test/schur_real.cpp

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2010,2012 Jitse Niesen <jitse@maths.leeds.ac.uk>
//
// 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

#include "main.h"
#include "fp_control.h"
#include <limits>
#include <Eigen/Eigenvalues>

template <typename MatrixType>
void verifyIsQuasiTriangular(const MatrixType& T) {
  const Index size = T.cols();
  typedef typename MatrixType::Scalar Scalar;

  // Check T is lower Hessenberg
  for (int row = 2; row < size; ++row) {
    for (int col = 0; col < row - 1; ++col) {
      VERIFY_IS_EQUAL(T(row, col), Scalar(0));
    }
  }

  // Check that any non-zero on the subdiagonal is followed by a zero and is
  // part of a 2x2 diagonal block with imaginary eigenvalues.
  for (int row = 1; row < size; ++row) {
    if (!numext::is_exactly_zero(T(row, row - 1))) {
      VERIFY(row == size - 1 || numext::is_exactly_zero(T(row + 1, row)));
      Scalar tr = T(row - 1, row - 1) + T(row, row);
      Scalar det = T(row - 1, row - 1) * T(row, row) - T(row - 1, row) * T(row, row - 1);
      VERIFY(4 * det > tr * tr);
    }
  }
}

template <typename MatrixType>
void schur(int size = MatrixType::ColsAtCompileTime) {
  // Test basic functionality: T is quasi-triangular and A = U T U*
  for (int counter = 0; counter < g_repeat; ++counter) {
    MatrixType A = MatrixType::Random(size, size);
    RealSchur<MatrixType> schurOfA(A);
    VERIFY_IS_EQUAL(schurOfA.info(), Success);
    MatrixType U = schurOfA.matrixU();
    MatrixType T = schurOfA.matrixT();
    verifyIsQuasiTriangular(T);
    VERIFY_IS_APPROX(A, U * T * U.transpose());
  }

  // Test asserts when not initialized
  RealSchur<MatrixType> rsUninitialized;
  VERIFY_RAISES_ASSERT(rsUninitialized.matrixT());
  VERIFY_RAISES_ASSERT(rsUninitialized.matrixU());
  VERIFY_RAISES_ASSERT(rsUninitialized.info());

  // Test whether compute() and constructor returns same result
  MatrixType A = MatrixType::Random(size, size);
  RealSchur<MatrixType> rs1;
  rs1.compute(A);
  RealSchur<MatrixType> rs2(A);
  VERIFY_IS_EQUAL(rs1.info(), Success);
  VERIFY_IS_EQUAL(rs2.info(), Success);
  VERIFY_IS_EQUAL(rs1.matrixT(), rs2.matrixT());
  VERIFY_IS_EQUAL(rs1.matrixU(), rs2.matrixU());

  // Test maximum number of iterations
  RealSchur<MatrixType> rs3;
  rs3.setMaxIterations(RealSchur<MatrixType>::m_maxIterationsPerRow * size).compute(A);
  VERIFY_IS_EQUAL(rs3.info(), Success);
  VERIFY_IS_EQUAL(rs3.matrixT(), rs1.matrixT());
  VERIFY_IS_EQUAL(rs3.matrixU(), rs1.matrixU());
  if (size > 2) {
    rs3.setMaxIterations(1).compute(A);
    VERIFY_IS_EQUAL(rs3.info(), NoConvergence);
    VERIFY_IS_EQUAL(rs3.getMaxIterations(), 1);
  }

  MatrixType Atriangular = A;
  Atriangular.template triangularView<StrictlyLower>().setZero();
  rs3.setMaxIterations(1).compute(Atriangular);  // triangular matrices do not need any iterations
  VERIFY_IS_EQUAL(rs3.info(), Success);
  VERIFY_IS_APPROX(rs3.matrixT(), Atriangular);  // approx because of scaling...
  VERIFY_IS_EQUAL(rs3.matrixU(), MatrixType::Identity(size, size));

