// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2009-2011 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/.
#ifndef EIGEN_MATRIX_FUNCTION
#define EIGEN_MATRIX_FUNCTION
#include "StemFunction.h"
#include "MatrixFunctionAtomic.h"
namespace Eigen {
/** \ingroup MatrixFunctions_Module
* \brief Class for computing matrix functions.
* \tparam MatrixType type of the argument of the matrix function,
* expected to be an instantiation of the Matrix class template.
* \tparam AtomicType type for computing matrix function of atomic blocks.
* \tparam IsComplex used internally to select correct specialization.
*
* This class implements the Schur-Parlett algorithm for computing matrix functions. The spectrum of the
* matrix is divided in clustered of eigenvalues that lies close together. This class delegates the
* computation of the matrix function on every block corresponding to these clusters to an object of type
* \p AtomicType and uses these results to compute the matrix function of the whole matrix. The class
* \p AtomicType should have a \p compute() member function for computing the matrix function of a block.
*
* \sa class MatrixFunctionAtomic, class MatrixLogarithmAtomic
*/
template <typename MatrixType,
typename AtomicType,
int IsComplex = NumTraits<typename internal::traits<MatrixType>::Scalar>::IsComplex>
class MatrixFunction
{
public:
/** \brief Constructor.
*
* \param[in] A argument of matrix function, should be a square matrix.
* \param[in] atomic class for computing matrix function of atomic blocks.
*
* The class stores references to \p A and \p atomic, so they should not be
* changed (or destroyed) before compute() is called.
*/
MatrixFunction(const MatrixType& A, AtomicType& atomic);
/** \brief Compute the matrix function.
*
* \param[out] result the function \p f applied to \p A, as
* specified in the constructor.
*
* See MatrixBase::matrixFunction() for details on how this computation
* is implemented.
*/
template <typename ResultType>
void compute(ResultType &result);
};
/** \internal \ingroup MatrixFunctions_Module
* \brief Partial specialization of MatrixFunction for real matrices
*/
template <typename MatrixType, typename AtomicType>
class MatrixFunction<MatrixType, AtomicType, 0>
{
private:
typedef internal::traits<MatrixType> Traits;
typedef typename Traits::Scalar Scalar;
static const int Rows = Traits::RowsAtCompileTime;
static const int Cols = Traits::ColsAtCompileTime;
static const int Options = MatrixType::Options;
static const int MaxRows = Traits::MaxRowsAtCompileTime;
static const int MaxCols = Traits::MaxColsAtCompileTime;
typedef std::complex<Scalar> ComplexScalar;
typedef Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols> ComplexMatrix;
public:
/** \brief Constructor.
*
* \param[in] A argument of matrix function, should be a square matrix.
* \param[in] atomic class for computing matrix function of atomic blocks.
*/
MatrixFunction(const MatrixType& A, AtomicType& atomic) : m_A(A), m_atomic(atomic) { }
/** \brief Compute the matrix function.
*
* \param[out] result the function \p f applied to \p A, as
* specified in the constructor.
*
* This function converts the real matrix \c A to a complex matrix,
* uses MatrixFunction<MatrixType,1> and then converts the result back to
* a real matrix.
