cholesky.cpp

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.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/.

#define TEST_ENABLE_TEMPORARY_TRACKING

#include <Eigen/Cholesky>
#include <Eigen/QR>
#include "main.h"
#include "solverbase.h"

template <typename MatrixType, int UpLo>
typename MatrixType::RealScalar matrix_l1_norm(const MatrixType& m) {
  if (m.cols() == 0) return typename MatrixType::RealScalar(0);
  MatrixType symm = m.template selfadjointView<UpLo>();
  return symm.cwiseAbs().colwise().sum().maxCoeff();
}

template <typename MatrixType, template <typename, int> class CholType>
void test_chol_update(const MatrixType& symm) {
  typedef typename MatrixType::Scalar Scalar;
  typedef typename MatrixType::RealScalar RealScalar;
  typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> VectorType;

  MatrixType symmLo = symm.template triangularView<Lower>();
  MatrixType symmUp = symm.template triangularView<Upper>();
  MatrixType symmCpy = symm;

  CholType<MatrixType, Lower> chollo(symmLo);
  CholType<MatrixType, Upper> cholup(symmUp);

  for (int k = 0; k < 10; ++k) {
    VectorType vec = VectorType::Random(symm.rows());
    RealScalar sigma = internal::random<RealScalar>();
    symmCpy += sigma * vec * vec.adjoint();

    // we are doing some downdates, so it might be the case that the matrix is
    // not SPD anymore
    CholType<MatrixType, Lower> chol(symmCpy);
    if (chol.info() != Success) break;

    chollo.rankUpdate(vec, sigma);
    VERIFY_IS_APPROX(symmCpy, chollo.reconstructedMatrix());

    cholup.rankUpdate(vec, sigma);
    VERIFY_IS_APPROX(symmCpy, cholup.reconstructedMatrix());
  }
}

template <typename MatrixType>
void cholesky(const MatrixType& m) {
  /* this test covers the following files:
     LLT.h LDLT.h
  */
  Index rows = m.rows();
  Index cols = m.cols();

  typedef typename MatrixType::Scalar Scalar;
  typedef typename NumTraits<Scalar>::Real RealScalar;
  typedef Matrix<Scalar, MatrixType::RowsAtCompileTime,
                 MatrixType::RowsAtCompileTime>
      SquareMatrixType;
  typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> VectorType;

  MatrixType a0 = MatrixType::Random(rows, cols);
  VectorType vecB = VectorType::Random(rows), vecX(rows);
  MatrixType matB = MatrixType::Random(rows, cols), matX(rows, cols);
  SquareMatrixType symm = a0 * a0.adjoint();
  // let's make sure the matrix is not singular or near singular
  for (int k = 0; k < 3; ++k) {
    MatrixType a1 = MatrixType::Random(rows, cols);
    symm += a1 * a1.adjoint();
  }

  {
    STATIC_CHECK(
        (internal::is_same<typename LLT<MatrixType, Lower>::StorageIndex,
                           int>::value));
    STATIC_CHECK(
        (internal::is_same<typename LLT<MatrixType, Upper>::StorageIndex,
                           int>::value));

    SquareMatrixType symmUp = symm.template triangularView<Upper>();
    SquareMatrixType symmLo = symm.template triangularView<Lower>();

    LLT<SquareMatrixType, Lower> chollo(symmLo);
    VERIFY_IS_APPROX(symm, chollo.reconstructedMatrix());

    check_solverbase<VectorType, VectorType>(symm, chollo, rows, rows, 1);
    check_solverbase<MatrixType, MatrixType>(symm, chollo, rows, cols, rows);

    const MatrixType symmLo_inverse =
        chollo.solve(MatrixType::Identity(rows, cols));
    RealScalar rcond =
        (RealScalar(1) / matrix_l1_norm<MatrixType, Lower>(symmLo)) /
        matrix_l1_norm<MatrixType, Lower>(symmLo_inverse);
    RealScalar rcond_est = chollo.rcond();
    // Verify that the estimated condition number is within a factor of 10 of
    // the
    // truth.
    VERIFY(rcond_est >= rcond / 10 && rcond_est <= rcond * 10);