  // Test computation of only T, not U
  RealSchur<MatrixType> rsOnlyT(A, false);
  VERIFY_IS_EQUAL(rsOnlyT.info(), Success);
  VERIFY_IS_EQUAL(rs1.matrixT(), rsOnlyT.matrixT());
  VERIFY_RAISES_ASSERT(rsOnlyT.matrixU());

  if (size > 2 && size < 20) {
    // Test matrix with NaN
    A(0, 0) = std::numeric_limits<typename MatrixType::Scalar>::quiet_NaN();
    RealSchur<MatrixType> rsNaN(A);
    VERIFY_IS_EQUAL(rsNaN.info(), NoConvergence);
  }
}

void test_bug2633() {
  Eigen::MatrixXd A(4, 4);
  A << 0, 0, 0, -2, 1, 0, 0, -0, 0, 1, 0, 2, 0, 0, 2, -0;
  RealSchur<Eigen::MatrixXd> schur(A);
  VERIFY(schur.info() == Eigen::Success);
}

void real_schur_power_of_two_scaling() {
  // Reciprocal scaling rounds the smaller diagonal entry up by one ULP.
  Matrix2f matrix = Matrix2f::Zero();
  matrix(0, 0) = numext::bit_cast<float>(numext::uint32_t(0x58f6aaed));
  matrix(0, 1) = numext::bit_cast<float>(numext::uint32_t(0x52123456));
  matrix(1, 1) = numext::bit_cast<float>(numext::uint32_t(0x537dcf0e));

  const RealSchur<Matrix2f> schur(matrix);
  VERIFY_IS_EQUAL(schur.info(), Success);
  VERIFY_IS_EQUAL(schur.matrixT(), matrix);

  // Probe arithmetic, not just loads: FTZ can preserve an input yet flush a subnormal result.
  volatile float normalMinInput = (std::numeric_limits<float>::min)();
  volatile float epsilonInput = std::numeric_limits<float>::epsilon();
  const float denormMin = normalMinInput * epsilonInput;
  if (!(denormMin > 0.0f)) return;
  matrix.setZero();
  matrix.diagonal() << 1.5f, denormMin;
  const RealSchur<Matrix2f> tailSchur(matrix);
  VERIFY_IS_EQUAL(tailSchur.matrixT()(1, 1), denormMin);
}

template <typename Scalar>
void real_schur_subnormal_restoration() {
  using Bits = typename numext::get_integer_by_size<sizeof(Scalar)>::unsigned_type;
  using MatrixType = Matrix<Scalar, 2, 2>;
  const Scalar normal_min = (std::numeric_limits<Scalar>::min)();
  const Bits half_min_bits = numext::bit_cast<Bits>(normal_min) >> 1;
  MatrixType matrix = MatrixType::Zero();
  matrix(0, 0) = normal_min;
  matrix(1, 1) = numext::bit_cast<Scalar>(half_min_bits);
  const ScopedFlushToZero flush;
  for (bool compute_u : {false, true}) {
    const RealSchur<MatrixType> schur(matrix, compute_u);
    VERIFY_IS_EQUAL(schur.info(), Success);
    // Compare representations: DAZ can make an erroneous zero compare equal to a subnormal.
    VERIFY_IS_EQUAL(numext::bit_cast<Bits>(schur.matrixT()(1, 1)), half_min_bits);
    VERIFY_IS_EQUAL(numext::bit_cast<Bits>(schur.matrixT()(0, 0)), numext::bit_cast<Bits>(normal_min));
  }
}

EIGEN_DECLARE_TEST(schur_real) {
  CALL_SUBTEST_1((schur<Matrix4f>()));
  CALL_SUBTEST_2((schur<MatrixXd>(internal::random<int>(1, EIGEN_TEST_MAX_SIZE / 4))));
  CALL_SUBTEST_3((schur<Matrix<float, 1, 1> >()));
  CALL_SUBTEST_4((schur<Matrix<double, 3, 3, Eigen::RowMajor> >()));

  // Test problem size constructors
  CALL_SUBTEST_5(RealSchur<MatrixXf>(10));

  CALL_SUBTEST_6((test_bug2633()));
  CALL_SUBTEST_6((real_schur_power_of_two_scaling()));
  CALL_SUBTEST_6(real_schur_subnormal_restoration<float>());
  CALL_SUBTEST_6(real_schur_subnormal_restoration<double>());
}