test/bdcsvd.cpp

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2013 Gauthier Brun <brun.gauthier@gmail.com>
// Copyright (C) 2013 Nicolas Carre <nicolas.carre@ensimag.fr>
// Copyright (C) 2013 Jean Ceccato <jean.ceccato@ensimag.fr>
// Copyright (C) 2013 Pierre Zoppitelli <pierre.zoppitelli@ensimag.fr>
//
// 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/
// SPDX-License-Identifier: MPL-2.0

// discard stack allocation as that too bypasses malloc
#define EIGEN_STACK_ALLOCATION_LIMIT 0
#define EIGEN_RUNTIME_NO_MALLOC

#include "main.h"
#include "fp_control.h"
#include "tridiag_test_matrices.h"
#include <Eigen/SVD>
#include <cstdlib>

#define SVD_DEFAULT(M) BDCSVD<M>
#define SVD_FOR_MIN_NORM(M) BDCSVD<M>
#define SVD_STATIC_OPTIONS(M, O) BDCSVD<M, O>
#include "svd_common.h"

template <typename MatrixType>
void bdcsvd_method() {
  enum { Size = MatrixType::RowsAtCompileTime };
  typedef typename MatrixType::RealScalar RealScalar;
  typedef Matrix<RealScalar, Size, 1> RealVecType;
  MatrixType m = MatrixType::Identity();
  VERIFY_IS_APPROX(m.bdcSvd().singularValues(), RealVecType::Ones());
  VERIFY_RAISES_ASSERT(m.bdcSvd().matrixU());
  VERIFY_RAISES_ASSERT(m.bdcSvd().matrixV());
}

// compare the Singular values returned with Jacobi and Bdc
template <typename MatrixType>
void compare_bdc_jacobi(const MatrixType& a = MatrixType(), int algoswap = 16, bool random = true) {
  MatrixType m = random ? MatrixType::Random(a.rows(), a.cols()) : a;

  BDCSVD<MatrixType> bdc_svd(m.rows(), m.cols());
  bdc_svd.setSwitchSize(algoswap);
  bdc_svd.compute(m);

  JacobiSVD<MatrixType> jacobi_svd(m);
  VERIFY_IS_APPROX(bdc_svd.singularValues(), jacobi_svd.singularValues());
}

#if defined(EIGEN_TEST_PART_46) || defined(EIGEN_TEST_PART_47) || defined(EIGEN_TEST_PART_48) || \
    defined(EIGEN_TEST_PART_49) || defined(EIGEN_TEST_PART_ALL)
// Verifies total deflation is **not** triggered.
template <typename = void>
void compare_bdc_jacobi_instance(bool structure_as_m, int algoswap = 16) {
  MatrixXd m(4, 3);
  if (structure_as_m) {
    // The first 3 rows are the reduced form of Matrix 1 as shown below, and it
    // has nonzero elements in the first column and diagonals only.
    m << 1.056293, 0, 0, -0.336468, 0.907359, 0, -1.566245, 0, 0.149150, -0.1, 0, 0;
  } else {
    // Matrix 1.
    m << 0.882336, 18.3914, -26.7921, -5.58135, 17.1931, -24.0892, -20.794, 8.68496, -4.83103, -8.4981, -10.5451,
        23.9072;
  }
  compare_bdc_jacobi(m, algoswap, false);
}
#endif

template <typename MatrixType>
void bdcsvd_thin_full_options(const MatrixType& input = MatrixType()) {
  svd_thin_full_option_checks<MatrixType, 0>(input);
  svd_thin_full_option_checks<MatrixType, PreconditionSquareMatrix>(input);
}

template <typename MatrixType>
void bdcsvd_asserts(const MatrixType& input = MatrixType()) {
  MatrixType m(input.rows(), input.cols());
  svd_fill_random(m);

  svd_verify_assert<MatrixType>(m);
}

template <typename MatrixType>
void bdcsvd_check_convergence(const MatrixType& input) {
  BDCSVD<MatrixType, Eigen::ComputeThinU | Eigen::ComputeThinV> svd(input);
  VERIFY(svd.info() == Eigen::Success);
  MatrixType D = svd.matrixU() * svd.singularValues().asDiagonal() * svd.matrixV().transpose();
  VERIFY_IS_APPROX(input, D);
}