    // test the upper mode
    LLT<SquareMatrixType, Upper> cholup(symmUp);
    VERIFY_IS_APPROX(symm, cholup.reconstructedMatrix());
    vecX = cholup.solve(vecB);
    VERIFY_IS_APPROX(symm * vecX, vecB);
    matX = cholup.solve(matB);
    VERIFY_IS_APPROX(symm * matX, matB);

    // Verify that the estimated condition number is within a factor of 10 of
    // the
    // truth.
    const MatrixType symmUp_inverse =
        cholup.solve(MatrixType::Identity(rows, cols));
    rcond = (RealScalar(1) / matrix_l1_norm<MatrixType, Upper>(symmUp)) /
            matrix_l1_norm<MatrixType, Upper>(symmUp_inverse);
    rcond_est = cholup.rcond();
    VERIFY(rcond_est >= rcond / 10 && rcond_est <= rcond * 10);

    MatrixType neg = -symmLo;
    chollo.compute(neg);
    VERIFY(neg.size() == 0 || chollo.info() == NumericalIssue);

    VERIFY_IS_APPROX(MatrixType(chollo.matrixL().transpose().conjugate()),
                     MatrixType(chollo.matrixU()));
    VERIFY_IS_APPROX(MatrixType(chollo.matrixU().transpose().conjugate()),
                     MatrixType(chollo.matrixL()));
    VERIFY_IS_APPROX(MatrixType(cholup.matrixL().transpose().conjugate()),
                     MatrixType(cholup.matrixU()));
    VERIFY_IS_APPROX(MatrixType(cholup.matrixU().transpose().conjugate()),
                     MatrixType(cholup.matrixL()));

    // test some special use cases of SelfCwiseBinaryOp:
    MatrixType m1 = MatrixType::Random(rows, cols), m2(rows, cols);
    m2 = m1;
    m2 += symmLo.template selfadjointView<Lower>().llt().solve(matB);
    VERIFY_IS_APPROX(
        m2, m1 + symmLo.template selfadjointView<Lower>().llt().solve(matB));
    m2 = m1;
    m2 -= symmLo.template selfadjointView<Lower>().llt().solve(matB);
    VERIFY_IS_APPROX(
        m2, m1 - symmLo.template selfadjointView<Lower>().llt().solve(matB));
    m2 = m1;
    m2.noalias() += symmLo.template selfadjointView<Lower>().llt().solve(matB);
    VERIFY_IS_APPROX(
        m2, m1 + symmLo.template selfadjointView<Lower>().llt().solve(matB));
    m2 = m1;
    m2.noalias() -= symmLo.template selfadjointView<Lower>().llt().solve(matB);
    VERIFY_IS_APPROX(
        m2, m1 - symmLo.template selfadjointView<Lower>().llt().solve(matB));
  }

  // LDLT
  {
    STATIC_CHECK(
        (internal::is_same<typename LDLT<MatrixType, Lower>::StorageIndex,
                           int>::value));
    STATIC_CHECK(
        (internal::is_same<typename LDLT<MatrixType, Upper>::StorageIndex,
                           int>::value));

    int sign = internal::random<int>() % 2 ? 1 : -1;

    if (sign == -1) {
      symm = -symm;  // test a negative matrix
    }

    SquareMatrixType symmUp = symm.template triangularView<Upper>();
    SquareMatrixType symmLo = symm.template triangularView<Lower>();

    LDLT<SquareMatrixType, Lower> ldltlo(symmLo);
    VERIFY(ldltlo.info() == Success);
    VERIFY_IS_APPROX(symm, ldltlo.reconstructedMatrix());

    check_solverbase<VectorType, VectorType>(symm, ldltlo, rows, rows, 1);
    check_solverbase<MatrixType, MatrixType>(symm, ldltlo, rows, cols, rows);

    const MatrixType symmLo_inverse =
        ldltlo.solve(MatrixType::Identity(rows, cols));
    RealScalar rcond =
        (RealScalar(1) / matrix_l1_norm<MatrixType, Lower>(symmLo)) /
        matrix_l1_norm<MatrixType, Lower>(symmLo_inverse);
    RealScalar rcond_est = ldltlo.rcond();
    // Verify that the estimated condition number is within a factor of 10 of
    // the
    // truth.
    VERIFY(rcond_est >= rcond / 10 && rcond_est <= rcond * 10);

