369 lines
14 KiB
C++
369 lines
14 KiB
C++
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// This file is part of Eigen, a lightweight C++ template library
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// for linear algebra.
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//
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// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
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// Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
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//
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// This Source Code Form is subject to the terms of the Mozilla
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// Public License v. 2.0. If a copy of the MPL was not distributed
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// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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#include "main.h"
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#include <Eigen/QR>
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#include <Eigen/SVD>
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#include "solverbase.h"
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template <typename MatrixType>
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void cod() {
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STATIC_CHECK(( internal::is_same<typename CompleteOrthogonalDecomposition<MatrixType>::StorageIndex,int>::value ));
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Index rows = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE);
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Index cols = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE);
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Index cols2 = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE);
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Index rank = internal::random<Index>(1, (std::min)(rows, cols) - 1);
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typedef typename MatrixType::Scalar Scalar;
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typedef Matrix<Scalar, MatrixType::RowsAtCompileTime,
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MatrixType::RowsAtCompileTime>
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MatrixQType;
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MatrixType matrix;
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createRandomPIMatrixOfRank(rank, rows, cols, matrix);
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CompleteOrthogonalDecomposition<MatrixType> cod(matrix);
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VERIFY(rank == cod.rank());
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VERIFY(cols - cod.rank() == cod.dimensionOfKernel());
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VERIFY(!cod.isInjective());
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VERIFY(!cod.isInvertible());
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VERIFY(!cod.isSurjective());
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MatrixQType q = cod.householderQ();
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VERIFY_IS_UNITARY(q);
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MatrixType z = cod.matrixZ();
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VERIFY_IS_UNITARY(z);
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MatrixType t;
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t.setZero(rows, cols);
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t.topLeftCorner(rank, rank) =
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cod.matrixT().topLeftCorner(rank, rank).template triangularView<Upper>();
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MatrixType c = q * t * z * cod.colsPermutation().inverse();
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VERIFY_IS_APPROX(matrix, c);
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check_solverbase<MatrixType, MatrixType>(matrix, cod, rows, cols, cols2);
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// Verify that we get the same minimum-norm solution as the SVD.
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MatrixType exact_solution = MatrixType::Random(cols, cols2);
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MatrixType rhs = matrix * exact_solution;
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MatrixType cod_solution = cod.solve(rhs);
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JacobiSVD<MatrixType> svd(matrix, ComputeThinU | ComputeThinV);
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MatrixType svd_solution = svd.solve(rhs);
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VERIFY_IS_APPROX(cod_solution, svd_solution);
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MatrixType pinv = cod.pseudoInverse();
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VERIFY_IS_APPROX(cod_solution, pinv * rhs);
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}
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template <typename MatrixType, int Cols2>
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void cod_fixedsize() {
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enum {
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Rows = MatrixType::RowsAtCompileTime,
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Cols = MatrixType::ColsAtCompileTime
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};
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typedef typename MatrixType::Scalar Scalar;
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typedef CompleteOrthogonalDecomposition<Matrix<Scalar, Rows, Cols> > COD;
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int rank = internal::random<int>(1, (std::min)(int(Rows), int(Cols)) - 1);
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Matrix<Scalar, Rows, Cols> matrix;
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createRandomPIMatrixOfRank(rank, Rows, Cols, matrix);
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COD cod(matrix);
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VERIFY(rank == cod.rank());
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VERIFY(Cols - cod.rank() == cod.dimensionOfKernel());
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VERIFY(cod.isInjective() == (rank == Rows));
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VERIFY(cod.isSurjective() == (rank == Cols));
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VERIFY(cod.isInvertible() == (cod.isInjective() && cod.isSurjective()));
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check_solverbase<Matrix<Scalar, Cols, Cols2>, Matrix<Scalar, Rows, Cols2> >(matrix, cod, Rows, Cols, Cols2);
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// Verify that we get the same minimum-norm solution as the SVD.
