// This file is part of Eigen, a lightweight C++ template library // for linear algebra. // // Copyright (C) 2008-2014 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/. #ifndef SVD_DEFAULT #error a macro SVD_DEFAULT(MatrixType) must be defined prior to including svd_common.h #endif #ifndef SVD_FOR_MIN_NORM #error a macro SVD_FOR_MIN_NORM(MatrixType) must be defined prior to including svd_common.h #endif #include "svd_fill.h" // Check that the matrix m is properly reconstructed and that the U and V factors are unitary // The SVD must have already been computed. template<typename SvdType, typename MatrixType> void svd_check_full(const MatrixType& m, const SvdType& svd) { typedef typename MatrixType::Index Index; Index rows = m.rows(); Index cols = m.cols(); enum { RowsAtCompileTime = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime }; typedef typename MatrixType::Scalar Scalar; typedef typename MatrixType::RealScalar RealScalar; typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime> MatrixUType; typedef Matrix<Scalar, ColsAtCompileTime, ColsAtCompileTime> MatrixVType; MatrixType sigma = MatrixType::Zero(rows,cols); sigma.diagonal() = svd.singularValues().template cast<Scalar>(); MatrixUType u = svd.matrixU(); MatrixVType v = svd.matrixV(); RealScalar scaling = m.cwiseAbs().maxCoeff(); if(scaling<=(std::numeric_limits<RealScalar>::min)()) scaling = RealScalar(1); VERIFY_IS_APPROX(m/scaling, u * (sigma/scaling) * v.adjoint()); VERIFY_IS_UNITARY(u); VERIFY_IS_UNITARY(v); } // Compare partial SVD defined by computationOptions to a full SVD referenceSvd template<typename SvdType, typename MatrixType> void svd_compare_to_full(const MatrixType& m, unsigned int computationOptions, const SvdType& referenceSvd) { typedef typename MatrixType::RealScalar RealScalar; Index rows = m.rows(); Index cols = m.cols(); Index diagSize = (std::min)(rows, cols); RealScalar prec = test_precision<RealScalar>(); SvdType svd(m, computationOptions); VERIFY_IS_APPROX(svd.singularValues(), referenceSvd.singularValues()); if(computationOptions & (ComputeFullV|ComputeThinV)) { VERIFY( (svd.matrixV().adjoint()*svd.matrixV()).isIdentity(prec) ); VERIFY_IS_APPROX( svd.matrixV().leftCols(diagSize) * svd.singularValues().asDiagonal() * svd.matrixV().leftCols(diagSize).adjoint(), referenceSvd.matrixV().leftCols(diagSize) * referenceSvd.singularValues().asDiagonal() * referenceSvd.matrixV().leftCols(diagSize).adjoint()); } if(computationOptions & (ComputeFullU|ComputeThinU)) { VERIFY( (svd.matrixU().adjoint()*svd.matrixU()).isIdentity(prec) ); VERIFY_IS_APPROX( svd.matrixU().leftCols(diagSize) * svd.singularValues().cwiseAbs2().asDiagonal() * svd.matrixU().leftCols(diagSize).adjoint(), referenceSvd.matrixU().leftCols(diagSize) * referenceSvd.singularValues().cwiseAbs2().asDiagonal() * referenceSvd.matrixU().leftCols(diagSize).adjoint()); } // The following checks are not critical. // For instance, with Dived&Conquer SVD, if only the factor 'V' is computedt then different matrix-matrix product implementation will be used // and the resulting 'V' factor might be significantly different when the SVD decomposition is not unique, especially with single precision float. ++g_test_level; if(computationOptions & ComputeFullU) VERIFY_IS_APPROX(svd.matrixU(), referenceSvd.matrixU()); if(computationOptions & ComputeThinU) VERIFY_IS_APPROX(svd.matrixU(), referenceSvd.matrixU().leftCols(diagSize)); if(computationOptions & ComputeFullV) VERIFY_IS_APPROX(svd.matrixV().cwiseAbs(), referenceSvd.matrixV().cwiseAbs()); if(computationOptions & ComputeThinV) VERIFY_IS_APPROX(svd.matrixV(), referenceSvd.matrixV().leftCols(diagSize)); --g_test_level; } // template<typename SvdType, typename MatrixType> void svd_least_square(const MatrixType& m, unsigned int computationOptions) { typedef typename MatrixType::Scalar Scalar; typedef typename MatrixType::RealScalar RealScalar; typedef typename MatrixType::Index Index; Index rows = m.rows(); Index cols = m.cols(); enum { RowsAtCompileTime = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime }; typedef Matrix<Scalar, RowsAtCompileTime, Dynamic> RhsType; typedef Matrix<Scalar, ColsAtCompileTime, Dynamic> SolutionType; RhsType rhs = RhsType::Random(rows, internal::random<Index>(1, cols)); SvdType svd(m, computationOptions); if(internal::is_same<RealScalar,double>::value) svd.setThreshold(1e-8); else if(internal::is_same<RealScalar,float>::value) svd.setThreshold(2e-4); SolutionType x = svd.solve(rhs); RealScalar residual = (m*x-rhs).norm(); RealScalar rhs_norm = rhs.norm(); if(!test_isMuchSmallerThan(residual,rhs.norm())) { // ^^^ If the residual is very small, then we have an exact solution, so we are already good. // evaluate normal equation which works also for least-squares solutions if(internal::is_same<RealScalar,double>::value || svd.rank()==m.diagonal().size()) { using std::sqrt; // This test is not stable with single precision. // This is probably because squaring m signicantly affects the precision. if(internal::is_same<RealScalar,float>::value) ++g_test_level; VERIFY_IS_APPROX(m.adjoint()*(m*x),m.adjoint()*rhs); if(internal::is_same<RealScalar,float>::value) --g_test_level; } // Check that there is no significantly better solution in the neighborhood of x for(Index k=0;k<x.rows();++k) { using std::abs; SolutionType y(x); y.row(k) = (1.+2*NumTraits<RealScalar>::epsilon())*x.row(k); RealScalar residual_y = (m*y-rhs).norm(); VERIFY( test_isMuchSmallerThan(abs(residual_y-residual), rhs_norm) || residual < residual_y ); if(internal::is_same<RealScalar,float>::value) ++g_test_level; VERIFY( test_isApprox(residual_y,residual) || residual < residual_y ); if(internal::is_same<RealScalar,float>::value) --g_test_level; y.row(k) = (1.-2*NumTraits<RealScalar>::epsilon())*x.row(k); residual_y = (m*y-rhs).norm(); VERIFY( test_isMuchSmallerThan(abs(residual_y-residual), rhs_norm) || residual < residual_y ); if(internal::is_same<RealScalar,float>::value) ++g_test_level; VERIFY( test_isApprox(residual_y,residual) || residual < residual_y ); if(internal::is_same<RealScalar,float>::value) --g_test_level; } } } // check minimal norm solutions, the inoput matrix m is only used to recover problem size template<typename MatrixType> void svd_min_norm(const MatrixType& m, unsigned int computationOptions) { typedef typename MatrixType::Scalar Scalar; typedef typename MatrixType::Index Index; Index cols = m.cols(); enum { ColsAtCompileTime = MatrixType::ColsAtCompileTime }; typedef Matrix<Scalar, ColsAtCompileTime, Dynamic> SolutionType; // generate a full-rank m x n problem with