    LDLT<SquareMatrixType, Upper> ldltup(symmUp);
    VERIFY(ldltup.info() == Success);
    VERIFY_IS_APPROX(symm, ldltup.reconstructedMatrix());
    vecX = ldltup.solve(vecB);
    VERIFY_IS_APPROX(symm * vecX, vecB);
    matX = ldltup.solve(matB);
    VERIFY_IS_APPROX(symm * matX, matB);

    // Verify that the estimated condition number is within a factor of 10 of
    // the
    // truth.
    const MatrixType symmUp_inverse =
        ldltup.solve(MatrixType::Identity(rows, cols));
    rcond = (RealScalar(1) / matrix_l1_norm<MatrixType, Upper>(symmUp)) /
            matrix_l1_norm<MatrixType, Upper>(symmUp_inverse);
    rcond_est = ldltup.rcond();
    VERIFY(rcond_est >= rcond / 10 && rcond_est <= rcond * 10);

    VERIFY_IS_APPROX(MatrixType(ldltlo.matrixL().transpose().conjugate()),
                     MatrixType(ldltlo.matrixU()));
    VERIFY_IS_APPROX(MatrixType(ldltlo.matrixU().transpose().conjugate()),
                     MatrixType(ldltlo.matrixL()));
    VERIFY_IS_APPROX(MatrixType(ldltup.matrixL().transpose().conjugate()),
                     MatrixType(ldltup.matrixU()));
    VERIFY_IS_APPROX(MatrixType(ldltup.matrixU().transpose().conjugate()),
                     MatrixType(ldltup.matrixL()));

    if (MatrixType::RowsAtCompileTime == Dynamic) {
      // note : each inplace permutation requires a small temporary vector
      // (mask)

      // check inplace solve
      matX = matB;
      VERIFY_EVALUATION_COUNT(matX = ldltlo.solve(matX), 0);
      VERIFY_IS_APPROX(matX, ldltlo.solve(matB).eval());

      matX = matB;
      VERIFY_EVALUATION_COUNT(matX = ldltup.solve(matX), 0);
      VERIFY_IS_APPROX(matX, ldltup.solve(matB).eval());
    }

    // restore
    if (sign == -1) symm = -symm;

    // check matrices coming from linear constraints with Lagrange multipliers
    if (rows >= 3) {
      SquareMatrixType A = symm;
      Index c = internal::random<Index>(0, rows - 2);
      A.bottomRightCorner(c, c).setZero();
      // Make sure a solution exists:
      vecX.setRandom();
      vecB = A * vecX;
      vecX.setZero();
      ldltlo.compute(A);
      VERIFY_IS_APPROX(A, ldltlo.reconstructedMatrix());
      vecX = ldltlo.solve(vecB);
      VERIFY_IS_APPROX(A * vecX, vecB);
    }

    // check non-full rank matrices
    if (rows >= 3) {
      Index r = internal::random<Index>(1, rows - 1);
      Matrix<Scalar, Dynamic, Dynamic> a =
          Matrix<Scalar, Dynamic, Dynamic>::Random(rows, r);
      SquareMatrixType A = a * a.adjoint();
      // Make sure a solution exists:
      vecX.setRandom();
      vecB = A * vecX;
      vecX.setZero();
      ldltlo.compute(A);
      VERIFY_IS_APPROX(A, ldltlo.reconstructedMatrix());
      vecX = ldltlo.solve(vecB);
      VERIFY_IS_APPROX(A * vecX, vecB);
    }

