InverseImpl.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008-2010 Benoit Jacob <jacob.benoit.1@gmail.com>
// Copyright (C) 2014 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/.

#ifndef EIGEN_INVERSE_IMPL_H
#define EIGEN_INVERSE_IMPL_H

// IWYU pragma: private
#include "./InternalHeaderCheck.h"

namespace Eigen {

namespace internal {

/**********************************
*** General case implementation ***
**********************************/

template <typename MatrixType, typename ResultType, int Size = MatrixType::RowsAtCompileTime>
struct compute_inverse {
  EIGEN_DEVICE_FUNC static inline void run(const MatrixType& matrix, ResultType& result) {
    result = matrix.partialPivLu().inverse();
  }
};

template <typename MatrixType, typename ResultType, int Size = MatrixType::RowsAtCompileTime>
struct compute_inverse_and_det_with_check { /* nothing! general case not supported. */
};

/****************************
*** Size 1 implementation ***
****************************/

template <typename MatrixType, typename ResultType>
struct compute_inverse<MatrixType, ResultType, 1> {
  EIGEN_DEVICE_FUNC static inline void run(const MatrixType& matrix, ResultType& result) {
    typedef typename MatrixType::Scalar Scalar;
    internal::evaluator<MatrixType> matrixEval(matrix);
    result.coeffRef(0, 0) = Scalar(1) / matrixEval.coeff(0, 0);
  }
};

template <typename MatrixType, typename ResultType>
struct compute_inverse_and_det_with_check<MatrixType, ResultType, 1> {
  EIGEN_DEVICE_FUNC static inline void run(const MatrixType& matrix,
                                           const typename MatrixType::RealScalar& absDeterminantThreshold,
                                           ResultType& result, typename ResultType::Scalar& determinant,
                                           bool& invertible) {
    using std::abs;
    determinant = matrix.coeff(0, 0);
    invertible = abs(determinant) > absDeterminantThreshold;
    if (invertible) result.coeffRef(0, 0) = typename ResultType::Scalar(1) / determinant;
  }
};

/****************************
*** Size 2 implementation ***
****************************/

template <typename MatrixType, typename ResultType>
EIGEN_DEVICE_FUNC inline void compute_inverse_size2_helper(const MatrixType& matrix,
                                                           const typename ResultType::Scalar& invdet,
                                                           ResultType& result) {
  typename ResultType::Scalar temp = matrix.coeff(0, 0);
  result.coeffRef(0, 0) = matrix.coeff(1, 1) * invdet;
  result.coeffRef(1, 0) = -matrix.coeff(1, 0) * invdet;
  result.coeffRef(0, 1) = -matrix.coeff(0, 1) * invdet;
  result.coeffRef(1, 1) = temp * invdet;
}

template <typename MatrixType, typename ResultType>
struct compute_inverse<MatrixType, ResultType, 2> {
  EIGEN_DEVICE_FUNC static inline void run(const MatrixType& matrix, ResultType& result) {
    typedef typename ResultType::Scalar Scalar;
    const Scalar invdet = typename MatrixType::Scalar(1) / matrix.determinant();
    compute_inverse_size2_helper(matrix, invdet, result);
  }
};

template <typename MatrixType, typename ResultType>
struct compute_inverse_and_det_with_check<MatrixType, ResultType, 2> {
  EIGEN_DEVICE_FUNC static inline void run(const MatrixType& matrix,
                                           const typename MatrixType::RealScalar& absDeterminantThreshold,
                                           ResultType& inverse, typename ResultType::Scalar& determinant,
                                           bool& invertible) {
    using std::abs;
    typedef typename ResultType::Scalar Scalar;
    determinant = matrix.determinant();
    invertible = abs(determinant) > absDeterminantThreshold;
    if (!invertible) return;
    const Scalar invdet = Scalar(1) / determinant;
    compute_inverse_size2_helper(matrix, invdet, inverse);
  }
};

