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eigen/test/svd_common.h

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// 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
// 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 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();
VERIFY_IS_APPROX(m, u * sigma * 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::Index Index;
Index rows = m.rows();
Index cols = m.cols();
Index diagSize = (std::min)(rows, cols);
SvdType svd(m, computationOptions);
VERIFY_IS_APPROX(svd.singularValues(), referenceSvd.singularValues());
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(), referenceSvd.matrixV());
if(computationOptions & ComputeThinV) VERIFY_IS_APPROX(svd.matrixV(), referenceSvd.matrixV().leftCols(diagSize));
}
//
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(1e-4);
SolutionType x = svd.solve(rhs);
RealScalar residual = (m*x-rhs).norm();
// Check that there is no significantly better solution in the neighborhood of x
if(!test_isMuchSmallerThan(residual,rhs.norm()))
{
// If the residual is very small, then we have an exact solution, so we are already good.
for(int k=0;k<x.rows();++k)
{
SolutionType y(x);
y.row(k).array() += 2*NumTraits<RealScalar>::epsilon();
RealScalar residual_y = (m*y-rhs).norm();
VERIFY( test_isApprox(residual_y,residual) || residual < residual_y );
y.row(k) = x.row(k).array() - 2*NumTraits<RealScalar>::epsilon();
residual_y = (m*y-rhs).norm();
VERIFY( test_isApprox(residual_y,residual) || residual < residual_y );
}
}
// evaluate normal equation which works also for least-squares solutions
if(internal::is_same<RealScalar,double>::value)
{
// This test is not stable with single precision.
// This is probably because squaring m signicantly affects the precision.
VERIFY_IS_APPROX(m.adjoint()*m*x,m.adjoint()*rhs);
}
}
// 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());
}
}
template<typename MatrixType>
void svd_fill_random(MatrixType &m)
{
typedef typename MatrixType::Scalar Scalar;
typedef typename MatrixType::RealScalar RealScalar;
typedef typename MatrixType::Index Index;
Index diagSize = (std::min)(m.rows(), m.cols());
RealScalar s = std::numeric_limits<RealScalar>::max_exponent10/4;
s = internal::random<RealScalar>(1,s);
Matrix<RealScalar,Dynamic,1> d = Matrix<RealScalar,Dynamic,1>::Random(diagSize);
for(Index k=0; k<diagSize; ++k)
d(k) = d(k)*std::pow(RealScalar(10),internal::random<RealScalar>(-s,s));
m = Matrix<Scalar,Dynamic,Dynamic>::Random(m.rows(),diagSize) * d.asDiagonal() * Matrix<Scalar,Dynamic,Dynamic>::Random(diagSize,m.cols());
// cancel some coeffs
Index n = internal::random<Index>(0,m.size()-1);
for(Index i=0; i<n; ++i)
m(internal::random<Index>(0,m.rows()-1), internal::random<Index>(0,m.cols()-1)) = Scalar(0);
}
// 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.
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());
}
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
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