    // check matrices with a wide spectrum
    if (rows >= 3) {
      using std::pow;
      using std::sqrt;
      RealScalar s =
          (std::min)(16, std::numeric_limits<RealScalar>::max_exponent10 / 8);
      Matrix<Scalar, Dynamic, Dynamic> a =
          Matrix<Scalar, Dynamic, Dynamic>::Random(rows, rows);
      Matrix<RealScalar, Dynamic, 1> d =
          Matrix<RealScalar, Dynamic, 1>::Random(rows);
      for (Index k = 0; k < rows; ++k)
        d(k) = d(k) * pow(RealScalar(10), internal::random<RealScalar>(-s, s));
      SquareMatrixType A = a * d.asDiagonal() * a.adjoint();
      // Make sure a solution exists:
      vecX.setRandom();
      vecB = A * vecX;
      vecX.setZero();
      ldltlo.compute(A);
      VERIFY_IS_APPROX(A, ldltlo.reconstructedMatrix());
      vecX = ldltlo.solve(vecB);

      if (ldltlo.vectorD().real().cwiseAbs().minCoeff() > RealScalar(0)) {
        VERIFY_IS_APPROX(A * vecX, vecB);
      } else {
        RealScalar large_tol = sqrt(test_precision<RealScalar>());
        VERIFY((A * vecX).isApprox(vecB, large_tol));

        ++g_test_level;
        VERIFY_IS_APPROX(A * vecX, vecB);
        --g_test_level;
      }
    }
  }

  // update/downdate
  CALL_SUBTEST((test_chol_update<SquareMatrixType, LLT>(symm)));
  CALL_SUBTEST((test_chol_update<SquareMatrixType, LDLT>(symm)));
}

template <typename MatrixType>
void cholesky_cplx(const MatrixType& m) {
  // classic test
  cholesky(m);

  // test mixing real/scalar types

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

  typedef typename MatrixType::Scalar Scalar;
  typedef typename NumTraits<Scalar>::Real RealScalar;
  typedef Matrix<RealScalar, MatrixType::RowsAtCompileTime,
                 MatrixType::RowsAtCompileTime>
      RealMatrixType;
  typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> VectorType;

  RealMatrixType a0 = RealMatrixType::Random(rows, cols);
  VectorType vecB = VectorType::Random(rows), vecX(rows);
  MatrixType matB = MatrixType::Random(rows, cols), matX(rows, cols);
  RealMatrixType symm = a0 * a0.adjoint();
  // let's make sure the matrix is not singular or near singular
  for (int k = 0; k < 3; ++k) {
    RealMatrixType a1 = RealMatrixType::Random(rows, cols);
    symm += a1 * a1.adjoint();
  }

  {
    RealMatrixType symmLo = symm.template triangularView<Lower>();

    LLT<RealMatrixType, Lower> chollo(symmLo);
    VERIFY_IS_APPROX(symm, chollo.reconstructedMatrix());

    check_solverbase<VectorType, VectorType>(symm, chollo, rows, rows, 1);
    // check_solverbase<MatrixType, MatrixType>(symm, chollo, rows, cols, rows);
  }

  // LDLT
  {
    int sign = internal::random<int>() % 2 ? 1 : -1;

    if (sign == -1) {
      symm = -symm;  // test a negative matrix
    }

    RealMatrixType symmLo = symm.template triangularView<Lower>();

    LDLT<RealMatrixType, Lower> ldltlo(symmLo);
    VERIFY(ldltlo.info() == Success);
    VERIFY_IS_APPROX(symm, ldltlo.reconstructedMatrix());

    check_solverbase<VectorType, VectorType>(symm, ldltlo, rows, rows, 1);
    // check_solverbase<MatrixType, MatrixType>(symm, ldltlo, rows, cols, rows);
  }
}

// regression test for bug 241
template <typename MatrixType>
void cholesky_bug241(const MatrixType& m) {
  eigen_assert(m.rows() == 2 && m.cols() == 2);

  typedef typename MatrixType::Scalar Scalar;
  typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> VectorType;

