test/svd_common.h

// 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

#ifndef SVD_STATIC_OPTIONS
#error a macro SVD_STATIC_OPTIONS(MatrixType, Options) must be defined prior to including svd_common.h
#endif

#include "svd_fill.h"
#include "solverbase.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)
{
  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)())
  {
    VERIFY(sigma.cwiseAbs().maxCoeff() <= (std::numeric_limits<RealScalar>::min)());
  }
  else
  {
    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 MatrixType, typename SvdType, int Options>
void svd_compare_to_full(const MatrixType& m, 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>();

  SVD_STATIC_OPTIONS(MatrixType, Options) svd(m);

  VERIFY_IS_APPROX(svd.singularValues(), referenceSvd.singularValues());

  if (Options & (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 (Options & (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 computed 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 (Options & ComputeFullU) VERIFY_IS_APPROX(svd.matrixU(), referenceSvd.matrixU());
  if (Options & ComputeThinU) VERIFY_IS_APPROX(svd.matrixU(), referenceSvd.matrixU().leftCols(diagSize));
  if (Options & ComputeFullV) VERIFY_IS_APPROX(svd.matrixV().cwiseAbs(), referenceSvd.matrixV().cwiseAbs());
  if (Options & 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) {
  typedef typename MatrixType::Scalar Scalar;
  typedef typename MatrixType::RealScalar RealScalar;
  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);

  if (internal::is_same<RealScalar, double>::value)  svd.setThreshold(RealScalar(1e-8));
  else if(internal::is_same<RealScalar,float>::value)  svd.setThreshold(RealScalar(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) = (RealScalar(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) = (RealScalar(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 input matrix m is only used to recover problem size
template <typename MatrixType, int Options>
void svd_min_norm(const MatrixType& m) {
  typedef typename MatrixType::Scalar Scalar;
  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_STATIC_OPTIONS(MatrixType2, Options) svd2(m2);
  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_STATIC_OPTIONS(MatrixType3, Options) svd3(m3);
  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);
}

template<typename MatrixType, typename SolverType>
void svd_test_solvers(const MatrixType& m, const SolverType& solver) {
    Index rows, cols, cols2;

    rows = m.rows();
    cols = m.cols();

    if(MatrixType::ColsAtCompileTime==Dynamic)
    {
      cols2 = internal::random<int>(2,EIGEN_TEST_MAX_SIZE);
    }
    else
    {
      cols2 = cols;
    }
    typedef Matrix<typename MatrixType::Scalar, MatrixType::ColsAtCompileTime, MatrixType::ColsAtCompileTime> CMatrixType;
    check_solverbase<CMatrixType, MatrixType>(m, solver, rows, cols, cols2);
}

// 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; }

// This function verifies we don't iterate infinitely on nan/inf values,
// and that info() returns InvalidInput.
template <typename MatrixType>
void svd_inf_nan() {
  SVD_STATIC_OPTIONS(MatrixType, ComputeFullU | ComputeFullV) 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));
  VERIFY(svd.info() == InvalidInput);

  Scalar nan = std::numeric_limits<Scalar>::quiet_NaN();
  VERIFY(nan != nan);
  svd.compute(MatrixType::Constant(10, 10, nan));
  VERIFY(svd.info() == InvalidInput);  

  MatrixType m = MatrixType::Zero(10,10);
  m(internal::random<int>(0,9), internal::random<int>(0,9)) = some_inf;
  svd.compute(m);
  VERIFY(svd.info() == InvalidInput);

  m = MatrixType::Zero(10,10);
  m(internal::random<int>(0,9), internal::random<int>(0,9)) = nan;
  svd.compute(m);
  VERIFY(svd.info() == InvalidInput);
  
  // 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);
  VERIFY(svd.info() == InvalidInput);
  
  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);
  VERIFY(svd.info() == InvalidInput);
}

// 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_STATIC_OPTIONS(Matrix2d, ComputeFullU | ComputeFullV) svd;
  svd.compute(M);
  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);
    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_STATIC_OPTIONS(Matrix3d, ComputeFullU | ComputeFullV) svd3;
  svd3.compute(M3);  // just check we don't loop indefinitely
  CALL_SUBTEST( svd_check_full(M3,svd3) );
}

template <typename MatrixType>
void svd_all_trivial_2x2(void (*cb)(const MatrixType&)) {
  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);

    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);
  VERIFY_RAISES_ASSERT(svd.matrixU());
  VERIFY_RAISES_ASSERT(svd.matrixV());

