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metaforce/gmm/gmm_dense_matrix_functions.h

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/* -*- c++ -*- (enables emacs c++ mode) */
/*===========================================================================
Copyright (C) 2014-2017 Konstantinos Poulios
This file is a part of GetFEM++
GetFEM++ is free software; you can redistribute it and/or modify it
under the terms of the GNU Lesser General Public License as published
by the Free Software Foundation; either version 3 of the License, or
(at your option) any later version along with the GCC Runtime Library
Exception either version 3.1 or (at your option) any later version.
This program is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
License and GCC Runtime Library Exception for more details.
You should have received a copy of the GNU Lesser General Public License
along with this program; if not, write to the Free Software Foundation,
Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA.
As a special exception, you may use this file as it is a part of a free
software library without restriction. Specifically, if other files
instantiate templates or use macros or inline functions from this file,
or you compile this file and link it with other files to produce an
executable, this file does not by itself cause the resulting executable
to be covered by the GNU Lesser General Public License. This exception
does not however invalidate any other reasons why the executable file
might be covered by the GNU Lesser General Public License.
===========================================================================*/
/**@file gmm_dense_matrix_functions.h
@author Konstantinos Poulios <poulios.konstantinos@gmail.com>
@date December 10, 2014.
@brief Common matrix functions for dense matrices.
*/
#ifndef GMM_DENSE_MATRIX_FUNCTIONS_H
#define GMM_DENSE_MATRIX_FUNCTIONS_H
namespace gmm {
/**
Matrix square root for upper triangular matrices (from GNU Octave).
*/
template <typename T>
void sqrtm_utri_inplace(dense_matrix<T>& A)
{
typedef typename number_traits<T>::magnitude_type R;
bool singular = false;
// The following code is equivalent to this triple loop:
//
// n = rows (A);
// for j = 1:n
// A(j,j) = sqrt (A(j,j));
// for i = j-1:-1:1
// A(i,j) /= (A(i,i) + A(j,j));
// k = 1:i-1;
// t storing a A(k,j) -= A(k,i) * A(i,j);
// endfor
// endfor
R tol = R(0); // default_tol(R()) * gmm::mat_maxnorm(A);
const size_type n = mat_nrows(A);
for (int j=0; j < int(n); j++) {
typename dense_matrix<T>::iterator colj = A.begin() + j*n;
if (gmm::abs(colj[j]) > tol)
colj[j] = gmm::sqrt(colj[j]);
else
singular = true;
for (int i=j-1; i >= 0; i--) {
typename dense_matrix<T>::const_iterator coli = A.begin() + i*n;
T colji = colj[i] = safe_divide(colj[i], (coli[i] + colj[j]));
for (int k = 0; k < i; k++)
colj[k] -= coli[k] * colji;
}
}
if (singular)
GMM_WARNING1("Matrix is singular, may not have a square root");
}
template <typename T>
void sqrtm(const dense_matrix<std::complex<T> >& A,
dense_matrix<std::complex<T> >& SQRTMA)
{
GMM_ASSERT1(gmm::mat_nrows(A) == gmm::mat_ncols(A),
"Matrix square root requires a square matrix");
gmm::resize(SQRTMA, gmm::mat_nrows(A), gmm::mat_ncols(A));
dense_matrix<std::complex<T> > S(A), Q(A), TMP(A);
#if defined(GMM_USES_LAPACK)