  MatrixType matA;
  matA << 1, 1, 1, 1;
  VectorType vecB;
  vecB << 1, 1;
  VectorType vecX = matA.ldlt().solve(vecB);
  VERIFY_IS_APPROX(matA * vecX, vecB);
}

// LDLT is not guaranteed to work for indefinite matrices, but happens to work
// fine if matrix is diagonal.
// This test checks that LDLT reports correctly that matrix is indefinite.
// See http://forum.kde.org/viewtopic.php?f=74&t=106942 and bug 736
template <typename MatrixType>
void cholesky_definiteness(const MatrixType& m) {
  eigen_assert(m.rows() == 2 && m.cols() == 2);
  MatrixType mat;
  LDLT<MatrixType> ldlt(2);

  {
    mat << 1, 0, 0, -1;
    ldlt.compute(mat);
    VERIFY(ldlt.info() == Success);
    VERIFY(!ldlt.isNegative());
    VERIFY(!ldlt.isPositive());
    VERIFY_IS_APPROX(mat, ldlt.reconstructedMatrix());
  }
  {
    mat << 1, 2, 2, 1;
    ldlt.compute(mat);
    VERIFY(ldlt.info() == Success);
    VERIFY(!ldlt.isNegative());
    VERIFY(!ldlt.isPositive());
    VERIFY_IS_APPROX(mat, ldlt.reconstructedMatrix());
  }
  {
    mat << 0, 0, 0, 0;
    ldlt.compute(mat);
    VERIFY(ldlt.info() == Success);
    VERIFY(ldlt.isNegative());
    VERIFY(ldlt.isPositive());
    VERIFY_IS_APPROX(mat, ldlt.reconstructedMatrix());
  }
  {
    mat << 0, 0, 0, 1;
    ldlt.compute(mat);
    VERIFY(ldlt.info() == Success);
    VERIFY(!ldlt.isNegative());
    VERIFY(ldlt.isPositive());
    VERIFY_IS_APPROX(mat, ldlt.reconstructedMatrix());
  }
  {
    mat << -1, 0, 0, 0;
    ldlt.compute(mat);
    VERIFY(ldlt.info() == Success);
    VERIFY(ldlt.isNegative());
    VERIFY(!ldlt.isPositive());
    VERIFY_IS_APPROX(mat, ldlt.reconstructedMatrix());
  }
}

template <typename>
void cholesky_faillure_cases() {
  MatrixXd mat;
  LDLT<MatrixXd> ldlt;

  {
    mat.resize(2, 2);
    mat << 0, 1, 1, 0;
    ldlt.compute(mat);
    VERIFY_IS_NOT_APPROX(mat, ldlt.reconstructedMatrix());
    VERIFY(ldlt.info() == NumericalIssue);
  }
#if (!EIGEN_ARCH_i386) || defined(EIGEN_VECTORIZE_SSE2)
  {
    mat.resize(3, 3);
    mat << -1, -3, 3, -3, -8.9999999999999999999, 1, 3, 1, 0;
    ldlt.compute(mat);
    VERIFY(ldlt.info() == NumericalIssue);
    VERIFY_IS_NOT_APPROX(mat, ldlt.reconstructedMatrix());
  }
#endif
  {
    mat.resize(3, 3);
    mat << 1, 2, 3, 2, 4, 1, 3, 1, 0;
    ldlt.compute(mat);
    VERIFY(ldlt.info() == NumericalIssue);
    VERIFY_IS_NOT_APPROX(mat, ldlt.reconstructedMatrix());
  }

  {
    mat.resize(8, 8);
    mat << 0.1, 0, -0.1, 0, 0, 0, 1, 0, 0, 4.24667, 0, 2.00333, 0, 0, 0, 0,
        -0.1, 0, 0.2, 0, -0.1, 0, 0, 0, 0, 2.00333, 0, 8.49333, 0, 2.00333, 0,
        0, 0, 0, -0.1, 0, 0.1, 0, 0, 1, 0, 0, 0, 2.00333, 0, 4.24667, 0, 0, 1,
        0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0;
    ldlt.compute(mat);
    VERIFY(ldlt.info() == NumericalIssue);
    VERIFY_IS_NOT_APPROX(mat, ldlt.reconstructedMatrix());
  }