  SVD_STATIC_OPTIONS(MatrixXf, ComputeFullU | ComputeFullV) 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_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);
}

template <typename MatrixType, int QRPreconditioner = 0>
void svd_verify_assert_full_only(const MatrixType& m = MatrixType()) {
  enum { RowsAtCompileTime = MatrixType::RowsAtCompileTime };

  typedef Matrix<typename MatrixType::Scalar, RowsAtCompileTime, 1> RhsType;
  RhsType rhs = RhsType::Zero(m.rows());

  SVD_STATIC_OPTIONS(MatrixType, QRPreconditioner) svd0;
  VERIFY_RAISES_ASSERT((svd0.matrixU()));
  VERIFY_RAISES_ASSERT((svd0.singularValues()));
  VERIFY_RAISES_ASSERT((svd0.matrixV()));
  VERIFY_RAISES_ASSERT((svd0.solve(rhs)));
  VERIFY_RAISES_ASSERT((svd0.transpose().solve(rhs)));
  VERIFY_RAISES_ASSERT((svd0.adjoint().solve(rhs)));

  SVD_STATIC_OPTIONS(MatrixType, QRPreconditioner) svd1(m);
  VERIFY_RAISES_ASSERT((svd1.matrixU()));
  VERIFY_RAISES_ASSERT((svd1.matrixV()));
  VERIFY_RAISES_ASSERT((svd1.solve(rhs)));

  SVD_STATIC_OPTIONS(MatrixType, QRPreconditioner | ComputeFullU) svdFullU(m);
  VERIFY_RAISES_ASSERT((svdFullU.matrixV()));
  VERIFY_RAISES_ASSERT((svdFullU.solve(rhs)));
  SVD_STATIC_OPTIONS(MatrixType, QRPreconditioner | ComputeFullV) svdFullV(m);
  VERIFY_RAISES_ASSERT((svdFullV.matrixU()));
  VERIFY_RAISES_ASSERT((svdFullV.solve(rhs)));
}

template <typename MatrixType, int QRPreconditioner = 0>
void svd_verify_assert(const MatrixType& m = MatrixType()) {
  enum { RowsAtCompileTime = MatrixType::RowsAtCompileTime };

  typedef Matrix<typename MatrixType::Scalar, RowsAtCompileTime, 1> RhsType;
  RhsType rhs = RhsType::Zero(m.rows());

  SVD_STATIC_OPTIONS(MatrixType, QRPreconditioner | ComputeThinU) svdThinU(m);
  VERIFY_RAISES_ASSERT((svdThinU.matrixV()));
  VERIFY_RAISES_ASSERT((svdThinU.solve(rhs)));
  SVD_STATIC_OPTIONS(MatrixType, QRPreconditioner | ComputeThinV) svdThinV(m);
  VERIFY_RAISES_ASSERT((svdThinV.matrixU()));
  VERIFY_RAISES_ASSERT((svdThinV.solve(rhs)));

  svd_verify_assert_full_only<MatrixType, QRPreconditioner>(m);
}

template <typename MatrixType, int Options>
void svd_compute_checks(const MatrixType& m) {
  typedef SVD_STATIC_OPTIONS(MatrixType, Options) SVDType;

  enum {
    RowsAtCompileTime = MatrixType::RowsAtCompileTime,
    ColsAtCompileTime = MatrixType::ColsAtCompileTime,
    DiagAtCompileTime = internal::min_size_prefer_dynamic(RowsAtCompileTime, ColsAtCompileTime),
    MatrixURowsAtCompileTime = SVDType::MatrixUType::RowsAtCompileTime,
    MatrixUColsAtCompileTime = SVDType::MatrixUType::ColsAtCompileTime,
    MatrixVRowsAtCompileTime = SVDType::MatrixVType::RowsAtCompileTime,
    MatrixVColsAtCompileTime = SVDType::MatrixVType::ColsAtCompileTime
  };

  SVDType staticSvd(m);