// Verify SVD of bidiagonal matrix given as diagonal + superdiagonal vectors.
template <typename RealScalar, int Options = ComputeFullU | ComputeFullV>
void verify_bidiagonal_svd(const Matrix<RealScalar, Dynamic, 1>& diag,
                           const Matrix<RealScalar, Dynamic, 1>& superdiag) {
  typedef Matrix<RealScalar, Dynamic, Dynamic> MatrixXr;
  typedef Matrix<RealScalar, Dynamic, 1> VectorXr;
  const Index n = diag.size();

  BDCSVD<MatrixXr, Options> bdcsvd(diag, superdiag);
  VERIFY(bdcsvd.info() == Success);

  const VectorXr& sv = bdcsvd.singularValues();

  // Singular values must be non-negative.
  for (Index i = 0; i < sv.size(); ++i) {
    VERIFY(sv(i) >= RealScalar(0));
  }

  // Singular values must be sorted descending.
  for (Index i = 1; i < sv.size(); ++i) {
    VERIFY(sv(i - 1) >= sv(i));
  }

  // Orthogonality of U and V.
  VERIFY_IS_APPROX(bdcsvd.matrixU().transpose() * bdcsvd.matrixU(), MatrixXr::Identity(n, n));
  VERIFY_IS_APPROX(bdcsvd.matrixV().transpose() * bdcsvd.matrixV(), MatrixXr::Identity(n, n));

  // Reconstruction: U * S * V^T should equal the original bidiagonal.
  MatrixXr B = MatrixXr::Zero(n, n);
  B.diagonal() = diag;
  if (n > 1) B.diagonal(1) = superdiag;
  MatrixXr recon = bdcsvd.matrixU() * sv.asDiagonal() * bdcsvd.matrixV().transpose();
  VERIFY_IS_APPROX(recon, B);

  // Cross-validate singular values against JacobiSVD.
  JacobiSVD<MatrixXr> jacobi(B);
  VERIFY_IS_APPROX(sv, jacobi.singularValues());
}

// Verify that bidiagonal API and matrix API produce matching singular values.
template <typename RealScalar, int Options = 0>
void verify_bidiagonal_vs_matrix_svd(const Matrix<RealScalar, Dynamic, 1>& diag,
                                     const Matrix<RealScalar, Dynamic, 1>& superdiag) {
  typedef Matrix<RealScalar, Dynamic, Dynamic> MatrixXr;
  const Index n = diag.size();

  // Build dense bidiagonal matrix.
  MatrixXr B = MatrixXr::Zero(n, n);
  B.diagonal() = diag;
  if (n > 1) B.diagonal(1) = superdiag;

  BDCSVD<MatrixXr, Options> bidiag_svd(diag, superdiag);
  BDCSVD<MatrixXr, Options> matrix_svd(B);

  VERIFY(bidiag_svd.info() == Success);
  VERIFY(matrix_svd.info() == Success);
  VERIFY_IS_APPROX(bidiag_svd.singularValues(), matrix_svd.singularValues());
}

template <typename RealScalar>
void bdcsvd_bidiagonal_hard_cases() {
  Eigen::internal::set_is_malloc_allowed(true);

  // Use the shared tridiagonal test matrix generators.
  // Each generator fills (diag, offdiag) which we treat as (diagonal, superdiagonal)
  // of a bidiagonal matrix.
  test::for_all_tridiag_test_matrices<RealScalar>([](const auto& diag, const auto& offdiag) {
    verify_bidiagonal_svd<RealScalar>(diag, offdiag);
    verify_bidiagonal_svd<RealScalar, PreconditionSquareMatrix | ComputeFullU | ComputeFullV>(diag, offdiag);
  });

  // Additional SVD-specific test: identity with cross-validation against full matrix SVD.
  test::for_tridiag_sizes<RealScalar>([](auto& diag, auto& offdiag) {
    test::tridiag_identity(diag, offdiag);
    verify_bidiagonal_vs_matrix_svd<RealScalar>(diag, offdiag);
  });

  // Additional SVD-specific test: scalar for n=1.
  {
    typedef Matrix<RealScalar, Dynamic, 1> VectorXr;
    VectorXr diag(1), offdiag(0);
    diag(0) = RealScalar(3.14);
    verify_bidiagonal_svd<RealScalar>(diag, offdiag);
  }
}

#if defined(EIGEN_TEST_PART_6) || defined(EIGEN_TEST_PART_ALL)
template <typename = void>
void bdcsvd_mixed_option_enum_regression() {
  using NoQrFullSVD = BDCSVD<MatrixXd, NoQRPreconditioner | ComputeFullU | ComputeFullV>;
  using ReversedMixedSVD = BDCSVD<MatrixXd, ComputeThinU | DisableQRDecomposition | ComputeFullV>;

