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qr_colpivoting.cpp
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Fri, Feb 21, 23:36

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

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