  VERIFY(MatrixURowsAtCompileTime == RowsAtCompileTime);
  VERIFY(MatrixVRowsAtCompileTime == ColsAtCompileTime);
  if (Options & ComputeThinU) VERIFY(MatrixUColsAtCompileTime == DiagAtCompileTime);
  if (Options & ComputeFullU) VERIFY(MatrixUColsAtCompileTime == RowsAtCompileTime);
  if (Options & ComputeThinV) VERIFY(MatrixVColsAtCompileTime == DiagAtCompileTime);
  if (Options & ComputeFullV) VERIFY(MatrixVColsAtCompileTime == ColsAtCompileTime);

  if (Options & (ComputeThinU | ComputeFullU))
    VERIFY(staticSvd.computeU());
  else
    VERIFY(!staticSvd.computeU());
  if (Options & (ComputeThinV | ComputeFullV))
    VERIFY(staticSvd.computeV());
  else
    VERIFY(!staticSvd.computeV());

  if (staticSvd.computeU()) VERIFY(staticSvd.matrixU().isUnitary());
  if (staticSvd.computeV()) VERIFY(staticSvd.matrixV().isUnitary());

  if (staticSvd.computeU() && staticSvd.computeV()) {
    svd_test_solvers(m, staticSvd);
    svd_least_square<SVDType, MatrixType>(m);
    // svd_min_norm generates non-square matrices so it can't be used with NoQRPreconditioner
    if ((Options & internal::QRPreconditionerBits) != NoQRPreconditioner) svd_min_norm<MatrixType, Options>(m);
  }
}

// Deprecated behavior.
template <typename SvdType, typename MatrixType>
void svd_check_runtime_options(const MatrixType& m, unsigned int computationOptions) {
  const bool fixedRowAndThinU = SvdType::RowsAtCompileTime != Dynamic && (computationOptions & ComputeThinU) != 0 && m.cols() < m.rows();
  const bool fixedColAndThinV = SvdType::ColsAtCompileTime != Dynamic && (computationOptions & ComputeThinV) != 0 && m.rows() < m.cols();
  if (fixedRowAndThinU || fixedColAndThinV) {
    VERIFY_RAISES_ASSERT(SvdType svd(m, computationOptions));
    return;
  }

  Index diagSize = (std::min)(m.rows(), m.cols());

  SvdType svd(m, computationOptions);
  if (svd.computeU()) {
    VERIFY(svd.matrixU().isUnitary());
    if (computationOptions & ComputeThinU) VERIFY(svd.matrixU().cols() == diagSize);
  }

  if (svd.computeV()) {
    VERIFY(svd.matrixV().isUnitary());
    if (computationOptions & ComputeThinV) VERIFY(svd.matrixV().cols() == diagSize);
  }
  if (svd.computeU() && svd.computeV()) {
    svd_test_solvers(m, svd);
    svd.matrixU().isUnitary();
    svd.matrixV().isUnitary();
  }
}

template <typename MatrixType, int QRPreconditioner = 0>
void svd_option_checks(const MatrixType& m) {
  svd_compute_checks<MatrixType, QRPreconditioner>(m);
  svd_compute_checks<MatrixType, QRPreconditioner | ComputeThinU>(m);
  svd_compute_checks<MatrixType, QRPreconditioner | ComputeThinV>(m);
  svd_compute_checks<MatrixType, QRPreconditioner | ComputeThinU | ComputeThinV>(m);

  svd_compute_checks<MatrixType, QRPreconditioner | ComputeFullU>(m);
  svd_compute_checks<MatrixType, QRPreconditioner | ComputeFullV>(m);
  svd_compute_checks<MatrixType, QRPreconditioner | ComputeFullU | ComputeFullV>(m);

  svd_compute_checks<MatrixType, QRPreconditioner | ComputeThinU | ComputeFullV>(m);
  svd_compute_checks<MatrixType, QRPreconditioner | ComputeFullU | ComputeThinV>(m);

  typedef SVD_STATIC_OPTIONS(MatrixType, QRPreconditioner | ComputeFullU | ComputeFullV) FullSvdType;
  FullSvdType fullSvd(m);
  svd_check_full(m, fullSvd);
  svd_compare_to_full<MatrixType, FullSvdType, QRPreconditioner | ComputeFullU | ComputeFullV>(m, fullSvd);

  // Deprecated behavior.
  typedef SVD_STATIC_OPTIONS(MatrixType, QRPreconditioner) DynamicSvd;
  svd_check_runtime_options<DynamicSvd>(m, 0);
  svd_check_runtime_options<DynamicSvd>(m, ComputeThinU);
  svd_check_runtime_options<DynamicSvd>(m, ComputeThinV);
  svd_check_runtime_options<DynamicSvd>(m, ComputeThinU | ComputeThinV);

  svd_check_runtime_options<DynamicSvd>(m, ComputeFullU);
  svd_check_runtime_options<DynamicSvd>(m, ComputeFullV);
  svd_check_runtime_options<DynamicSvd>(m, ComputeFullU | ComputeFullV);

  svd_check_runtime_options<DynamicSvd>(m, ComputeThinU | ComputeFullV);
  svd_check_runtime_options<DynamicSvd>(m, ComputeFullU | ComputeThinV);
}

template <typename MatrixType, int QRPreconditioner = 0>
void svd_option_checks_full_only(const MatrixType& m) {
  svd_compute_checks<MatrixType, QRPreconditioner | ComputeFullU>(m);
  svd_compute_checks<MatrixType, QRPreconditioner | ComputeFullV>(m);
  svd_compute_checks<MatrixType, QRPreconditioner | ComputeFullU | ComputeFullV>(m);

