// 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).
//---------------------------------------------------------
void umSOLVE_CH(const DMat& mat, const DVec& b, DVec& x)
//---------------------------------------------------------
{
// check args
assert(mat.is_square()); // symmetric
assert(b.size() >= mat.num_rows()); // is b consistent?
assert(b.size() <= x.size()); // can x store solution?
DMat A(mat); // work with copy of input
x = b; // allocate solution vector
int rows=A.num_rows(), LDA=A.num_rows(), cols=A.num_cols();
int LDB=b.size(), NRHS=1, info=0;
if (rows<1) {umWARNING("umSOLVE_CH()", "system is empty"); return;}
// 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);
}
}
// 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);
}
}
//---------------------------------------------------------
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
}