  STATIC_CHECK((int(NoQrFullSVD::QRDecomposition) == int(NoQRPreconditioner)));
  STATIC_CHECK((NoQrFullSVD::ComputationOptions == (ComputeFullU | ComputeFullV)));

  STATIC_CHECK((int(ReversedMixedSVD::QRDecomposition) == int(DisableQRDecomposition)));
  STATIC_CHECK((ReversedMixedSVD::ComputationOptions == (ComputeThinU | ComputeFullV)));

  // Runtime unitary options must reach the JacobiSVD used below the switch size when Options holds only QR bits.
  const MatrixXd m = MatrixXd::Random(5, 5);
  EIGEN_DIAGNOSTICS(push)
  EIGEN_DISABLE_DEPRECATED_WARNING
  const BDCSVD<MatrixXd, DisableQRDecomposition> svd(m, ComputeFullU | ComputeFullV);
  EIGEN_DIAGNOSTICS(pop)
  svd_check_full(m, svd);
}
#endif

#if defined(EIGEN_TEST_PART_65) || defined(EIGEN_TEST_PART_ALL)
template <typename = void>
void bdcsvd_switch_size() {
  const MatrixXd m = MatrixXd::Random(20, 20);
  BDCSVD<MatrixXd, ComputeFullU | ComputeFullV> svd(m.rows(), m.cols());
  svd.setSwitchSize(32);
  svd.compute(m);
  svd_check_full(m, svd);
  svd.setSwitchSize(4);
  svd.compute(m);
  svd_check_full(m, svd);
  svd.setSwitchSize(32);
  svd.compute(m);
  svd_check_full(m, svd);

  // The bidiagonal entry point allocates through the same path.
  const VectorXd diag = VectorXd::Random(20);
  const VectorXd superdiag = VectorXd::Random(19);
  MatrixXd bidiag = MatrixXd::Zero(20, 20);
  bidiag.diagonal() = diag;
  bidiag.diagonal(1) = superdiag;
  BDCSVD<MatrixXd, ComputeFullU | ComputeFullV> bidiag_svd(bidiag.rows(), bidiag.cols());
  bidiag_svd.setSwitchSize(32);
  bidiag_svd.compute(diag, superdiag);
  svd_check_full(bidiag, bidiag_svd);
}
#endif

#if defined(EIGEN_TEST_PART_62) || defined(EIGEN_TEST_PART_ALL)
// Below the switch size, the bidiagonal API forwards PreconditionSquareMatrix to BDCSVD's JacobiSVD.
template <typename RealScalar>
void bdcsvd_precondition_square_matrix_bidiagonal(Index n) {
  using VectorXr = Matrix<RealScalar, Dynamic, 1>;
  const VectorXr diagonal = VectorXr::Random(n);
  const VectorXr superdiagonal = VectorXr::Random(n - 1);
  verify_bidiagonal_vs_matrix_svd<RealScalar, PreconditionSquareMatrix>(diagonal, superdiagonal);
}
#endif

#if defined(EIGEN_TEST_PART_53) || defined(EIGEN_TEST_PART_ALL)
template <typename = void>
void bdcsvd_extreme_scale_regressions() {
  typedef Matrix<double, 6, 6> Matrix6d;
  const double kTolerance = 16 * Matrix6d::RowsAtCompileTime * NumTraits<double>::epsilon();

  const auto verify_decomposition = [kTolerance](const Matrix6d& matrix) {
    BDCSVD<Matrix6d, ComputeFullU | ComputeFullV> svd;
    svd.setSwitchSize(3);
    svd.compute(matrix);

    VERIFY(svd.info() == Success);

    const Matrix6d reconstruction = svd.matrixU() * svd.singularValues().asDiagonal() * svd.matrixV().transpose();
    VERIFY((reconstruction - matrix).stableNorm() <= kTolerance * matrix.stableNorm());

    const Matrix6d identity = Matrix6d::Identity();
    VERIFY((svd.matrixU().transpose() * svd.matrixU() - identity).stableNorm() <= kTolerance);
    VERIFY((svd.matrixV().transpose() * svd.matrixV() - identity).stableNorm() <= kTolerance);

    // Also exercise the values-only path, which uses a compact m_naiveU and
    // linear workspace during divide-and-conquer merges.
    BDCSVD<Matrix6d> valuesOnlySvd;
    valuesOnlySvd.setSwitchSize(3);
    valuesOnlySvd.compute(matrix);
    VERIFY(valuesOnlySvd.info() == Success);
    VERIFY_IS_APPROX(valuesOnlySvd.singularValues(), svd.singularValues());
  };

