matrix_power.cpp

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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2012 Chen-Pang He <jdh8@ms63.hinet.net>
//
// 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/.

#include "matrix_functions.h"

template <typename MatrixType, int IsComplex = NumTraits<typename MatrixType::Scalar>::IsComplex>
struct generateTriangularMatrix;

// for real matrices, make sure none of the eigenvalues are negative
template <typename MatrixType>
struct generateTriangularMatrix<MatrixType,0>
{
  static void run(MatrixType& result, typename MatrixType::Index size)
  {
    result.resize(size, size);
    result.template triangularView<Upper>() = MatrixType::Random(size, size);
    for (typename MatrixType::Index i = 0; i < size; ++i)
      result.coeffRef(i,i) = std::abs(result.coeff(i,i));
  }
};

// for complex matrices, any matrix is fine
template <typename MatrixType>
struct generateTriangularMatrix<MatrixType,1>
{
  static void run(MatrixType& result, typename MatrixType::Index size)
  {
    result.resize(size, size);
    result.template triangularView<Upper>() = MatrixType::Random(size, size);
  }
};

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

  A << 0, 1, -1, 0;
  MatrixPower<Matrix<T,2,2> > Apow(A);

  for (int i=0; i<=20; ++i) {
    angle = pow(10, (i-10) / 5.);
    c = std::cos(angle);
    s = std::sin(angle);
    B << c, s, -s, c;

    C = Apow(std::ldexp(angle,1) / M_PI);
    std::cout << "test2dRotation: i = " << i << "   error powerm = " << relerr(C,B) << '\n';
    VERIFY(C.isApprox(B, static_cast<T>(tol)));
  }
}

template<typename T>
void test2dHyperbolicRotation(double tol)
{
  Matrix<std::complex<T>,2,2> A, B, C;
  T angle, ch = std::cosh((T)1);
  std::complex<T> ish(0, std::sinh((T)1));

  A << ch, ish, -ish, ch;
  MatrixPower<Matrix<std::complex<T>,2,2> > Apow(A);

  for (int i=0; i<=20; ++i) {
    angle = std::ldexp(static_cast<T>(i-10), -1);
    ch = std::cosh(angle);
    ish = std::complex<T>(0, std::sinh(angle));
    B << ch, ish, -ish, ch;

    C = Apow(angle);
    std::cout << "test2dHyperbolicRotation: i = " << i << "   error powerm = " << relerr(C,B) << '\n';
    VERIFY(C.isApprox(B, static_cast<T>(tol)));
  }
}

template<typename MatrixType>
void testExponentLaws(const MatrixType& m, double tol)
{
  typedef typename MatrixType::RealScalar RealScalar;
  MatrixType m1, m2, m3, m4, m5;
  RealScalar x, y;

  for (int i=0; i < g_repeat; ++i) {
    generateTestMatrix<MatrixType>::run(m1, m.rows());
    MatrixPower<MatrixType> mpow(m1);

    x = internal::random<RealScalar>();
    y = internal::random<RealScalar>();
    m2 = mpow(x);
    m3 = mpow(y);

    m4 = mpow(x+y);
    m5.noalias() = m2 * m3;
    VERIFY(m4.isApprox(m5, static_cast<RealScalar>(tol)));

    m4 = mpow(x*y);
    m5 = m2.pow(y);
    VERIFY(m4.isApprox(m5, static_cast<RealScalar>(tol)));

    m4 = (std::abs(x) * m1).pow(y);
    m5 = std::pow(std::abs(x), y) * m3;
    VERIFY(m4.isApprox(m5, static_cast<RealScalar>(tol)));
  }
}

typedef Matrix<double,3,3,RowMajor>         Matrix3dRowMajor;
typedef Matrix<long double,Dynamic,Dynamic> MatrixXe;
 
void test_matrix_power()
{
  CALL_SUBTEST_2(test2dRotation<double>(1e-13));
  CALL_SUBTEST_1(test2dRotation<float>(2e-5));  // was 1e-5, relaxed for clang 2.8 / linux / x86-64
  CALL_SUBTEST_9(test2dRotation<long double>(1e-13)); 
  CALL_SUBTEST_2(test2dHyperbolicRotation<double>(1e-14));
  CALL_SUBTEST_1(test2dHyperbolicRotation<float>(1e-5));
  CALL_SUBTEST_9(test2dHyperbolicRotation<long double>(1e-14));

  CALL_SUBTEST_2(testExponentLaws(Matrix2d(),         1e-13));
  CALL_SUBTEST_7(testExponentLaws(Matrix3dRowMajor(), 1e-13));
  CALL_SUBTEST_3(testExponentLaws(Matrix4cd(),        1e-13));
  CALL_SUBTEST_4(testExponentLaws(MatrixXd(8,8),      2e-12));
  CALL_SUBTEST_1(testExponentLaws(Matrix2f(),         1e-4));
  CALL_SUBTEST_5(testExponentLaws(Matrix3cf(),        1e-4));
  CALL_SUBTEST_8(testExponentLaws(Matrix4f(),         1e-4));
  CALL_SUBTEST_6(testExponentLaws(MatrixXf(2,2),      1e-3)); // see bug 614
  CALL_SUBTEST_9(testExponentLaws(MatrixXe(7,7),      1e-13));
}