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Matrix<Scalar, Cols, Cols2> exact_solution;
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exact_solution.setRandom(Cols, Cols2);
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Matrix<Scalar, Rows, Cols2> rhs = matrix * exact_solution;
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Matrix<Scalar, Cols, Cols2> cod_solution = cod.solve(rhs);
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JacobiSVD<MatrixType> svd(matrix, ComputeFullU | ComputeFullV);
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Matrix<Scalar, Cols, Cols2> svd_solution = svd.solve(rhs);
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VERIFY_IS_APPROX(cod_solution, svd_solution);
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typename Inverse<COD>::PlainObject pinv = cod.pseudoInverse();
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VERIFY_IS_APPROX(cod_solution, pinv * rhs);
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}
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template<typename MatrixType> void qr()
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{
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using std::sqrt;
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STATIC_CHECK(( internal::is_same<typename ColPivHouseholderQR<MatrixType>::StorageIndex,int>::value ));
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Index rows = internal::random<Index>(2,EIGEN_TEST_MAX_SIZE), cols = internal::random<Index>(2,EIGEN_TEST_MAX_SIZE), cols2 = internal::random<Index>(2,EIGEN_TEST_MAX_SIZE);
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Index rank = internal::random<Index>(1, (std::min)(rows, cols)-1);
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typedef typename MatrixType::Scalar Scalar;
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typedef typename MatrixType::RealScalar RealScalar;
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typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, MatrixType::RowsAtCompileTime> MatrixQType;
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MatrixType m1;
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createRandomPIMatrixOfRank(rank,rows,cols,m1);
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ColPivHouseholderQR<MatrixType> qr(m1);
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VERIFY_IS_EQUAL(rank, qr.rank());
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VERIFY_IS_EQUAL(cols - qr.rank(), qr.dimensionOfKernel());
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VERIFY(!qr.isInjective());
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VERIFY(!qr.isInvertible());
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VERIFY(!qr.isSurjective());
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MatrixQType q = qr.householderQ();
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VERIFY_IS_UNITARY(q);
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MatrixType r = qr.matrixQR().template triangularView<Upper>();
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MatrixType c = q * r * qr.colsPermutation().inverse();
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VERIFY_IS_APPROX(m1, c);
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// Verify that the absolute value of the diagonal elements in R are
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// non-increasing until they reach the singularity threshold.
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RealScalar threshold =
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sqrt(RealScalar(rows)) * numext::abs(r(0, 0)) * NumTraits<Scalar>::epsilon();
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for (Index i = 0; i < (std::min)(rows, cols) - 1; ++i) {
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RealScalar x = numext::abs(r(i, i));
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RealScalar y = numext::abs(r(i + 1, i + 1));
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if (x < threshold && y < threshold) continue;
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if (!test_isApproxOrLessThan(y, x)) {
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for (Index j = 0; j < (std::min)(rows, cols); ++j) {
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std::cout << "i = " << j << ", |r_ii| = " << numext::abs(r(j, j)) << std::endl;
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}
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std::cout << "Failure at i=" << i << ", rank=" << rank
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<< ", threshold=" << threshold << std::endl;
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}
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VERIFY_IS_APPROX_OR_LESS_THAN(y, x);
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}
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check_solverbase<MatrixType, MatrixType>(m1, qr, rows, cols, cols2);
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{
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MatrixType m2, m3;
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Index size = rows;
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do {