m<n enum { RankAtCompileTime2 = ColsAtCompileTime==Dynamic ? Dynamic : (ColsAtCompileTime)/2+1, RowsAtCompileTime3 = ColsAtCompileTime==Dynamic ? Dynamic : ColsAtCompileTime+1 }; typedef Matrix<Scalar, RankAtCompileTime2, ColsAtCompileTime> MatrixType2; typedef Matrix<Scalar, RankAtCompileTime2, 1> RhsType2; typedef Matrix<Scalar, ColsAtCompileTime, RankAtCompileTime2> MatrixType2T; Index rank = RankAtCompileTime2==Dynamic ? internal::random<Index>(1,cols) : Index(RankAtCompileTime2); MatrixType2 m2(rank,cols); int guard = 0; do { m2.setRandom(); } while(SVD_FOR_MIN_NORM(MatrixType2)(m2).setThreshold(test_precision<Scalar>()).rank()!=rank && (++guard)<10); VERIFY(guard<10); RhsType2 rhs2 = RhsType2::Random(rank); // use QR to find a reference minimal norm solution HouseholderQR<MatrixType2T> qr(m2.adjoint()); Matrix<Scalar,Dynamic,1> tmp = qr.matrixQR().topLeftCorner(rank,rank).template triangularView<Upper>().adjoint().solve(rhs2); tmp.conservativeResize(cols); tmp.tail(cols-rank).setZero(); SolutionType x21 = qr.householderQ() * tmp; // now check with SVD SVD_FOR_MIN_NORM(MatrixType2) svd2(m2, computationOptions); SolutionType x22 = svd2.solve(rhs2); VERIFY_IS_APPROX(m2*x21, rhs2); VERIFY_IS_APPROX(m2*x22, rhs2); VERIFY_IS_APPROX(x21, x22); // Now check with a rank deficient matrix typedef Matrix<Scalar, RowsAtCompileTime3, ColsAtCompileTime> MatrixType3; typedef Matrix<Scalar, RowsAtCompileTime3, 1> RhsType3; Index rows3 = RowsAtCompileTime3==Dynamic ? internal::random<Index>(rank+1,2*cols) : Index(RowsAtCompileTime3); Matrix<Scalar,RowsAtCompileTime3,Dynamic> C = Matrix<Scalar,RowsAtCompileTime3,Dynamic>::Random(rows3,rank); MatrixType3 m3 = C * m2; RhsType3 rhs3 = C * rhs2; SVD_FOR_MIN_NORM(MatrixType3) svd3(m3, computationOptions); SolutionType x3 = svd3.solve(rhs3); VERIFY_IS_APPROX(m3*x3, rhs3); VERIFY_IS_APPROX(m3*x21, rhs3); VERIFY_IS_APPROX(m2*x3, rhs2); VERIFY_IS_APPROX(x21, x3); } // Check full, compare_to_full, least_square, and min_norm for all possible compute-options template<typename SvdType, typename MatrixType> void svd_test_all_computation_options(const MatrixType& m, bool full_only) { // if (QRPreconditioner == NoQRPreconditioner && m.rows() != m.cols()) // return; SvdType fullSvd(m, ComputeFullU|ComputeFullV); CALL_SUBTEST(( svd_check_full(m, fullSvd) )); CALL_SUBTEST(( svd_least_square<SvdType>(m, ComputeFullU | ComputeFullV) )); CALL_SUBTEST(( svd_min_norm(m, ComputeFullU | ComputeFullV) )); #if defined __INTEL_COMPILER // remark #111: statement is unreachable #pragma warning disable 111 #endif if(full_only) return; CALL_SUBTEST(( svd_compare_to_full(m, ComputeFullU, fullSvd) )); CALL_SUBTEST(( svd_compare_to_full(m, ComputeFullV, fullSvd) )); CALL_SUBTEST(( svd_compare_to_full(m, 0, fullSvd) )); if (MatrixType::ColsAtCompileTime == Dynamic) { // thin U/V are only available with dynamic number of columns CALL_SUBTEST(( svd_compare_to_full(m, ComputeFullU|ComputeThinV, fullSvd) )); CALL_SUBTEST(( svd_compare_to_full(m, ComputeThinV, fullSvd) )); CALL_SUBTEST(( svd_compare_to_full(m, ComputeThinU|ComputeFullV, fullSvd) )); CALL_SUBTEST(( svd_compare_to_full(m, ComputeThinU , fullSvd) )); CALL_SUBTEST(( svd_compare_to_full(m, ComputeThinU|ComputeThinV, fullSvd) )); CALL_SUBTEST(( svd_least_square<SvdType>(m, ComputeFullU | ComputeThinV) )); CALL_SUBTEST(( svd_least_square<SvdType>(m, ComputeThinU | ComputeFullV) )); CALL_SUBTEST(( svd_least_square<SvdType>(m, ComputeThinU | ComputeThinV) )); CALL_SUBTEST(( svd_min_norm(m, ComputeFullU | ComputeThinV) )); CALL_SUBTEST(( svd_min_norm(m, ComputeThinU | ComputeFullV) )); CALL_SUBTEST(( svd_min_norm(m, ComputeThinU | ComputeThinV) )); // test reconstruction typedef typename MatrixType::Index Index; Index diagSize = (std::min)(m.rows(), m.cols()); SvdType svd(m, ComputeThinU | ComputeThinV); VERIFY_IS_APPROX(m, svd.matrixU().leftCols(diagSize) * svd.singularValues().asDiagonal() * svd.matrixV().leftCols(diagSize).adjoint()); } } // work around stupid msvc error when constructing at compile time an expression that involves // a division by zero, even if the numeric type has floating point template<typename Scalar> EIGEN_DONT_INLINE Scalar zero() { return Scalar(0); } // workaround aggressive optimization in ICC template<typename T> EIGEN_DONT_INLINE T sub(T a, T b) { return a - b; } // all this function does is verify we don't iterate infinitely on nan/inf values template<typename SvdType, typename MatrixType> void svd_inf_nan() { SvdType svd; typedef typename MatrixType::Scalar Scalar; Scalar some_inf = Scalar(1) / zero<Scalar>(); VERIFY(sub(some_inf, some_inf) != sub(some_inf, some_inf)); svd.compute(MatrixType::Constant(10,10,some_inf), ComputeFullU | ComputeFullV); Scalar nan = std::numeric_limits<Scalar>::quiet_NaN(); VERIFY(nan != nan); svd.compute(MatrixType::Constant(10,10,nan), ComputeFullU | ComputeFullV); MatrixType m = MatrixType::Zero(10,10); m(internal::random<int>(0,9), internal::random<int>(0,9)) = some_inf; svd.compute(m, ComputeFullU | ComputeFullV); m = MatrixType::Zero(10,10); m(internal::random<int>(0,9), internal::random<int>(0,9)) = nan; svd.compute(m, ComputeFullU | ComputeFullV); // regression test for bug 791 m.resize(3,3); m << 0, 2*NumTraits<Scalar>::epsilon(), 0.5, 0, -0.5, 0, nan, 0, 0; svd.compute(m, ComputeFullU | ComputeFullV); m.resize(4,4); m << 1, 0, 0, 0, 0, 3, 1, 2e-308, 1, 0, 1, nan, 0, nan, nan, 0; svd.compute(m, ComputeFullU | ComputeFullV); } // Regression test for bug 286: JacobiSVD loops indefinitely with some // matrices containing denormal numbers. template<typename> void svd_underoverflow() { #if defined __INTEL_COMPILER // shut up warning #239: floating point underflow #pragma warning push #pragma warning disable 239 #endif Matrix2d M; M << -7.90884e-313, -4.94e-324, 0, 5.60844e-313; SVD_DEFAULT(Matrix2d) svd; svd.compute(M,ComputeFullU|ComputeFullV); CALL_SUBTEST( svd_check_full(M,svd) ); // Check all 2x2 matrices made with the following coefficients: VectorXd value_set(9); value_set << 0, 1, -1, 5.60844e-313, -5.60844e-313, 4.94e-324, -4.94e-324, -4.94e-223, 4.94e-223; Array4i id(0,0,0,0); int k = 0; do { M << value_set(id(0)), value_set(id(1)), value_set(id(2)), value_set(id(3)); svd.compute(M,ComputeFullU|ComputeFullV); CALL_SUBTEST( svd_check_full(M,svd) ); id(k)++; if(id(k)>=value_set.size()) { while(k<3 && id(k)>=value_set.size()) id(++k)++; id.head(k).setZero(); k=0; } } while((id<int(value_set.size())).all()); #if defined __INTEL_COMPILER #pragma warning pop #endif // Check for overflow: Matrix3d M3; M3 << 4.4331978442502944e+307, -5.8585363752028680e+307, 