*/
template <typename ResultType>
void compute(ResultType& result)
{
ComplexMatrix CA = m_A.template cast<ComplexScalar>();
ComplexMatrix Cresult;
MatrixFunction<ComplexMatrix, AtomicType> mf(CA, m_atomic);
mf.compute(Cresult);
result = Cresult.real();
}
private:
typename internal::nested<MatrixType>::type m_A; /**< \brief Reference to argument of matrix function. */
AtomicType& m_atomic; /**< \brief Class for computing matrix function of atomic blocks. */
MatrixFunction& operator=(const MatrixFunction&);
};
/** \internal \ingroup MatrixFunctions_Module
* \brief Partial specialization of MatrixFunction for complex matrices
*/
template <typename MatrixType, typename AtomicType>
class MatrixFunction<MatrixType, AtomicType, 1>
{
private:
typedef internal::traits<MatrixType> Traits;
typedef typename MatrixType::Scalar Scalar;
typedef typename MatrixType::Index Index;
static const int RowsAtCompileTime = Traits::RowsAtCompileTime;
static const int ColsAtCompileTime = Traits::ColsAtCompileTime;
static const int Options = MatrixType::Options;
typedef typename NumTraits<Scalar>::Real RealScalar;
typedef Matrix<Scalar, Traits::RowsAtCompileTime, 1> VectorType;
typedef Matrix<Index, Traits::RowsAtCompileTime, 1> IntVectorType;
typedef Matrix<Index, Dynamic, 1> DynamicIntVectorType;
typedef std::list<Scalar> Cluster;
typedef std::list<Cluster> ListOfClusters;
typedef Matrix<Scalar, Dynamic, Dynamic, Options, RowsAtCompileTime, ColsAtCompileTime> DynMatrixType;
public:
MatrixFunction(const MatrixType& A, AtomicType& atomic);
template <typename ResultType> void compute(ResultType& result);
private:
void computeSchurDecomposition();
void partitionEigenvalues();
typename ListOfClusters::iterator findCluster(Scalar key);
void computeClusterSize();
void computeBlockStart();
void constructPermutation();
void permuteSchur();
void swapEntriesInSchur(Index index);
void computeBlockAtomic();
Block<MatrixType> block(MatrixType& A, Index i, Index j);
void computeOffDiagonal();
DynMatrixType solveTriangularSylvester(const DynMatrixType& A, const DynMatrixType& B, const DynMatrixType& C);
typename internal::nested<MatrixType>::type m_A; /**< \brief Reference to argument of matrix function. */
AtomicType& m_atomic; /**< \brief Class for computing matrix function of atomic blocks. */
MatrixType m_T; /**< \brief Triangular part of Schur decomposition */
MatrixType m_U; /**< \brief Unitary part of Schur decomposition */
MatrixType m_fT; /**< \brief %Matrix function applied to #m_T */
ListOfClusters m_clusters; /**< \brief Partition of eigenvalues into clusters of ei'vals "close" to each other */
DynamicIntVectorType m_eivalToCluster; /**< \brief m_eivalToCluster[i] = j means i-th ei'val is in j-th cluster */
DynamicIntVectorType m_clusterSize; /**< \brief Number of eigenvalues in each clusters */
DynamicIntVectorType m_blockStart; /**< \brief Row index at which block corresponding to i-th cluster starts */
IntVectorType m_permutation; /**< \brief Permutation which groups ei'vals in the same cluster together */
/** \brief Maximum distance allowed between eigenvalues to be considered "close".
*
* This is morally a \c static \c const \c Scalar, but only
* integers can be static constant class members in C++. The
* separation constant is set to 0.1, a value taken from the
* paper by Davies and Higham. */
static const RealScalar separation() { return static_cast<RealScalar>(0.1); }
MatrixFunction& operator=(const MatrixFunction&);
};
/** \brief Constructor.
*
* \param[in] A argument of matrix function, should be a square matrix.
* \param[in] atomic class for computing matrix function of atomic blocks.
*/
template <typename MatrixType, typename AtomicType>
MatrixFunction<MatrixType,AtomicType,1>::MatrixFunction(const MatrixType& A, AtomicType& atomic)
: m_A(A), m_atomic(atomic)
{
/* empty body */
}
/** \brief Compute the matrix function.
*
* \param[out] result the function \p f applied to \p A, as
* specified in the constructor.