  Matrix6d matrix = Matrix6d::Zero();
  const double kSubnormal1040 = std::numeric_limits<double>::denorm_min() * 17179869184.0;  // 2^-1040
  const double kSubnormal1060 = std::numeric_limits<double>::denorm_min() * 16384.0;        // 2^-1060
  const double kSmallestNormal = (std::numeric_limits<double>::min)();                      // 2^-1022
  const double kNormal1000 = kSmallestNormal * 4194304.0;                                   // 2^-1000

  // The merge combines normal and subnormal couplings. Squaring the two
  // coupling terms directly used to underflow, which later left perturbCol0
  // without a predecessor and made the decomposition report NumericalIssue.
  matrix.diagonal() << kSubnormal1040, -kSubnormal1060, kSmallestNormal, 0.5, 1.0, kSmallestNormal;
  matrix.diagonal(1) << -kNormal1000, kNormal1000, kSubnormal1040, kSubnormal1060, -8.0;
  verify_decomposition(matrix);

  // A singular-vector coefficient grows to about 2^570 here. Its squared
  // norm overflows even though the vector has a finite, well-scaled
  // normalization.
  using std::ldexp;
  matrix.setZero();
  matrix.diagonal() << 0.0, 0.0, ldexp(1.0, -487), -1.0, 0.0, 0.0;
  matrix.diagonal(1) << 0.0, ldexp(1.0, -453), -ldexp(1.0, -627), 0.0, 0.0;
  verify_decomposition(matrix);
}
#endif

#if defined(EIGEN_TEST_PART_54) || defined(EIGEN_TEST_PART_ALL)
template <typename = void>
void bdcsvd_fast_math_regression_1588() {
  const Index n = 500;
  MatrixXd matrix = MatrixXd::Zero(n, n);

  std::srand(1);
  for (Index k = 0; k < 5000; ++k) {
    const Index row = std::rand() % n;
    const Index col = std::rand() % n;
    matrix(row, col) = static_cast<double>(std::rand()) / static_cast<double>(RAND_MAX);
  }
  matrix = matrix * matrix;

  BDCSVD<MatrixXd, ComputeThinU | ComputeThinV> svd(matrix);
  VERIFY(svd.info() == Success);

  MatrixXd reconstruction = svd.matrixU() * svd.singularValues().asDiagonal() * svd.matrixV().transpose();
  const double relative_error = (reconstruction - matrix).norm() / matrix.norm();
  // Deterministic input (fixed seed); the reconstruction is backward stable, so the relative error stays near eps.
  VERIFY(relative_error < 64 * NumTraits<double>::epsilon());
}
#endif

// An all-subnormal matrix, which a SIMD unit that flushes subnormal inputs (ARMv7 NEON, Arm FZ, DAZ) reads as zero in
// the maxCoeff() selecting the scale: recovering the maximum from the representation and scaling through integer
// significands keeps the decomposition, and the singular values come back through the subnormal range exactly. The
// switch size below the dimensions keeps the divide-and-conquer path rather than the JacobiSVD fallback.
template <typename MatrixType>
void bdcsvd_flushed_subnormal_matrix(Index rows, Index cols) {
  using RealScalar = typename MatrixType::RealScalar;
  const MatrixType m = svd_subnormal_fixture<MatrixType>(rows, cols);
  const int k = 60;
  const MatrixType ms = m.unaryExpr(internal::scale_by_exponent_op<RealScalar>(k));
  BDCSVD<MatrixType, ComputeFullU | ComputeFullV> scaled;
  scaled.setSwitchSize(8);
  scaled.compute(ms);
  forEachFlushToZeroMode([&](FlushToZeroMode) {
    BDCSVD<MatrixType, ComputeFullU | ComputeFullV> svd;
    svd.setSwitchSize(8);
    svd.compute(m);
    svd_check_flushed_subnormal(svd, scaled, m, k);
  });
}

