// 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);
}
}
// 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);
}
}
}