
X = 0.0;
umERROR("umSOLVE_CH(A,B, X)",
"Error in input argument (%d)\nNo solution computed.", -info);
} else if (info > 0) {
X = 0.0;
umERROR("umSOLVE_CH(A,B, X)",
"\nINFO = %d. The leading minor of order %d of A"
"\nis not positive definite, so the factorization"
"\ncould not be completed. No solution computed.",
info, info);
}
}
// DGESV uses the LU factorization to compute solution
// to a real system of linear equations, A * X = B,
// where A is square (N,N) and X, B are (N,NRHS).
//
// If the system is over or under-determined,
// (i.e. A is not square), then pass the problem
// to the Least-squares solver (DGELSS) below.
//---------------------------------------------------------
void umSOLVE(const DMat& mat, const DMat& B, DMat& X)
//---------------------------------------------------------
{
if (!mat.ok()) {umWARNING("umSOLVE()", "system is empty"); return;}
if (!mat.is_square()) {
umSOLVE_LS(mat, B, X); // return a least-squares solution.
return;
}
DMat A(mat); // work with copy of input
X = B; // initialize result with RHS
int rows=A.num_rows(), LDA=A.num_rows(), cols=A.num_cols();
int LDB=B.num_rows(), NRHS=B.num_cols(), info=0;
if (rows<1) {umWARNING("umSOLVE()", "system is empty"); return;}
IVec ipiv(rows);
// Solve the system.
GESV(rows, NRHS, A.data(), LDA, ipiv.data(), X.data(), LDB, info);
if (info < 0) {
X = 0.0;
umERROR("umSOLVE(A,B, X)",
"Error in input argument (%d)\nNo solution computed.", -info);
} else if (info > 0) {
X = 0.0;
umERROR("umSOLVE(A,B, X)",
"\nINFO = %d. U(%d,%d) was exactly zero."
"\nThe factorization has been completed, but the factor U is "
"\nexactly singular, so the solution could not be computed.",
info, info, info);
}
}
//---------------------------------------------------------
bool chol_solve(const DMat& ch, const DMat& B, DMat& X)
//---------------------------------------------------------
{
// Solve a set of linear systems using Cholesky-factored
// symmetric positive-definite matrix, A = U^T U.
if (FACT_CHOL != ch.get_factmode()) {umERROR("chol_solve(ch,B,X)", "matrix is not factored.");}
int M =ch.num_rows(), lda=ch.num_rows();
int ldb=B.num_rows(), nrhs=B.num_cols(); assert(ldb == M);
char uplo = 'U'; int info=0;
double* ch_data = const_cast<double*>(ch.data());
X = B; // overwrite X with RHS's, then solutions
POTRS (uplo, M, nrhs, ch_data, lda, X.data(), ldb, info);
if (info) { umERROR("chol_solve(ch,B,X)", "dpotrs reports: info = %d", info); }
return true;
}
//---------------------------------------------------------
void umAxB(const DMat& A, const DMat& B, DMat& C)
//---------------------------------------------------------
{
//-------------------------
// C = A * B
//-------------------------
// A = op(A) is (M,K)
// B = op(B) is (K,N)
// C is (M,N)
//-------------------------
int M=A.num_rows(), K=A.num_cols(), N=B.num_cols();
int LDA=M, LDB=K, LDC=M;
double one=1.0, zero=0.0;
if (B.num_rows() != K) { umERROR("umAxB(A,B,C)", "wrong dimensions"); }
C.resize(M,N);
GEMM ('N','N',M,N,K, one,A.data(),LDA,
B.data(),LDB,
zero,C.data(),LDC);
}
// DGELSS computes minimum norm solution to a real linear
// least squares problem: Minimize 2-norm(| b - A*x |).
// using the singular value decomposition (SVD) of A.
// A is an M-by-N matrix which may be rank-deficient.
//---------------------------------------------------------
void umSOLVE_LS(const DMat& mat, const DMat& B, DMat& X)
//---------------------------------------------------------
{
if (!mat.ok()) {umWARNING("umSOLVE_LS()", "system is empty"); return;}
DMat A(mat); // work with copy of input.
int rows=A.num_rows(), cols=A.num_cols(), mmn=A.min_mn();
int LDB=A.max_mn(), NRHS=B.num_cols();
if (rows!=B.num_rows()) {umERROR("umSOLVE_LS(A,B)", "Inconsistant matrix sizes.");}
DVec s(mmn); // allocate array for singular values
// X must be big enough to store various results.
// Resize X so that its leading dimension = max(M,N),
// then load the set of right hand sides.
X.resize(LDB,NRHS, true, 0.0);
for (int j=1; j<=NRHS; ++j) // loop across colums
for (int i=1; i<=rows; ++i) // loop down rows
X(i,j) = B(i,j);
// 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.
//double rcond = 1.0 / 1.0e16;
double rcond = -1.0;
// NBN: ACML does not use the work vector.
int mnLo=A.min_mn(), mnHi=A.max_mn(), rank=1, info=1;
int lwork = 10*mnLo + std::max(2*mnLo, std::max(mnHi, NRHS));
DVec work(lwork);
// Solve the system
GELSS (rows, cols, NRHS, A.data(), rows, X.data(), LDB, s.data(), rcond, rank, work.data(), lwork, info);
//---------------------------------------------
// Report:
//---------------------------------------------
if (info == 0) {
umLOG(1, "umSOLVE_LS reports successful LS-solution."
"\nRCOND = %0.6e, "
"\nOptimal length of work array was %d\n", rcond, lwork);
}
else
{
if (info < 0) {
X = 0.0;
umERROR("umSOLVE_LS(DMat&, DMat&)",
"Error in input argument (%d)\nNo solution or error bounds computed.", -info);
} else if (info > 0) {
X = 0.0;
umERROR("umSOLVE_LS(DMat&, DMat&)",
"\nThe algorithm for computing the SVD failed to converge.\n"
"\n%d off-diagonal elements of an intermediate "
"\nbidiagonal form did not converge to zero.\n "
"\nRCOND = %0.6e, "
"\nOptimal length of work array was %d.\n", info, rcond, lwork);
}
}
}