// The bidiagonal entry point, on both sides of the switch size: divide and conquer above it, JacobiSVD below.
template <typename RealScalar>
void bdcsvd_flushed_subnormal_bidiagonal(Index n, int switchSize) {
  using Vec = Matrix<RealScalar, Dynamic, 1>;
  using Mat = Matrix<RealScalar, Dynamic, Dynamic>;
  const Vec diagonal = svd_subnormal_fixture<Vec>(n, 1), superdiagonal = svd_subnormal_fixture<Vec>(n - 1, 1);
  const int k = 60;
  const Vec ds = diagonal.unaryExpr(internal::scale_by_exponent_op<RealScalar>(k));
  const Vec ss = superdiagonal.unaryExpr(internal::scale_by_exponent_op<RealScalar>(k));
  Mat B = Mat::Zero(n, n);
  B.diagonal() = diagonal;
  B.diagonal(1) = superdiagonal;
  BDCSVD<Mat, ComputeFullU | ComputeFullV> scaled;
  scaled.setSwitchSize(switchSize);
  scaled.compute(ds, ss);
  forEachFlushToZeroMode([&](FlushToZeroMode) {
    BDCSVD<Mat, ComputeFullU | ComputeFullV> svd;
    svd.setSwitchSize(switchSize);
    svd.compute(diagonal, superdiagonal);
    svd_check_flushed_subnormal(svd, scaled, B, k);
  });
}

template <typename = void>
void bdcsvd_power_of_two_scaling() {
  // Reciprocal scaling rounds the smaller singular value down by one ULP in both entry paths.
  const Index size = 20;
  VectorXf diagonal = VectorXf::Zero(size);
  diagonal(0) = numext::bit_cast<float>(numext::uint32_t(0x13afd4a1));
  diagonal(1) = numext::bit_cast<float>(numext::uint32_t(0x11525720));

  MatrixXf matrix = MatrixXf::Zero(size, size + 1);
  matrix.leftCols(size).diagonal() = diagonal;
  BDCSVD<MatrixXf> denseSvd;
  denseSvd.setSwitchSize(8);
  denseSvd.compute(matrix);
  VERIFY_IS_EQUAL(denseSvd.singularValues()(0), diagonal(0));
  VERIFY_IS_EQUAL(denseSvd.singularValues()(1), diagonal(1));

  BDCSVD<MatrixXf> bidiagonalSvd;
  bidiagonalSvd.setSwitchSize(8);
  bidiagonalSvd.compute(diagonal, VectorXf::Zero(size - 1));
  VERIFY_IS_EQUAL(bidiagonalSvd.singularValues()(0), diagonal(0));
  VERIFY_IS_EQUAL(bidiagonalSvd.singularValues()(1), diagonal(1));

  // A subnormal coupling remains significant relative to a uniformly tiny bidiagonal matrix. Scale it without first
  // feeding the subnormal operand to packet arithmetic, which flushes it on ARMv7 NEON.
  const auto checkSubnormalCoupling = [&]() {
    volatile float normalMin = (std::numeric_limits<float>::min)();
    const float largest = 256.0f * normalMin;
    // Construct min_normal / 2 without arithmetic that FTZ would flush before it reaches the solver.
    const float coupling = numext::bit_cast<float>(numext::uint32_t(0x00400000));
    diagonal.setZero();
    diagonal(0) = largest;
    VectorXf superdiagonal = VectorXf::Zero(size - 1);
    superdiagonal(0) = coupling;
    VERIFY_IS_EQUAL(numext::bit_cast<numext::uint32_t>(superdiagonal(0)), numext::uint32_t(0x00400000));
    const float expectedRatio = numext::sqrt(1.0f + 1.0f / (512.0f * 512.0f));
    for (int switchSize : {8, 32}) {
      bidiagonalSvd.setSwitchSize(switchSize);
      bidiagonalSvd.compute(diagonal, superdiagonal);
      VERIFY(numext::abs(bidiagonalSvd.singularValues()(0) / largest - expectedRatio) <=
             2.0f * NumTraits<float>::epsilon());
    }
  };
  checkSubnormalCoupling();
  ScopedFlushToZero ftz;
  checkSubnormalCoupling();
}