/****************************
*** Size 3 implementation ***
****************************/

template <typename MatrixType, int i, int j>
EIGEN_DEVICE_FUNC inline typename MatrixType::Scalar cofactor_3x3(const MatrixType& m) {
  enum { i1 = (i + 1) % 3, i2 = (i + 2) % 3, j1 = (j + 1) % 3, j2 = (j + 2) % 3 };
  return m.coeff(i1, j1) * m.coeff(i2, j2) - m.coeff(i1, j2) * m.coeff(i2, j1);
}

template <typename MatrixType, typename ResultType>
EIGEN_DEVICE_FUNC inline void compute_inverse_size3_helper(
    const MatrixType& matrix, const typename ResultType::Scalar& invdet,
    const Matrix<typename ResultType::Scalar, 3, 1>& cofactors_col0, ResultType& result) {
  // Compute cofactors in a way that avoids aliasing issues.
  typedef typename ResultType::Scalar Scalar;
  const Scalar c01 = cofactor_3x3<MatrixType, 0, 1>(matrix) * invdet;
  const Scalar c11 = cofactor_3x3<MatrixType, 1, 1>(matrix) * invdet;
  const Scalar c02 = cofactor_3x3<MatrixType, 0, 2>(matrix) * invdet;
  result.coeffRef(1, 2) = cofactor_3x3<MatrixType, 2, 1>(matrix) * invdet;
  result.coeffRef(2, 1) = cofactor_3x3<MatrixType, 1, 2>(matrix) * invdet;
  result.coeffRef(2, 2) = cofactor_3x3<MatrixType, 2, 2>(matrix) * invdet;
  result.coeffRef(1, 0) = c01;
  result.coeffRef(1, 1) = c11;
  result.coeffRef(2, 0) = c02;
  result.row(0) = cofactors_col0 * invdet;
}

template <typename MatrixType, typename ResultType>
struct compute_inverse<MatrixType, ResultType, 3> {
  EIGEN_DEVICE_FUNC static inline void run(const MatrixType& matrix, ResultType& result) {
    typedef typename ResultType::Scalar Scalar;
    Matrix<typename MatrixType::Scalar, 3, 1> cofactors_col0;
    cofactors_col0.coeffRef(0) = cofactor_3x3<MatrixType, 0, 0>(matrix);
    cofactors_col0.coeffRef(1) = cofactor_3x3<MatrixType, 1, 0>(matrix);
    cofactors_col0.coeffRef(2) = cofactor_3x3<MatrixType, 2, 0>(matrix);
    const Scalar det = (cofactors_col0.cwiseProduct(matrix.col(0))).sum();
    const Scalar invdet = Scalar(1) / det;
    compute_inverse_size3_helper(matrix, invdet, cofactors_col0, result);
  }
};

template <typename MatrixType, typename ResultType>
struct compute_inverse_and_det_with_check<MatrixType, ResultType, 3> {
  EIGEN_DEVICE_FUNC static inline void run(const MatrixType& matrix,
                                           const typename MatrixType::RealScalar& absDeterminantThreshold,
                                           ResultType& inverse, typename ResultType::Scalar& determinant,
                                           bool& invertible) {
    typedef typename ResultType::Scalar Scalar;
    Matrix<Scalar, 3, 1> cofactors_col0;
    cofactors_col0.coeffRef(0) = cofactor_3x3<MatrixType, 0, 0>(matrix);
    cofactors_col0.coeffRef(1) = cofactor_3x3<MatrixType, 1, 0>(matrix);
    cofactors_col0.coeffRef(2) = cofactor_3x3<MatrixType, 2, 0>(matrix);
    determinant = (cofactors_col0.cwiseProduct(matrix.col(0))).sum();
    invertible = Eigen::numext::abs(determinant) > absDeterminantThreshold;
    if (!invertible) return;
    const Scalar invdet = Scalar(1) / determinant;
    compute_inverse_size3_helper(matrix, invdet, cofactors_col0, inverse);
  }
};