schur(TMP, S, Q);
#else
GMM_ASSERT1(false, "Please recompile with lapack and blas librairies "
"to use sqrtm matrix function.");
#endif
sqrtm_utri_inplace(S);
gmm::mult(Q, S, TMP);
gmm::mult(TMP, gmm::transposed(Q), SQRTMA);
}
template <typename T>
void sqrtm(const dense_matrix<T>& A,
dense_matrix<std::complex<T> >& SQRTMA)
{
dense_matrix<std::complex<T> > cA(mat_nrows(A), mat_ncols(A));
gmm::copy(A, gmm::real_part(cA));
sqrtm(cA, SQRTMA);
}
template <typename T>
void sqrtm(const dense_matrix<T>& A, dense_matrix<T>& SQRTMA)
{
dense_matrix<std::complex<T> > cA(mat_nrows(A), mat_ncols(A));
gmm::copy(A, gmm::real_part(cA));
dense_matrix<std::complex<T> > cSQRTMA(cA);
sqrtm(cA, cSQRTMA);
gmm::resize(SQRTMA, gmm::mat_nrows(A), gmm::mat_ncols(A));
gmm::copy(gmm::real_part(cSQRTMA), SQRTMA);
// dense_matrix<std::complex<T1> >::const_reference
// it = cSQRTMA.begin(), ite = cSQRTMA.end();
// dense_matrix<std::complex<T1> >::reference
// rit = SQRTMA.begin();
// for (; it != ite; ++it, ++rit) *rit = it->real();
}
/**
Matrix logarithm for upper triangular matrices (from GNU/Octave)
*/
template <typename T>
void logm_utri_inplace(dense_matrix<T>& S)
{
typedef typename number_traits<T>::magnitude_type R;
size_type n = gmm::mat_nrows(S);
GMM_ASSERT1(n == gmm::mat_ncols(S),
"Matrix logarithm is not defined for non-square matrices");
for (size_type i=0; i < n-1; ++i)
if (gmm::abs(S(i+1,i)) > default_tol(T())) {
GMM_ASSERT1(false, "An upper triangular matrix is expected");
break;
}
for (size_type i=0; i < n-1; ++i)
if (gmm::real(S(i,i)) <= -default_tol(R()) &&
gmm::abs(gmm::imag(S(i,i))) <= default_tol(R())) {
GMM_ASSERT1(false, "Principal matrix logarithm is not defined "
"for matrices with negative eigenvalues");
break;
}
// Algorithm 11.9 in "Function of matrices", by N. Higham
R theta[] = { R(0),R(0),R(1.61e-2),R(5.38e-2),R(1.13e-1),R(1.86e-1),R(2.6429608311114350e-1) };
R scaling(1);
size_type p(0), m(6), opt_iters(100);
for (size_type k=0; k < opt_iters; ++k, scaling *= R(2)) {
dense_matrix<T> auxS(S);
for (size_type i = 0; i < n; ++i) auxS(i,i) -= R(1);
R tau = gmm::mat_norm1(auxS);
if (tau <= theta[6]) {
++p;
size_type j1(6), j2(6);
for (size_type j=0; j < 6; ++j)
if (tau <= theta[j]) { j1 = j; break; }
for (size_type j=0; j < j1; ++j)
if (tau <= 2*theta[j]) { j2 = j; break; }
if (j1 - j2 <= 1 || p == 2) { m = j1; break; }
}
sqrtm_utri_inplace(S);
if (k == opt_iters-1)
GMM_WARNING1 ("Maximum number of square roots exceeded; "
"the calculated matrix logarithm may still be accurate");
}
for (size_type i = 0; i < n; ++i) S(i,i) -= R(1);
if (m > 0) {
std::vector<R> nodes, wts;
switch(m) {
case 0: {
R nodes_[] = { R(0.5) };
R wts_[] = { R(1) };
nodes.assign(nodes_, nodes_+m+1);
wts.assign(wts_, wts_+m+1);
} break;
case 1: {
R nodes_[] = { R(0.211324865405187),R(0.788675134594813) };
R wts_[] = { R(0.5),R(0.5) };
nodes.assign(nodes_, nodes_+m+1);
wts.assign(wts_, wts_+m+1);
} break;
case 2: {
R nodes_[] = { R(0.112701665379258),R(0.500000000000000),R(0.887298334620742) };
R wts_[] = { R(0.277777777777778),R(0.444444444444444),R(0.277777777777778) };
nodes.assign(nodes_, nodes_+m+1);
wts.assign(wts_, wts_+m+1);
} break;
case 3: {
R nodes_[] = { R(0.0694318442029737),R(0.3300094782075718),R(0.6699905217924281),R(0.9305681557970263) };