  SVD_STATIC_OPTIONS(MatrixType, QRPreconditioner | ComputeFullU | ComputeFullV) fullSvd(m);
  svd_check_full(m, fullSvd);
}

template <typename MatrixType, int QRPreconditioner = 0>
void svd_check_max_size_matrix(int initialRows, int initialCols) {
  enum {
    MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
    MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
  };

  int rows = MaxRowsAtCompileTime == Dynamic ? initialRows : (std::min)(initialRows, (int)MaxRowsAtCompileTime);
  int cols = MaxColsAtCompileTime == Dynamic ? initialCols : (std::min)(initialCols, (int)MaxColsAtCompileTime);

  MatrixType m(rows, cols);
  SVD_STATIC_OPTIONS(MatrixType, QRPreconditioner | ComputeThinU | ComputeThinV) thinSvd(m);
  SVD_STATIC_OPTIONS(MatrixType, QRPreconditioner | ComputeThinU | ComputeFullV) mixedSvd1(m);
  SVD_STATIC_OPTIONS(MatrixType, QRPreconditioner | ComputeFullU | ComputeThinV) mixedSvd2(m);
  SVD_STATIC_OPTIONS(MatrixType, QRPreconditioner | ComputeFullU | ComputeFullV) fullSvd(m);

  MatrixType n(MaxRowsAtCompileTime, MaxColsAtCompileTime);
  thinSvd.compute(n);
  mixedSvd1.compute(n);
  mixedSvd2.compute(n);
  fullSvd.compute(n);

  MatrixX<typename MatrixType::Scalar> dynamicMatrix(MaxRowsAtCompileTime + 1, MaxColsAtCompileTime + 1);

  VERIFY_RAISES_ASSERT(thinSvd.compute(dynamicMatrix));
  VERIFY_RAISES_ASSERT(mixedSvd1.compute(dynamicMatrix));
  VERIFY_RAISES_ASSERT(mixedSvd2.compute(dynamicMatrix));
  VERIFY_RAISES_ASSERT(fullSvd.compute(dynamicMatrix));
}

template <typename SvdType, typename MatrixType>
void svd_verify_constructor_options_assert(const MatrixType& m, bool fullOnly = false) {
  typedef typename MatrixType::Scalar Scalar;
  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))
  VERIFY_RAISES_ASSERT(svd.transpose().solve(rhs))
  VERIFY_RAISES_ASSERT(svd.adjoint().solve(rhs))

  MatrixType a = MatrixType::Zero(rows, cols);
  SvdType svd2(a, 0);
  VERIFY_RAISES_ASSERT(svd2.matrixU())
  VERIFY_RAISES_ASSERT(svd2.matrixV())
  svd2.singularValues();
  VERIFY_RAISES_ASSERT(svd2.solve(rhs))

  // Deprecated behavior.
  SvdType svd3(a, ComputeFullU);
  svd3.matrixU();
  VERIFY_RAISES_ASSERT(svd3.matrixV())
  VERIFY_RAISES_ASSERT(svd3.solve(rhs))

  SvdType svd4(a, ComputeFullV);
  svd4.matrixV();
  VERIFY_RAISES_ASSERT(svd4.matrixU())
  VERIFY_RAISES_ASSERT(svd4.solve(rhs))

  if (!fullOnly && ColsAtCompileTime == Dynamic)
  {
    SvdType svd5(a, ComputeThinU);
    svd5.matrixU();
    VERIFY_RAISES_ASSERT(svd5.matrixV())
    VERIFY_RAISES_ASSERT(svd5.solve(rhs))

    SvdType svd6(a, ComputeThinV);
    svd6.matrixV();
    VERIFY_RAISES_ASSERT(svd6.matrixU())
    VERIFY_RAISES_ASSERT(svd6.solve(rhs))
  }
}

#undef SVD_DEFAULT
#undef SVD_FOR_MIN_NORM
#undef SVD_STATIC_OPTIONS