template <typename RealScalar>
void bdcsvd_secular_extrapolation() {
  using Mat = Matrix<RealScalar, Dynamic, Dynamic>;
  using Vec = Matrix<RealScalar, Dynamic, 1>;
  for (Index n : {31, 32, 33, 63, 64, 65, 127, 128, 129}) {
    Mat bidiagonal = Mat::Zero(n, n);
    Vec diagonal = Vec::Random(n), superdiagonal = Vec::Random(n - 1);
    for (int kind = 0; kind < 3; ++kind) {
      if (kind == 1) {
        diagonal.setOnes();
        superdiagonal *= numext::sqrt(NumTraits<RealScalar>::epsilon());
      } else if (kind == 2) {
        diagonal = Vec::LinSpaced(n, RealScalar(0), RealScalar(1));
        superdiagonal.setRandom();
        for (Index i = 0; i + 1 < n; i += 7) superdiagonal(i) = RealScalar(0);
      }
      bidiagonal.diagonal() = diagonal;
      bidiagonal.template diagonal<1>() = superdiagonal;
      BDCSVD<Mat, ComputeFullU | ComputeFullV> svd(diagonal, superdiagonal);
      VERIFY_IS_EQUAL(svd.info(), Success);
      // Reconstruction and orthogonality each accumulate O(n * epsilon) rounding.
      const RealScalar bound = RealScalar(64 * n) * NumTraits<RealScalar>::epsilon();
      VERIFY((bidiagonal - svd.matrixU() * svd.singularValues().asDiagonal() * svd.matrixV().transpose()).norm() <=
             bound * bidiagonal.norm());
      const Mat identity = Mat::Identity(n, n);
      VERIFY((svd.matrixU().transpose() * svd.matrixU() - identity).norm() <= bound);
      VERIFY((svd.matrixV().transpose() * svd.matrixV() - identity).norm() <= bound);
      JacobiSVD<Mat> reference(bidiagonal);
      VERIFY((svd.singularValues() - reference.singularValues()).norm() <= bound * bidiagonal.norm());
      BDCSVD<Mat> valuesOnly(diagonal, superdiagonal);
      VERIFY_IS_EQUAL(valuesOnly.info(), Success);
      VERIFY((valuesOnly.singularValues() - reference.singularValues()).norm() <= bound * bidiagonal.norm());
    }
  }
}

template <typename Scalar, int StorageOrder>
void bdcsvd_qr_crossover() {
  using RealScalar = typename NumTraits<Scalar>::Real;
  using Mat = Matrix<Scalar, Dynamic, Dynamic, StorageOrder>;
  constexpr int options = ComputeThinU | ComputeThinV;
  for (Index n : {16, 31}) {
    for (Index rows : {4 * n - 1, 4 * n, 4 * n + 3, 4 * n + 4}) {
      Mat tall = Mat::Random(rows, n);
      tall.col(1) = tall.col(0);
      for (bool transpose : {false, true}) {
        Mat a;
        if (transpose)
          a = tall.adjoint();
        else
          a = tall;
        BDCSVD<Mat, options> svd(a);
        BDCSVD<Mat, options | int(DisableQRDecomposition)> direct(a);
        VERIFY_IS_EQUAL(svd.info(), Success);
        VERIFY_IS_EQUAL(direct.info(), Success);
        const RealScalar bound = RealScalar(64 * rows) * NumTraits<RealScalar>::epsilon();
        VERIFY((a - svd.matrixU() * svd.singularValues().asDiagonal() * svd.matrixV().adjoint()).norm() <=
               bound * a.norm());
        const Mat identity = Mat::Identity(n, n);
        VERIFY((svd.matrixU().adjoint() * svd.matrixU() - identity).norm() <= bound);
        VERIFY((svd.matrixV().adjoint() * svd.matrixV() - identity).norm() <= bound);
        VERIFY((svd.singularValues() - direct.singularValues()).norm() <= bound * a.norm());
      }
    }
  }
}

EIGEN_DECLARE_TEST(bdcsvd) {
  CALL_SUBTEST_60((svd_normal_equation_roundoff<float, ColMajor>()));
  CALL_SUBTEST_60((svd_normal_equation_roundoff<double, RowMajor>()));
  CALL_SUBTEST_61((svd_normal_equation_roundoff<std::complex<float>, RowMajor>()));
  CALL_SUBTEST_61((svd_normal_equation_roundoff<std::complex<double>, ColMajor>()));

  CALL_SUBTEST_2((bdcsvd_asserts<Matrix4d>()));
  CALL_SUBTEST_3((bdcsvd_asserts<Matrix<float, 10, 7>>()));
  CALL_SUBTEST_4((bdcsvd_asserts<Matrix<float, 7, 10>>()));
  CALL_SUBTEST_5((bdcsvd_asserts<Matrix<std::complex<double>, 6, 9>>()));
  CALL_SUBTEST_6((bdcsvd_mixed_option_enum_regression<>()));

  CALL_SUBTEST_7((bdcsvd_thin_full_options<Matrix2cd>()));
  CALL_SUBTEST_9((bdcsvd_thin_full_options<Matrix2d>()));

  for (int i = 0; i < g_repeat; i++) {
    int r = internal::random<int>(1, EIGEN_TEST_MAX_SIZE / 2), c = internal::random<int>(1, EIGEN_TEST_MAX_SIZE / 2);

