contrib/test/matrix_exponential.cpp

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2009 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 "matrix_functions.h"
#include "CustomComplex.h"

double binom(int n, int k) {
  double res = 1;
  for (int i = 0; i < k; i++) res = res * (n - k + i + 1) / (i + 1);
  return res;
}

template <typename T>
T expfn(T x, int) {
  return std::exp(x);
}

template <typename T>
void test2dRotation(T tol) {
  Matrix<T, 2, 2> A, B, C;
  T angle;

  A << 0, 1, -1, 0;
  for (int i = 0; i <= 20; i++) {
    angle = static_cast<T>(pow(10, i / 5. - 2));
    B << std::cos(angle), std::sin(angle), -std::sin(angle), std::cos(angle);

    C = (angle * A).matrixFunction(expfn);
    std::cout << "test2dRotation: i = " << i << "   error funm = " << relerr(C, B);
    VERIFY(C.isApprox(B, tol));

    C = (angle * A).exp();
    std::cout << "   error expm = " << relerr(C, B) << "\n";
    VERIFY(C.isApprox(B, tol));
  }
}

template <typename T>
void test2dHyperbolicRotation(T tol) {
  Matrix<std::complex<T>, 2, 2> A, B, C;
  std::complex<T> imagUnit(0, 1);
  T angle, ch, sh;

  for (int i = 0; i <= 20; i++) {
    angle = static_cast<T>((i - 10) / 2.0);
    ch = std::cosh(angle);
    sh = std::sinh(angle);
    A << 0, angle * imagUnit, -angle * imagUnit, 0;
    B << ch, sh * imagUnit, -sh * imagUnit, ch;

    C = A.matrixFunction(expfn);
    std::cout << "test2dHyperbolicRotation: i = " << i << "   error funm = " << relerr(C, B);
    VERIFY(C.isApprox(B, tol));

    C = A.exp();
    std::cout << "   error expm = " << relerr(C, B) << "\n";
    VERIFY(C.isApprox(B, tol));
  }
}

template <typename T>
void testPascal(T tol) {
  for (int size = 1; size < 20; size++) {
    Matrix<T, Dynamic, Dynamic> A(size, size), B(size, size), C(size, size);
    A.setZero();
    for (int i = 0; i < size - 1; i++) A(i + 1, i) = static_cast<T>(i + 1);
    B.setZero();
    for (int i = 0; i < size; i++)
      for (int j = 0; j <= i; j++) B(i, j) = static_cast<T>(binom(i, j));

    C = A.matrixFunction(expfn);
    std::cout << "testPascal: size = " << size << "   error funm = " << relerr(C, B);
    VERIFY(C.isApprox(B, tol));

    C = A.exp();
    std::cout << "   error expm = " << relerr(C, B) << "\n";
    VERIFY(C.isApprox(B, tol));
  }
}

template <typename MatrixType>
void randomTest(const MatrixType& m,
                const typename NumTraits<typename internal::traits<MatrixType>::Scalar>::Real& tol) {
  /* this test covers the following files:
     Inverse.h
  */
  typename MatrixType::Index rows = m.rows();
  typename MatrixType::Index cols = m.cols();
  MatrixType m1(rows, cols), m2(rows, cols), identity = MatrixType::Identity(rows, cols);

  for (int i = 0; i < g_repeat; i++) {
    m1 = MatrixType::Random(rows, cols);

    m2 = m1.matrixFunction(expfn) * (-m1).matrixFunction(expfn);
    std::cout << "randomTest: error funm = " << relerr(identity, m2);
    VERIFY(identity.isApprox(m2, tol));

    m2 = m1.exp() * (-m1).exp();
    std::cout << "   error expm = " << relerr(identity, m2) << "\n";
    VERIFY(identity.isApprox(m2, tol));
  }
}

template <int Options>
void testComplexScalingPath() {
  using Scalar = std::complex<double>;
  using MatrixType = Matrix<Scalar, 3, 3, Options>;
  MatrixType A = MatrixType::Zero();
  A.diagonal() << Scalar(-1.0, 64.0), Scalar(0.5, -32.0), Scalar(-0.25, 16.0);

  MatrixType expected = MatrixType::Zero();
  for (Index i = 0; i < A.rows(); ++i) expected(i, i) = std::exp(A(i, i));

  const double tol = 100.0 * NumTraits<double>::epsilon();
  VERIFY(A.exp().isApprox(expected, tol));
}

void testCustomComplexScalingPath() {
  using Scalar = CustomComplex<double>;
  using MatrixType = Matrix<Scalar, 3, 3>;
  static_assert(!internal::complex_array_access<Scalar>::value, "test must exercise the scalar scaling fallback");