*/
template <typename MatrixType, typename AtomicType>
template <typename ResultType>
void MatrixFunction<MatrixType,AtomicType,1>::compute(ResultType& result)
{
computeSchurDecomposition();
partitionEigenvalues();
computeClusterSize();
computeBlockStart();
constructPermutation();
permuteSchur();
computeBlockAtomic();
computeOffDiagonal();
result = m_U * (m_fT.template triangularView<Upper>() * m_U.adjoint());
}
/** \brief Store the Schur decomposition of #m_A in #m_T and #m_U */
template <typename MatrixType, typename AtomicType>
void MatrixFunction<MatrixType,AtomicType,1>::computeSchurDecomposition()
{
const ComplexSchur<MatrixType> schurOfA(m_A);
m_T = schurOfA.matrixT();
m_U = schurOfA.matrixU();
}
/** \brief Partition eigenvalues in clusters of ei'vals close to each other
*
* This function computes #m_clusters. This is a partition of the
* eigenvalues of #m_T in clusters, such that
* # Any eigenvalue in a certain cluster is at most separation() away
* from another eigenvalue in the same cluster.
* # The distance between two eigenvalues in different clusters is
* more than separation().
* The implementation follows Algorithm 4.1 in the paper of Davies
* and Higham.
*/
template <typename MatrixType, typename AtomicType>
void MatrixFunction<MatrixType,AtomicType,1>::partitionEigenvalues()
{
using std::abs;
const Index rows = m_T.rows();
VectorType diag = m_T.diagonal(); // contains eigenvalues of A
for (Index i=0; i<rows; ++i) {
// Find set containing diag(i), adding a new set if necessary
typename ListOfClusters::iterator qi = findCluster(diag(i));
if (qi == m_clusters.end()) {
Cluster l;
l.push_back(diag(i));
m_clusters.push_back(l);
qi = m_clusters.end();
--qi;
}
// Look for other element to add to the set
for (Index j=i+1; j<rows; ++j) {
if (abs(diag(j) - diag(i)) <= separation() && std::find(qi->begin(), qi->end(), diag(j)) == qi->end()) {
typename ListOfClusters::iterator qj = findCluster(diag(j));
if (qj == m_clusters.end()) {
qi->push_back(diag(j));
} else {
qi->insert(qi->end(), qj->begin(), qj->end());
m_clusters.erase(qj);
}
}
}
}
}
/** \brief Find cluster in #m_clusters containing some value
* \param[in] key Value to find
* \returns Iterator to cluster containing \c key, or
* \c m_clusters.end() if no cluster in m_clusters contains \c key.
*/
template <typename MatrixType, typename AtomicType>
typename MatrixFunction<MatrixType,AtomicType,1>::ListOfClusters::iterator MatrixFunction<MatrixType,AtomicType,1>::findCluster(Scalar key)
{
typename Cluster::iterator j;
for (typename ListOfClusters::iterator i = m_clusters.begin(); i != m_clusters.end(); ++i) {
j = std::find(i->begin(), i->end(), key);
if (j != i->end())
return i;
}
return m_clusters.end();
}
/** \brief Compute #m_clusterSize and #m_eivalToCluster using #m_clusters */
template <typename MatrixType, typename AtomicType>
void MatrixFunction<MatrixType,AtomicType,1>::computeClusterSize()
{
const Index rows = m_T.rows();
VectorType diag = m_T.diagonal();
const Index numClusters = static_cast<Index>(m_clusters.size());
m_clusterSize.setZero(numClusters);
m_eivalToCluster.resize(rows);
Index clusterIndex = 0;
for (typename ListOfClusters::const_iterator cluster = m_clusters.begin(); cluster != m_clusters.end(); ++cluster) {
for (Index i = 0; i < diag.rows(); ++i) {
if (std::find(cluster->begin(), cluster->end(), diag(i)) != cluster->end()) {
++m_clusterSize[clusterIndex];
m_eivalToCluster[i] = clusterIndex;
}
}
++clusterIndex;
}
}
/** \brief Compute #m_blockStart using #m_clusterSize */
template <typename MatrixType, typename AtomicType>