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m1 = MatrixType::Random(size,size);
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qr.compute(m1);
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} while(!qr.isInvertible());
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MatrixType m1_inv = qr.inverse();
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m3 = m1 * MatrixType::Random(size,cols2);
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m2 = qr.solve(m3);
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VERIFY_IS_APPROX(m2, m1_inv*m3);
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}
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}
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template<typename MatrixType, int Cols2> void qr_fixedsize()
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{
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using std::sqrt;
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using std::abs;
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enum { Rows = MatrixType::RowsAtCompileTime, Cols = MatrixType::ColsAtCompileTime };
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typedef typename MatrixType::Scalar Scalar;
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typedef typename MatrixType::RealScalar RealScalar;
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int rank = internal::random<int>(1, (std::min)(int(Rows), int(Cols))-1);
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Matrix<Scalar,Rows,Cols> m1;
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createRandomPIMatrixOfRank(rank,Rows,Cols,m1);
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ColPivHouseholderQR<Matrix<Scalar,Rows,Cols> > qr(m1);
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VERIFY_IS_EQUAL(rank, qr.rank());
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VERIFY_IS_EQUAL(Cols - qr.rank(), qr.dimensionOfKernel());
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VERIFY_IS_EQUAL(qr.isInjective(), (rank == Rows));
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VERIFY_IS_EQUAL(qr.isSurjective(), (rank == Cols));
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VERIFY_IS_EQUAL(qr.isInvertible(), (qr.isInjective() && qr.isSurjective()));
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Matrix<Scalar,Rows,Cols> r = qr.matrixQR().template triangularView<Upper>();
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Matrix<Scalar,Rows,Cols> c = qr.householderQ() * r * qr.colsPermutation().inverse();
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VERIFY_IS_APPROX(m1, c);
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check_solverbase<Matrix<Scalar,Cols,Cols2>, Matrix<Scalar,Rows,Cols2> >(m1, qr, Rows, Cols, Cols2);
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// Verify that the absolute value of the diagonal elements in R are
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// non-increasing until they reache the singularity threshold.
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RealScalar threshold =
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sqrt(RealScalar(Rows)) * (std::abs)(r(0, 0)) * NumTraits<Scalar>::epsilon();
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for (Index i = 0; i < (std::min)(int(Rows), int(Cols)) - 1; ++i) {
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RealScalar x = numext::abs(r(i, i));
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RealScalar y = numext::abs(r(i + 1, i + 1));
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if (x < threshold && y < threshold) continue;
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if (!test_isApproxOrLessThan(y, x)) {
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for (Index j = 0; j < (std::min)(int(Rows), int(Cols)); ++j) {
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std::cout << "i = " << j << ", |r_ii| = " << numext::abs(r(j, j)) << std::endl;
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}
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std::cout << "Failure at i=" << i << ", rank=" << rank
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<< ", threshold=" << threshold << std::endl;
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}
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VERIFY_IS_APPROX_OR_LESS_THAN(y, x);
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}
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}
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// This test is meant to verify that pivots are chosen such that
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// even for a graded matrix, the diagonal of R falls of roughly
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// monotonically until it reaches the threshold for singularity.
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// We use the so-called Kahan matrix, which is a famous counter-example
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// for rank-revealing QR. See
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// http://www.netlib.org/lapack/lawnspdf/lawn176.pdf
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// page 3 for more detail.
