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/* Copyright (c) 1997-2024
Ewgenij Gawrilow, Michael Joswig, and the polymake team
Technische Universität Berlin, Germany
https://polymake.org
This program is free software; you can redistribute it and/or modify it
under the terms of the GNU General Public License as published by the
Free Software Foundation; either version 2, or (at your option) any
later version: http://www.gnu.org/licenses/gpl.txt.
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 General Public License for more details.
--------------------------------------------------------------------------------
*/
#pragma once
#if defined(__clang__)
#pragma clang diagnostic push
#pragma clang diagnostic ignored "-Wzero-as-null-pointer-constant"
#elif defined(__GNUC__)
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wzero-as-null-pointer-constant"
#endif
#include "boost/numeric/ublas/lu.hpp"
#if defined(__clang__)
#pragma clang diagnostic pop
#elif defined(__GNUC__)
#pragma GCC diagnostic pop
#endif
#include "polymake/SparseMatrix.h"
#include "polymake/Matrix.h"
#include "polymake/Vector.h"
#include "polymake/internal/linalg_exceptions.h"
#include "polymake/internal/sparse_linalg.h"
#include "polymake/internal/dense_linalg.h"
namespace pm {
/*
We solve the matrix equation AX=B for X by transforming the system
____d___ ___e____
| a1 | | b1 |
| a2 | ___e___ | b2 |
| | | | | |
n| | |x1 xe| = | |
| | d | | |
| | |_____| | |
| an | | bn |
|________| |______|
into
-a1- 0 0 | b11
0 -a1- 0 x1 b12
0 0 -a1- | b1e
-a2- 0 0 | |
0 -a2- 0 b2
0 0 -a2- | = |
-an- 0 0 | |
0 -an- 0 xe bn
0 0 -an- | |
*/
template <typename TMatrix1, typename TMatrix2, typename E>
std::enable_if_t<is_field<E>::value, std::pair<SparseMatrix<E>, Vector<E>>>
augmented_system(const GenericMatrix<TMatrix1, E>& A, const GenericMatrix<TMatrix2, E>& B)
{
const Int n(A.rows()), d(A.cols()), e(B.cols());
SparseMatrix<E> A_aug(n*e, d*e);
Vector<E> rhs(n*e);
auto rhs_it = rhs.begin();
for (Int i = 0; i < n; ++i) {
for (Int j = 0; j < e; ++j, ++rhs_it) {
A_aug.minor(scalar2set(i*e+j), sequence(d*j, d)) = A.minor(scalar2set(i),All);
*rhs_it = B[i][j];
}
}
return std::make_pair(A_aug, rhs);
}
template <typename TMatrix1, typename TMatrix2, typename E>
std::enable_if_t<is_field<E>::value, Matrix<E>>
solve_right(const GenericMatrix<TMatrix1, E>& A, const GenericMatrix<TMatrix2, E>& B)
{
if (POLYMAKE_DEBUG || is_wary<TMatrix1>() || is_wary<TMatrix2>()) {
if (B.rows() != A.rows())
throw std::runtime_error("solve_right - mismatch in number of rows");
}
const auto aug_rhs(augmented_system(A,B));
return T(Matrix<E>(B.cols(), A.cols(), lin_solve<E,false>(aug_rhs.first, aug_rhs.second).begin()));
}
/*
In the floating-point case, to solve the matrix equation AX = B for X, we use the LU factorization provided by boost.
By expressing A as a product A = LU of a Lower and an Upper triangular matrix, we can find the solution to
A X = L U X = B
via
L Y = B // solve for Y
U X = Y // solve for X
In fact, we use partial pivoting, so the LU decomposition reads PA = LU, so to solve AX = B we instead solve
P A X = L U X = P B
via
L Y = P B
U X = Y
Because boost::ublas's LU solver only works for square matrices (even if this is not documented),
we need additional steps to process rectangular matrices.
(1) If A has more rows than columns, instead of solving
A X = B
we solve
A^T A X == A^T B
by LU-decomposing the (small) matrix A^T A. Numerically, it would be better to use QR decomposition here, see
https://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_inverse#The_QR_method ,
but boost::ublas also doesn't provide this.
(2) The case where A has more columns than rows needs SVD decomposition; this is easy implement and will be done when the need arises.
*/
template <typename TMatrix1, typename TMatrix2>
Matrix<double>
solve_right(const GenericMatrix<TMatrix1, double>& A, const GenericMatrix<TMatrix2, double>& B)
{
if (POLYMAKE_DEBUG || is_wary<TMatrix1>() || is_wary<TMatrix2>()) {
if (B.rows() != A.rows())
throw std::runtime_error("solve_right: mismatch in number of rows");
}
if (A.cols() > A.rows()) {
throw std::runtime_error("solve_right: the case A.cols() > A.rows() is not implemented yet.");
}
const bool square( A.cols() == A.rows() );
const Int
Arows ( square ? A.rows() : A.cols() ),
Brows ( square ? B.rows() : A.cols() );
boost::numeric::ublas::matrix<double> ublasA (Arows, A.cols());
if (square)
copy_range(entire(concat_rows(Matrix<double>(A))), ublasA.data().begin());
else
copy_range(entire(concat_rows(Matrix<double>(T(A)*A))), ublasA.data().begin());
boost::numeric::ublas::matrix<double> ublasB (Brows, B.cols());
if (square)
copy_range(entire(concat_rows(Matrix<double>(B))), ublasB.data().begin());
else
copy_range(entire(concat_rows(Matrix<double>(T(A)*B))), ublasB.data().begin());
boost::numeric::ublas::permutation_matrix<> ublasP(Arows); // permutation matrix for LU factorization
boost::numeric::ublas::lu_factorize(ublasA, ublasP); // now ublasA is factored in-place into L and U
boost::numeric::ublas::lu_substitute(ublasA, ublasP, ublasB); // now ublasB contains the solution
Matrix<double> sol(Brows, B.cols());
for (Int i = 0; i < Brows; ++i)
for (Int j = 0; j < B.cols(); ++j) {
const double b = ublasB(i,j);
sol(i,j) = fabs(b) < 10.0 * std::numeric_limits<double>::epsilon()
? 0
: b;
}
return sol;
}
// solve the matrix equation X A = B for X by reducing it to A^T X^T = B^T
template <typename TMatrix1, typename TMatrix2, typename E>
std::enable_if_t<is_field<E>::value, Matrix<E>>
solve_left(const GenericMatrix<TMatrix1, E>& A, const GenericMatrix<TMatrix2, E>& B)
{
return T(solve_right(T(A), T(B)));
}
} // end namespace pm
// Local Variables:
// mode:C++
// c-basic-offset:3
// indent-tabs-mode:nil
// End:
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