/****************************
*** Size 4 implementation ***
****************************/

template <typename Derived>
EIGEN_DEVICE_FUNC inline const typename Derived::Scalar general_det3_helper(const MatrixBase<Derived>& matrix, int i1,
                                                                            int i2, int i3, int j1, int j2, int j3) {
  return matrix.coeff(i1, j1) *
         (matrix.coeff(i2, j2) * matrix.coeff(i3, j3) - matrix.coeff(i2, j3) * matrix.coeff(i3, j2));
}

template <typename MatrixType, int i, int j>
EIGEN_DEVICE_FUNC inline typename MatrixType::Scalar cofactor_4x4(const MatrixType& matrix) {
  enum { i1 = (i + 1) % 4, i2 = (i + 2) % 4, i3 = (i + 3) % 4, j1 = (j + 1) % 4, j2 = (j + 2) % 4, j3 = (j + 3) % 4 };
  return general_det3_helper(matrix, i1, i2, i3, j1, j2, j3) + general_det3_helper(matrix, i2, i3, i1, j1, j2, j3) +
         general_det3_helper(matrix, i3, i1, i2, j1, j2, j3);
}

template <int Arch, typename Scalar, typename MatrixType, typename ResultType>
struct compute_inverse_size4 {
  EIGEN_DEVICE_FUNC static void run(const MatrixType& matrix, ResultType& result) {
    result.coeffRef(0, 0) = cofactor_4x4<MatrixType, 0, 0>(matrix);
    result.coeffRef(1, 0) = -cofactor_4x4<MatrixType, 0, 1>(matrix);
    result.coeffRef(2, 0) = cofactor_4x4<MatrixType, 0, 2>(matrix);
    result.coeffRef(3, 0) = -cofactor_4x4<MatrixType, 0, 3>(matrix);
    result.coeffRef(0, 2) = cofactor_4x4<MatrixType, 2, 0>(matrix);
    result.coeffRef(1, 2) = -cofactor_4x4<MatrixType, 2, 1>(matrix);
    result.coeffRef(2, 2) = cofactor_4x4<MatrixType, 2, 2>(matrix);
    result.coeffRef(3, 2) = -cofactor_4x4<MatrixType, 2, 3>(matrix);
    result.coeffRef(0, 1) = -cofactor_4x4<MatrixType, 1, 0>(matrix);
    result.coeffRef(1, 1) = cofactor_4x4<MatrixType, 1, 1>(matrix);
    result.coeffRef(2, 1) = -cofactor_4x4<MatrixType, 1, 2>(matrix);
    result.coeffRef(3, 1) = cofactor_4x4<MatrixType, 1, 3>(matrix);
    result.coeffRef(0, 3) = -cofactor_4x4<MatrixType, 3, 0>(matrix);
    result.coeffRef(1, 3) = cofactor_4x4<MatrixType, 3, 1>(matrix);
    result.coeffRef(2, 3) = -cofactor_4x4<MatrixType, 3, 2>(matrix);
    result.coeffRef(3, 3) = cofactor_4x4<MatrixType, 3, 3>(matrix);
    result /= (matrix.col(0).cwiseProduct(result.row(0).transpose())).sum();
  }
};

template <typename MatrixType, typename ResultType>
struct compute_inverse<MatrixType, ResultType, 4>
    : compute_inverse_size4<Architecture::Target, typename MatrixType::Scalar, MatrixType, ResultType> {};

template <typename MatrixType, typename ResultType>
struct compute_inverse_and_det_with_check<MatrixType, ResultType, 4> {
  EIGEN_DEVICE_FUNC static inline void run(const MatrixType& matrix,
                                           const typename MatrixType::RealScalar& absDeterminantThreshold,
                                           ResultType& inverse, typename ResultType::Scalar& determinant,
                                           bool& invertible) {
    using std::abs;
    determinant = matrix.determinant();
    invertible = abs(determinant) > absDeterminantThreshold;
    if (invertible && extract_data(matrix) != extract_data(inverse)) {
      compute_inverse<MatrixType, ResultType>::run(matrix, inverse);
    } else if (invertible) {
      MatrixType matrix_t = matrix;
      compute_inverse<MatrixType, ResultType>::run(matrix_t, inverse);
    }
  }
};