R wts_[] = { R(0.173927422568727),R(0.326072577431273),R(0.326072577431273),R(0.173927422568727) };
nodes.assign(nodes_, nodes_+m+1);
wts.assign(wts_, wts_+m+1);
} break;
case 4: {
R nodes_[] = { R(0.0469100770306681),R(0.2307653449471584),R(0.5000000000000000),R(0.7692346550528415),R(0.9530899229693319) };
R wts_[] = { R(0.118463442528095),R(0.239314335249683),R(0.284444444444444),R(0.239314335249683),R(0.118463442528094) };
nodes.assign(nodes_, nodes_+m+1);
wts.assign(wts_, wts_+m+1);
} break;
case 5: {
R nodes_[] = { R(0.0337652428984240),R(0.1693953067668678),R(0.3806904069584015),R(0.6193095930415985),R(0.8306046932331322),R(0.9662347571015761) };
R wts_[] = { R(0.0856622461895853),R(0.1803807865240693),R(0.2339569672863452),R(0.2339569672863459),R(0.1803807865240693),R(0.0856622461895852) };
nodes.assign(nodes_, nodes_+m+1);
wts.assign(wts_, wts_+m+1);
} break;
case 6: {
R nodes_[] = { R(0.0254460438286208),R(0.1292344072003028),R(0.2970774243113015),R(0.4999999999999999),R(0.7029225756886985),R(0.8707655927996973),R(0.9745539561713792) };
R wts_[] = { R(0.0647424830844348),R(0.1398526957446384),R(0.1909150252525594),R(0.2089795918367343),R(0.1909150252525595),R(0.1398526957446383),R(0.0647424830844349) };
nodes.assign(nodes_, nodes_+m+1);
wts.assign(wts_, wts_+m+1);
} break;
}
dense_matrix<T> auxS1(S), auxS2(S);
std::vector<T> auxvec(n);
gmm::clear(S);
for (size_type j=0; j <= m; ++j) {
gmm::copy(gmm::scaled(auxS1, nodes[j]), auxS2);
gmm::add(gmm::identity_matrix(), auxS2);
// S += wts[i] * auxS1 * inv(auxS2)
for (size_type i=0; i < n; ++i) {
gmm::copy(gmm::mat_row(auxS1, i), auxvec);
gmm::lower_tri_solve(gmm::transposed(auxS2), auxvec, false);
gmm::add(gmm::scaled(auxvec, wts[j]), gmm::mat_row(S, i));
}
}
}
gmm::scale(S, scaling);
}
/**
Matrix logarithm (from GNU/Octave)
*/
template <typename T>
void logm(const dense_matrix<T>& A, dense_matrix<T>& LOGMA)
{
typedef typename number_traits<T>::magnitude_type R;
size_type n = gmm::mat_nrows(A);
GMM_ASSERT1(n == gmm::mat_ncols(A),
"Matrix logarithm is not defined for non-square matrices");
dense_matrix<T> S(A), Q(A);
#if defined(GMM_USES_LAPACK)
schur(A, S, Q); // A = Q * S * Q^T
#else
GMM_ASSERT1(false, "Please recompile with lapack and blas librairies "
"to use logm matrix function.");
#endif
bool convert_to_complex(false);
if (!is_complex(T()))
for (size_type i=0; i < n-1; ++i)
if (gmm::abs(S(i+1,i)) > default_tol(T())) {
convert_to_complex = true;
break;
}
gmm::resize(LOGMA, n, n);
if (convert_to_complex) {
dense_matrix<std::complex<R> > cS(n,n), cQ(n,n), auxmat(n,n);
gmm::copy(gmm::real_part(S), gmm::real_part(cS));
gmm::copy(gmm::real_part(Q), gmm::real_part(cQ));
block2x2_reduction(cS, cQ, default_tol(R())*R(3));
for (size_type j=0; j < n-1; ++j)
for (size_type i=j+1; i < n; ++i)
cS(i,j) = T(0);
logm_utri_inplace(cS);
gmm::mult(cQ, cS, auxmat);
gmm::mult(auxmat, gmm::transposed(cQ), cS);
// Remove small complex values which may have entered calculation
gmm::copy(gmm::real_part(cS), LOGMA);
// GMM_ASSERT1(gmm::mat_norm1(gmm::imag_part(cS)) < n*default_tol(T()),
// "Internal error, imag part should be zero");
} else {
dense_matrix<T> auxmat(n,n);
logm_utri_inplace(S);
gmm::mult(Q, S, auxmat);
gmm::mult(auxmat, gmm::transposed(Q), LOGMA);
}
}
}
#endif
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