    TEST_SET_BUT_UNUSED_VARIABLE(r);
    TEST_SET_BUT_UNUSED_VARIABLE(c);

    CALL_SUBTEST_11((compare_bdc_jacobi<MatrixXf>(MatrixXf(r, c))));
    CALL_SUBTEST_12((compare_bdc_jacobi<MatrixXd>(MatrixXd(r, c))));
    CALL_SUBTEST_13((compare_bdc_jacobi<MatrixXcd>(MatrixXcd(r, c))));
    // Test on inf/nan matrix
    CALL_SUBTEST_14((svd_inf_nan<MatrixXf>()));
    CALL_SUBTEST_15((svd_inf_nan<MatrixXd>()));

    // Verify some computations using all combinations of the Options template parameter.
    CALL_SUBTEST_16((bdcsvd_thin_full_options<Matrix3f>()));
    CALL_SUBTEST_18((bdcsvd_thin_full_options<Matrix<float, 2, 3>>()));
    CALL_SUBTEST_20((bdcsvd_thin_full_options<MatrixXd>(MatrixXd(20, 17))));
    CALL_SUBTEST_22((bdcsvd_thin_full_options<MatrixXd>(MatrixXd(17, 20))));
    CALL_SUBTEST_24((bdcsvd_thin_full_options<Matrix<double, Dynamic, 15>>(Matrix<double, Dynamic, 15>(r, 15))));
    CALL_SUBTEST_26((bdcsvd_thin_full_options<Matrix<double, 13, Dynamic>>(Matrix<double, 13, Dynamic>(13, c))));
    CALL_SUBTEST_28((bdcsvd_thin_full_options<MatrixXf>(MatrixXf(r, c))));
    CALL_SUBTEST_30((bdcsvd_thin_full_options<MatrixXcd>(MatrixXcd(r, c))));
    CALL_SUBTEST_32((bdcsvd_thin_full_options<MatrixXd>(MatrixXd(r, c))));
    CALL_SUBTEST_34((bdcsvd_thin_full_options<Matrix<double, Dynamic, Dynamic, RowMajor>>(
        Matrix<double, Dynamic, Dynamic, RowMajor>(20, 27))));
    CALL_SUBTEST_36((bdcsvd_thin_full_options<Matrix<double, Dynamic, Dynamic, RowMajor>>(
        Matrix<double, Dynamic, Dynamic, RowMajor>(27, 20))));
    CALL_SUBTEST_38((
        svd_check_max_size_matrix<Matrix<float, Dynamic, Dynamic, ColMajor, 20, 35>, ColPivHouseholderQRPreconditioner>(
            r, c)));
    CALL_SUBTEST_39(
        (svd_check_max_size_matrix<Matrix<float, Dynamic, Dynamic, ColMajor, 35, 20>, HouseholderQRPreconditioner>(r,
                                                                                                                   c)));
    CALL_SUBTEST_40((
        svd_check_max_size_matrix<Matrix<float, Dynamic, Dynamic, RowMajor, 20, 35>, ColPivHouseholderQRPreconditioner>(
            r, c)));
    CALL_SUBTEST_41(
        (svd_check_max_size_matrix<Matrix<float, Dynamic, Dynamic, RowMajor, 35, 20>, HouseholderQRPreconditioner>(r,
                                                                                                                   c)));

    const Index smallSize = internal::random<Index>(2, 15);
    TEST_SET_BUT_UNUSED_VARIABLE(smallSize);
    CALL_SUBTEST_62((bdcsvd_precondition_square_matrix_bidiagonal<double>(smallSize)));
    CALL_SUBTEST_62((bdcsvd_precondition_square_matrix_bidiagonal<float>(smallSize)));
    CALL_SUBTEST_62((svd_precondition_square_matrix_accuracy<Matrix2f>(2)));
    CALL_SUBTEST_62((svd_precondition_square_matrix_accuracy<Matrix4d>(4)));
    CALL_SUBTEST_62((svd_precondition_square_matrix_accuracy<MatrixXf>(12)));
    CALL_SUBTEST_62((svd_precondition_square_matrix_accuracy<MatrixXd>(12)));
    CALL_SUBTEST_62((svd_precondition_square_matrix_accuracy<Matrix<double, Dynamic, Dynamic, RowMajor>>(12)));
    CALL_SUBTEST_63((svd_precondition_square_matrix_accuracy<Matrix2cd>(2)));
    CALL_SUBTEST_63((svd_precondition_square_matrix_accuracy<Matrix4cf>(4)));
    CALL_SUBTEST_63((svd_precondition_square_matrix_accuracy<MatrixXcf>(12)));
    CALL_SUBTEST_63((svd_precondition_square_matrix_accuracy<MatrixXcd>(12)));
    CALL_SUBTEST_64((svd_precondition_square_matrix_runtime_options<MatrixXf>(12)));
    CALL_SUBTEST_64((svd_precondition_square_matrix_runtime_options<MatrixXcd>(internal::random<Index>(1, 30))));
    CALL_SUBTEST_64((svd_precondition_square_matrix_runtime_options<Matrix2f>(2)));
    CALL_SUBTEST_64((bdcsvd_thin_full_options<Matrix<double, Dynamic, Dynamic, RowMajor>>(
        Matrix<double, Dynamic, Dynamic, RowMajor>(8, 8))));
    // smallSvd keeps its QR preconditioner, so small non-square inputs work with DisableQRDecomposition too.
    CALL_SUBTEST_64(
        (svd_thin_full_option_checks<MatrixXd, DisableQRDecomposition | PreconditionSquareMatrix>(MatrixXd(10, 5))));
    CALL_SUBTEST_64(
        (svd_thin_full_option_checks<MatrixXd, DisableQRDecomposition | PreconditionSquareMatrix>(MatrixXd(5, 10))));
  }