/*************************
*** MatrixBase methods ***
*************************/

}  // end namespace internal

namespace internal {

// Specialization for "dense = dense_xpr.inverse()"
template <typename DstXprType, typename XprType>
struct Assignment<DstXprType, Inverse<XprType>,
                  internal::assign_op<typename DstXprType::Scalar, typename XprType::Scalar>, Dense2Dense> {
  typedef Inverse<XprType> SrcXprType;
  EIGEN_DEVICE_FUNC static void run(DstXprType& dst, const SrcXprType& src,
                                    const internal::assign_op<typename DstXprType::Scalar, typename XprType::Scalar>&) {
    Index dstRows = src.rows();
    Index dstCols = src.cols();
    if ((dst.rows() != dstRows) || (dst.cols() != dstCols)) dst.resize(dstRows, dstCols);

    const int Size = plain_enum_min(XprType::ColsAtCompileTime, DstXprType::ColsAtCompileTime);
    EIGEN_ONLY_USED_FOR_DEBUG(Size);
    eigen_assert(((Size <= 1) || (Size > 4) || (extract_data(src.nestedExpression()) != extract_data(dst))) &&
                 "Aliasing problem detected in inverse(), you need to do inverse().eval() here.");

    typedef typename internal::nested_eval<XprType, XprType::ColsAtCompileTime>::type ActualXprType;
    typedef internal::remove_all_t<ActualXprType> ActualXprTypeCleanded;

    ActualXprType actual_xpr(src.nestedExpression());

    compute_inverse<ActualXprTypeCleanded, DstXprType>::run(actual_xpr, dst);
  }
};

}  // end namespace internal

template <typename Derived>
EIGEN_DEVICE_FUNC inline const Inverse<Derived> MatrixBase<Derived>::inverse() const {
  EIGEN_STATIC_ASSERT(!NumTraits<Scalar>::IsInteger, THIS_FUNCTION_IS_NOT_FOR_INTEGER_NUMERIC_TYPES)
  eigen_assert(rows() == cols());
  return Inverse<Derived>(derived());
}

template <typename Derived>
template <typename ResultType>
inline void MatrixBase<Derived>::computeInverseAndDetWithCheck(ResultType& inverse,
                                                               typename ResultType::Scalar& determinant,
                                                               bool& invertible,
                                                               const RealScalar& absDeterminantThreshold) const {
  // i'd love to put some static assertions there, but SFINAE means that they have no effect...
  eigen_assert(rows() == cols());
  // for 2x2, it's worth giving a chance to avoid evaluating.
  // for larger sizes, evaluating has negligible cost and limits code size.
  typedef std::conditional_t<RowsAtCompileTime == 2,
                             internal::remove_all_t<typename internal::nested_eval<Derived, 2>::type>, PlainObject>
      MatrixType;
  internal::compute_inverse_and_det_with_check<MatrixType, ResultType>::run(derived(), absDeterminantThreshold, inverse,
                                                                            determinant, invertible);
}

template <typename Derived>
template <typename ResultType>
inline void MatrixBase<Derived>::computeInverseWithCheck(ResultType& inverse, bool& invertible,
                                                         const RealScalar& absDeterminantThreshold) const {
  Scalar determinant;
  // i'd love to put some static assertions there, but SFINAE means that they have no effect...
  eigen_assert(rows() == cols());
  computeInverseAndDetWithCheck(inverse, determinant, invertible, absDeterminantThreshold);
}

}  // end namespace Eigen

#endif  // EIGEN_INVERSE_IMPL_H