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template<typename MatrixType> void qr_kahan_matrix()
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{
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using std::sqrt;
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using std::abs;
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typedef typename MatrixType::Scalar Scalar;
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typedef typename MatrixType::RealScalar RealScalar;
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Index rows = 300, cols = rows;
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MatrixType m1;
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m1.setZero(rows,cols);
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RealScalar s = std::pow(NumTraits<RealScalar>::epsilon(), 1.0 / rows);
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RealScalar c = std::sqrt(1 - s*s);
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RealScalar pow_s_i(1.0); // pow(s,i)
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for (Index i = 0; i < rows; ++i) {
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m1(i, i) = pow_s_i;
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m1.row(i).tail(rows - i - 1) = -pow_s_i * c * MatrixType::Ones(1, rows - i - 1);
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pow_s_i *= s;
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}
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m1 = (m1 + m1.transpose()).eval();
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ColPivHouseholderQR<MatrixType> qr(m1);
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MatrixType r = qr.matrixQR().template triangularView<Upper>();
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RealScalar threshold =
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std::sqrt(RealScalar(rows)) * numext::abs(r(0, 0)) * NumTraits<Scalar>::epsilon();
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for (Index i = 0; i < (std::min)(rows, cols) - 1; ++i) {
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RealScalar x = numext::abs(r(i, i));
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RealScalar y = numext::abs(r(i + 1, i + 1));
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if (x < threshold && y < threshold) continue;
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if (!test_isApproxOrLessThan(y, x)) {
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for (Index j = 0; j < (std::min)(rows, cols); ++j) {
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std::cout << "i = " << j << ", |r_ii| = " << numext::abs(r(j, j)) << std::endl;
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}
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std::cout << "Failure at i=" << i << ", rank=" << qr.rank()
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<< ", threshold=" << threshold << std::endl;
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}
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VERIFY_IS_APPROX_OR_LESS_THAN(y, x);
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}
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}
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template<typename MatrixType> void qr_invertible()
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{
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using std::log;
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using std::abs;
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typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
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typedef typename MatrixType::Scalar Scalar;
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int size = internal::random<int>(10,50);
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MatrixType m1(size, size), m2(size, size), m3(size, size);
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m1 = MatrixType::Random(size,size);
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if (internal::is_same<RealScalar,float>::value)
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{
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// let's build a matrix more stable to inverse
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MatrixType a = MatrixType::Random(size,size*2);
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m1 += a * a.adjoint();
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}
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ColPivHouseholderQR<MatrixType> qr(m1);
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check_solverbase<MatrixType, MatrixType>(m1, qr, size, size, size);
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// now construct a matrix with prescribed determinant
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m1.setZero();
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for(int i = 0; i < size; i++) m1(i,i) = internal::random<Scalar>();
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RealScalar absdet = abs(m1.diagonal().prod());
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m3 = qr.householderQ(); // get a unitary
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m1 = m3 * m1 * m3;
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qr.compute(m1);