  // test matrixbase method
  CALL_SUBTEST_42((bdcsvd_method<Matrix2cd>()));
  CALL_SUBTEST_43((bdcsvd_method<Matrix3f>()));

  // Check that preallocation avoids subsequent mallocs
  // Disabled because not supported by BDCSVD
  // CALL_SUBTEST_9( svd_preallocate<void>() );

  CALL_SUBTEST_45(svd_underoverflow<void>());
  CALL_SUBTEST_45(bdcsvd_power_of_two_scaling<>());
  CALL_SUBTEST_45((svd_zero_matrix_solvers<Matrix3d>(3, 3)));
  CALL_SUBTEST_45((svd_zero_matrix_solvers<Matrix<float, 2, 5>>(2, 5)));
  CALL_SUBTEST_45((svd_zero_matrix_solvers<MatrixXcf>(5, 3)));

  // Without total deflation issues.
  CALL_SUBTEST_46((compare_bdc_jacobi_instance<>(true)));
  CALL_SUBTEST_47((compare_bdc_jacobi_instance<>(false)));

  // With total deflation issues before, when it shouldn't be triggered.
  CALL_SUBTEST_48((compare_bdc_jacobi_instance<>(true, 3)));
  CALL_SUBTEST_49((compare_bdc_jacobi_instance<>(false, 3)));

  // Convergence for large constant matrix (https://gitlab.com/libeigen/eigen/-/issues/2491)
  CALL_SUBTEST_50(bdcsvd_check_convergence<MatrixXf>(MatrixXf::Constant(500, 500, 1)));

  // Bidiagonal SVD hard test cases
  CALL_SUBTEST_51((bdcsvd_bidiagonal_hard_cases<float>()));
  CALL_SUBTEST_52((bdcsvd_bidiagonal_hard_cases<double>()));
  CALL_SUBTEST_53((bdcsvd_extreme_scale_regressions<>()));
  CALL_SUBTEST_54((bdcsvd_fast_math_regression_1588<>()));
  CALL_SUBTEST_55((bdcsvd_secular_extrapolation<float>()));
  CALL_SUBTEST_56((bdcsvd_secular_extrapolation<double>()));
  CALL_SUBTEST_57((bdcsvd_qr_crossover<float, ColMajor>()));
  CALL_SUBTEST_58((bdcsvd_qr_crossover<double, RowMajor>()));
  CALL_SUBTEST_59((bdcsvd_qr_crossover<std::complex<double>, ColMajor>()));
  CALL_SUBTEST_60((bdcsvd_flushed_subnormal_matrix<MatrixXf>(12, 10)));
  CALL_SUBTEST_60((bdcsvd_flushed_subnormal_matrix<MatrixXd>(10, 12)));
  CALL_SUBTEST_60((bdcsvd_flushed_subnormal_matrix<MatrixXcd>(10, 10)));
  CALL_SUBTEST_60((bdcsvd_flushed_subnormal_bidiagonal<float>(12, 8)));
  CALL_SUBTEST_60((bdcsvd_flushed_subnormal_bidiagonal<double>(16, 8)));
  CALL_SUBTEST_60((bdcsvd_flushed_subnormal_bidiagonal<double>(6, 8)));
  CALL_SUBTEST_65((bdcsvd_switch_size<>()));
}