  // bug 1479
  {
    mat.resize(4, 4);
    mat << 1, 2, 0, 1, 2, 4, 0, 2, 0, 0, 0, 1, 1, 2, 1, 1;
    ldlt.compute(mat);
    VERIFY(ldlt.info() == NumericalIssue);
    VERIFY_IS_NOT_APPROX(mat, ldlt.reconstructedMatrix());
  }
}

template <typename MatrixType>
void cholesky_verify_assert() {
  MatrixType tmp;

  LLT<MatrixType> llt;
  VERIFY_RAISES_ASSERT(llt.matrixL())
  VERIFY_RAISES_ASSERT(llt.matrixU())
  VERIFY_RAISES_ASSERT(llt.solve(tmp))
  VERIFY_RAISES_ASSERT(llt.transpose().solve(tmp))
  VERIFY_RAISES_ASSERT(llt.adjoint().solve(tmp))
  VERIFY_RAISES_ASSERT(llt.solveInPlace(tmp))

  LDLT<MatrixType> ldlt;
  VERIFY_RAISES_ASSERT(ldlt.matrixL())
  VERIFY_RAISES_ASSERT(ldlt.transpositionsP())
  VERIFY_RAISES_ASSERT(ldlt.vectorD())
  VERIFY_RAISES_ASSERT(ldlt.isPositive())
  VERIFY_RAISES_ASSERT(ldlt.isNegative())
  VERIFY_RAISES_ASSERT(ldlt.solve(tmp))
  VERIFY_RAISES_ASSERT(ldlt.transpose().solve(tmp))
  VERIFY_RAISES_ASSERT(ldlt.adjoint().solve(tmp))
  VERIFY_RAISES_ASSERT(ldlt.solveInPlace(tmp))
}

EIGEN_DECLARE_TEST(cholesky) {
  int s = 0;
  for (int i = 0; i < g_repeat; i++) {
    CALL_SUBTEST_1(cholesky(Matrix<double, 1, 1>()));
    CALL_SUBTEST_3(cholesky(Matrix2d()));
    CALL_SUBTEST_3(cholesky_bug241(Matrix2d()));
    CALL_SUBTEST_3(cholesky_definiteness(Matrix2d()));
    CALL_SUBTEST_4(cholesky(Matrix3f()));
    CALL_SUBTEST_5(cholesky(Matrix4d()));

    s = internal::random<int>(1, EIGEN_TEST_MAX_SIZE);
    CALL_SUBTEST_2(cholesky(MatrixXd(s, s)));
    TEST_SET_BUT_UNUSED_VARIABLE(s)

    s = internal::random<int>(1, EIGEN_TEST_MAX_SIZE / 2);
    CALL_SUBTEST_6(cholesky_cplx(MatrixXcd(s, s)));
    TEST_SET_BUT_UNUSED_VARIABLE(s)
  }
  // empty matrix, regression test for Bug 785:
  CALL_SUBTEST_2(cholesky(MatrixXd(0, 0)));

  // This does not work yet:
  // CALL_SUBTEST_2( cholesky(Matrix<double,0,0>()) );

  CALL_SUBTEST_4(cholesky_verify_assert<Matrix3f>());
  CALL_SUBTEST_7(cholesky_verify_assert<Matrix3d>());
  CALL_SUBTEST_8(cholesky_verify_assert<MatrixXf>());
  CALL_SUBTEST_2(cholesky_verify_assert<MatrixXd>());

  // Test problem size constructors
  CALL_SUBTEST_9(LLT<MatrixXf>(10));
  CALL_SUBTEST_9(LDLT<MatrixXf>(10));

  CALL_SUBTEST_2(cholesky_faillure_cases<void>());

  TEST_SET_BUT_UNUSED_VARIABLE(nb_temporaries)
}