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VERIFY_IS_APPROX(absdet, qr.absDeterminant());
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VERIFY_IS_APPROX(log(absdet), qr.logAbsDeterminant());
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}
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template<typename MatrixType> void qr_verify_assert()
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{
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MatrixType tmp;
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ColPivHouseholderQR<MatrixType> qr;
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VERIFY_RAISES_ASSERT(qr.matrixQR())
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VERIFY_RAISES_ASSERT(qr.solve(tmp))
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VERIFY_RAISES_ASSERT(qr.transpose().solve(tmp))
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VERIFY_RAISES_ASSERT(qr.adjoint().solve(tmp))
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VERIFY_RAISES_ASSERT(qr.householderQ())
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VERIFY_RAISES_ASSERT(qr.dimensionOfKernel())
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VERIFY_RAISES_ASSERT(qr.isInjective())
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VERIFY_RAISES_ASSERT(qr.isSurjective())
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VERIFY_RAISES_ASSERT(qr.isInvertible())
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VERIFY_RAISES_ASSERT(qr.inverse())
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VERIFY_RAISES_ASSERT(qr.absDeterminant())
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VERIFY_RAISES_ASSERT(qr.logAbsDeterminant())
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}
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template<typename MatrixType> void cod_verify_assert()
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{
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MatrixType tmp;
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CompleteOrthogonalDecomposition<MatrixType> cod;
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VERIFY_RAISES_ASSERT(cod.matrixQTZ())
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VERIFY_RAISES_ASSERT(cod.solve(tmp))
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VERIFY_RAISES_ASSERT(cod.transpose().solve(tmp))
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VERIFY_RAISES_ASSERT(cod.adjoint().solve(tmp))
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VERIFY_RAISES_ASSERT(cod.householderQ())
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VERIFY_RAISES_ASSERT(cod.dimensionOfKernel())
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VERIFY_RAISES_ASSERT(cod.isInjective())
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VERIFY_RAISES_ASSERT(cod.isSurjective())
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VERIFY_RAISES_ASSERT(cod.isInvertible())
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VERIFY_RAISES_ASSERT(cod.pseudoInverse())
|
||
|
|
VERIFY_RAISES_ASSERT(cod.absDeterminant())
|
||
|
|
VERIFY_RAISES_ASSERT(cod.logAbsDeterminant())
|
||
|
|
}
|
||
|
|
|
||
|
|
EIGEN_DECLARE_TEST(qr_colpivoting)
|
||
|
|
{
|
||
|
|
for(int i = 0; i < g_repeat; i++) {
|
||
|
|
CALL_SUBTEST_1( qr<MatrixXf>() );
|
||
|
|
CALL_SUBTEST_2( qr<MatrixXd>() );
|
||
|
|
CALL_SUBTEST_3( qr<MatrixXcd>() );
|
||
|
|
CALL_SUBTEST_4(( qr_fixedsize<Matrix<float,3,5>, 4 >() ));
|
||
|
|
CALL_SUBTEST_5(( qr_fixedsize<Matrix<double,6,2>, 3 >() ));
|
||
|
|
CALL_SUBTEST_5(( qr_fixedsize<Matrix<double,1,1>, 1 >() ));
|
||
|
|
}
|
||
|
|
|
||
|
|
for(int i = 0; i < g_repeat; i++) {
|
||
|
|
CALL_SUBTEST_1( cod<MatrixXf>() );
|
||
|
|
CALL_SUBTEST_2( cod<MatrixXd>() );
|
||
|
|
CALL_SUBTEST_3( cod<MatrixXcd>() );
|
||
|
|
CALL_SUBTEST_4(( cod_fixedsize<Matrix<float,3,5>, 4 >() ));
|
||
|
|
CALL_SUBTEST_5(( cod_fixedsize<Matrix<double,6,2>, 3 >() ));
|
||
|
|
CALL_SUBTEST_5(( cod_fixedsize<Matrix<double,1,1>, 1 >() ));
|
||
|
|
}
|
||
|
|
|
||
|
|
for(int i = 0; i < g_repeat; i++) {
|
||
|
|
CALL_SUBTEST_1( qr_invertible<MatrixXf>() );
|
||
|
|
CALL_SUBTEST_2( qr_invertible<MatrixXd>() );
|
||
|
|
CALL_SUBTEST_6( qr_invertible<MatrixXcf>() );
|
||
|
|
CALL_SUBTEST_3( qr_invertible<MatrixXcd>() );
|
||
|
|
}
|
||
|
|
|
||
|
|
CALL_SUBTEST_7(qr_verify_assert<Matrix3f>());
|
||
|
|
CALL_SUBTEST_8(qr_verify_assert<Matrix3d>());
|
||
|
|
CALL_SUBTEST_1(qr_verify_assert<MatrixXf>());
|
||
|
|
CALL_SUBTEST_2(qr_verify_assert<MatrixXd>());
|
||
|
|
CALL_SUBTEST_6(qr_verify_assert<MatrixXcf>());
|
||
|
|
CALL_SUBTEST_3(qr_verify_assert<MatrixXcd>());
|
||
|
|
|
||
|
|
CALL_SUBTEST_7(cod_verify_assert<Matrix3f>());
|
||
|
|
CALL_SUBTEST_8(cod_verify_assert<Matrix3d>());
|
||
|
|
CALL_SUBTEST_1(cod_verify_assert<MatrixXf>());
|
||
|
|
CALL_SUBTEST_2(cod_verify_assert<MatrixXd>());
|
||
|
|
CALL_SUBTEST_6(cod_verify_assert<MatrixXcf>());
|
||
|
|
CALL_SUBTEST_3(cod_verify_assert<MatrixXcd>());
|
||
|
|
|
||
|
|
// Test problem size constructors
|
||
|
|
CALL_SUBTEST_9(ColPivHouseholderQR<MatrixXf>(10, 20));
|
||
|
|
|
||
|
|
CALL_SUBTEST_1( qr_kahan_matrix<MatrixXf>() );
|
||
|
|
CALL_SUBTEST_2( qr_kahan_matrix<MatrixXd>() );
|
||
|
|
}
|