  MatrixType A = MatrixType::Zero();
  A.diagonal() << Scalar(-1.0, 64.0), Scalar(0.5, -32.0), Scalar(-0.25, 16.0);

  const int squarings = 4;
  const MatrixType scaled = internal::matrix_exp_scale<MatrixType>(A, squarings);
  for (Index i = 0; i < A.size(); ++i) {
    using std::ldexp;
    VERIFY_IS_EQUAL(numext::real(scaled(i)), ldexp(numext::real(A(i)), -squarings));
    VERIFY_IS_EQUAL(numext::imag(scaled(i)), ldexp(numext::imag(A(i)), -squarings));
  }
}

template <typename Scalar>
void testAtomicTaylorConvergence() {
  using MatrixType = Matrix<Scalar, 2, 2>;
  // The eigenvalues +-a are 1/16 apart, within matrix_function_separation, so A is a single atomic block.
  const Scalar a = Scalar(1) / Scalar(32);
  MatrixType A;
  A << a, 1, 0, -a;
  // A^2 = a^2 I, hence exp(A) = cosh(a) I + sinh(a)/a A. Reference: MPFR, 256 bits.
  MatrixType expected;
  expected << Scalar(1.031743407499102670938747815281507144194498326641816096008L),
      Scalar(1.000162768364137425450217952562469769436361163924174806184L), Scalar(0),
      Scalar(0.969233234476344081848109193246352783604725753896555170622L);
  const MatrixType actual = A.matrixFunction(expfn);
  const Scalar bound = 8 * NumTraits<Scalar>::epsilon() * expected.cwiseAbs().maxCoeff();
  VERIFY(actual.allFinite());
  VERIFY((actual - expected).cwiseAbs().maxCoeff() <= bound);
}

EIGEN_DECLARE_TEST(matrix_exponential) {
  CALL_SUBTEST_1(testAtomicTaylorConvergence<float>());
  CALL_SUBTEST_2(testAtomicTaylorConvergence<double>());
  CALL_SUBTEST_9(testAtomicTaylorConvergence<long double>());
  // matrixFunction() dominates the largest cases. The factors retain at least 1.6x headroom over the largest relative
  // errors observed in extended GCC and Clang test runs.
  CALL_SUBTEST_2(test2dRotation<double>(256 * NumTraits<double>::epsilon()));
  CALL_SUBTEST_1(test2dRotation<float>(128 * NumTraits<float>::epsilon()));
  CALL_SUBTEST_8(test2dRotation<long double>(256 * NumTraits<long double>::epsilon()));
  CALL_SUBTEST_2(test2dHyperbolicRotation<double>(32 * NumTraits<double>::epsilon()));
  CALL_SUBTEST_1(test2dHyperbolicRotation<float>(32 * NumTraits<float>::epsilon()));
  CALL_SUBTEST_8(test2dHyperbolicRotation<long double>(32 * NumTraits<long double>::epsilon()));
  CALL_SUBTEST_6(testPascal<float>(4 * NumTraits<float>::epsilon()));
  CALL_SUBTEST_5(testPascal<double>(4 * NumTraits<double>::epsilon()));
  CALL_SUBTEST_2(randomTest(Matrix2d(), 384 * NumTraits<double>::epsilon()));
  CALL_SUBTEST_7(randomTest(Matrix<double, 3, 3, RowMajor>(), 384 * NumTraits<double>::epsilon()));
  CALL_SUBTEST_3(randomTest(Matrix4cd(), 384 * NumTraits<std::complex<double>>::epsilon()));
  CALL_SUBTEST_3((testComplexScalingPath<ColMajor>()));
  CALL_SUBTEST_3((testComplexScalingPath<RowMajor>()));
  CALL_SUBTEST_3(testCustomComplexScalingPath());
  CALL_SUBTEST_4(randomTest(MatrixXd(8, 8), 384 * NumTraits<double>::epsilon()));
  CALL_SUBTEST_1(randomTest(Matrix2f(), 384 * NumTraits<float>::epsilon()));
  CALL_SUBTEST_5(randomTest(Matrix3cf(), 384 * NumTraits<std::complex<float>>::epsilon()));
  CALL_SUBTEST_1(randomTest(Matrix4f(), 384 * NumTraits<float>::epsilon()));
  CALL_SUBTEST_6(randomTest(MatrixXf(8, 8), 384 * NumTraits<float>::epsilon()));
  CALL_SUBTEST_9(randomTest(Matrix<long double, Dynamic, Dynamic>(7, 7), 384 * NumTraits<long double>::epsilon()));
}