6.4527017443412964e+307, 3.7841695601406358e+307, 2.4331702789740617e+306, -3.5235707140272905e+307, -8.7190887618028355e+307, -7.3453213709232193e+307, -2.4367363684472105e+307; SVD_DEFAULT(Matrix3d) svd3; svd3.compute(M3,ComputeFullU|ComputeFullV); // just check we don't loop indefinitely CALL_SUBTEST( svd_check_full(M3,svd3) ); } // void jacobisvd(const MatrixType& a = MatrixType(), bool pickrandom = true) template<typename MatrixType> void svd_all_trivial_2x2( void (*cb)(const MatrixType&,bool) ) { MatrixType M; VectorXd value_set(3); value_set << 0, 1, -1; Array4i id(0,0,0,0); int k = 0; do { M << value_set(id(0)), value_set(id(1)), value_set(id(2)), value_set(id(3)); cb(M,false); id(k)++; if(id(k)>=value_set.size()) { while(k<3 && id(k)>=value_set.size()) id(++k)++; id.head(k).setZero(); k=0; } } while((id<int(value_set.size())).all()); } template<typename> void svd_preallocate() { Vector3f v(3.f, 2.f, 1.f); MatrixXf m = v.asDiagonal(); internal::set_is_malloc_allowed(false); VERIFY_RAISES_ASSERT(VectorXf tmp(10);) SVD_DEFAULT(MatrixXf) svd; internal::set_is_malloc_allowed(true); svd.compute(m); VERIFY_IS_APPROX(svd.singularValues(), v); SVD_DEFAULT(MatrixXf) svd2(3,3); internal::set_is_malloc_allowed(false); svd2.compute(m); internal::set_is_malloc_allowed(true); VERIFY_IS_APPROX(svd2.singularValues(), v); VERIFY_RAISES_ASSERT(svd2.matrixU()); VERIFY_RAISES_ASSERT(svd2.matrixV()); svd2.compute(m, ComputeFullU | ComputeFullV); VERIFY_IS_APPROX(svd2.matrixU(), Matrix3f::Identity()); VERIFY_IS_APPROX(svd2.matrixV(), Matrix3f::Identity()); internal::set_is_malloc_allowed(false); svd2.compute(m); internal::set_is_malloc_allowed(true); SVD_DEFAULT(MatrixXf) svd3(3,3,ComputeFullU|ComputeFullV); internal::set_is_malloc_allowed(false); svd2.compute(m); internal::set_is_malloc_allowed(true); VERIFY_IS_APPROX(svd2.singularValues(), v); VERIFY_IS_APPROX(svd2.matrixU(), Matrix3f::Identity()); VERIFY_IS_APPROX(svd2.matrixV(), Matrix3f::Identity()); internal::set_is_malloc_allowed(false); svd2.compute(m, ComputeFullU|ComputeFullV); internal::set_is_malloc_allowed(true); } template<typename SvdType,typename MatrixType> void svd_verify_assert(const MatrixType& m) { typedef typename MatrixType::Scalar Scalar; typedef typename MatrixType::Index Index; Index rows = m.rows(); Index cols = m.cols(); enum { RowsAtCompileTime = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime }; typedef Matrix<Scalar, RowsAtCompileTime, 1> RhsType; RhsType rhs(rows); SvdType svd; VERIFY_RAISES_ASSERT(svd.matrixU()) VERIFY_RAISES_ASSERT(svd.singularValues()) VERIFY_RAISES_ASSERT(svd.matrixV()) VERIFY_RAISES_ASSERT(svd.solve(rhs)) MatrixType a = MatrixType::Zero(rows, cols); a.setZero(); svd.compute(a, 0); VERIFY_RAISES_ASSERT(svd.matrixU()) VERIFY_RAISES_ASSERT(svd.matrixV()) svd.singularValues(); VERIFY_RAISES_ASSERT(svd.solve(rhs)) if (ColsAtCompileTime == Dynamic) { svd.compute(a, ComputeThinU); svd.matrixU(); VERIFY_RAISES_ASSERT(svd.matrixV()) VERIFY_RAISES_ASSERT(svd.solve(rhs)) svd.compute(a, ComputeThinV); svd.matrixV(); VERIFY_RAISES_ASSERT(svd.matrixU()) VERIFY_RAISES_ASSERT(svd.solve(rhs)) } else { VERIFY_RAISES_ASSERT(svd.compute(a, ComputeThinU)) VERIFY_RAISES_ASSERT(svd.compute(a, ComputeThinV)) } } #undef SVD_DEFAULT #undef SVD_FOR_MIN_NORM