void MatrixFunction<MatrixType,AtomicType,1>::computeBlockStart()
{
m_blockStart.resize(m_clusterSize.rows());
m_blockStart(0) = 0;
for (Index i = 1; i < m_clusterSize.rows(); i++) {
m_blockStart(i) = m_blockStart(i-1) + m_clusterSize(i-1);
}
}
/** \brief Compute #m_permutation using #m_eivalToCluster and #m_blockStart */
template <typename MatrixType, typename AtomicType>
void MatrixFunction<MatrixType,AtomicType,1>::constructPermutation()
{
DynamicIntVectorType indexNextEntry = m_blockStart;
m_permutation.resize(m_T.rows());
for (Index i = 0; i < m_T.rows(); i++) {
Index cluster = m_eivalToCluster[i];
m_permutation[i] = indexNextEntry[cluster];
++indexNextEntry[cluster];
}
}
/** \brief Permute Schur decomposition in #m_U and #m_T according to #m_permutation */
template <typename MatrixType, typename AtomicType>
void MatrixFunction<MatrixType,AtomicType,1>::permuteSchur()
{
IntVectorType p = m_permutation;
for (Index i = 0; i < p.rows() - 1; i++) {
Index j;
for (j = i; j < p.rows(); j++) {
if (p(j) == i) break;
}
eigen_assert(p(j) == i);
for (Index k = j-1; k >= i; k--) {
swapEntriesInSchur(k);
std::swap(p.coeffRef(k), p.coeffRef(k+1));
}
}
}
/** \brief Swap rows \a index and \a index+1 in Schur decomposition in #m_U and #m_T */
template <typename MatrixType, typename AtomicType>
void MatrixFunction<MatrixType,AtomicType,1>::swapEntriesInSchur(Index index)
{
JacobiRotation<Scalar> rotation;
rotation.makeGivens(m_T(index, index+1), m_T(index+1, index+1) - m_T(index, index));
m_T.applyOnTheLeft(index, index+1, rotation.adjoint());
m_T.applyOnTheRight(index, index+1, rotation);
m_U.applyOnTheRight(index, index+1, rotation);
}
/** \brief Compute block diagonal part of #m_fT.
*
* This routine computes the matrix function applied to the block diagonal part of #m_T, with the blocking
* given by #m_blockStart. The matrix function of each diagonal block is computed by #m_atomic. The
* off-diagonal parts of #m_fT are set to zero.
*/
template <typename MatrixType, typename AtomicType>
void MatrixFunction<MatrixType,AtomicType,1>::computeBlockAtomic()
{
m_fT.resize(m_T.rows(), m_T.cols());
m_fT.setZero();
for (Index i = 0; i < m_clusterSize.rows(); ++i) {
block(m_fT, i, i) = m_atomic.compute(block(m_T, i, i));
}
}
/** \brief Return block of matrix according to blocking given by #m_blockStart */
template <typename MatrixType, typename AtomicType>
Block<MatrixType> MatrixFunction<MatrixType,AtomicType,1>::block(MatrixType& A, Index i, Index j)
{
return A.block(m_blockStart(i), m_blockStart(j), m_clusterSize(i), m_clusterSize(j));
}
/** \brief Compute part of #m_fT above block diagonal.
*
* This routine assumes that the block diagonal part of #m_fT (which
* equals the matrix function applied to #m_T) has already been computed and computes
* the part above the block diagonal. The part below the diagonal is
* zero, because #m_T is upper triangular.
*/
template <typename MatrixType, typename AtomicType>
void MatrixFunction<MatrixType,AtomicType,1>::computeOffDiagonal()
{
for (Index diagIndex = 1; diagIndex < m_clusterSize.rows(); diagIndex++) {
for (Index blockIndex = 0; blockIndex < m_clusterSize.rows() - diagIndex; blockIndex++) {
// compute (blockIndex, blockIndex+diagIndex) block
DynMatrixType A = block(m_T, blockIndex, blockIndex);
DynMatrixType B = -block(m_T, blockIndex+diagIndex, blockIndex+diagIndex);
DynMatrixType C = block(m_fT, blockIndex, blockIndex) * block(m_T, blockIndex, blockIndex+diagIndex);
C -= block(m_T, blockIndex, blockIndex+diagIndex) * block(m_fT, blockIndex+diagIndex, blockIndex+diagIndex);
for (Index k = blockIndex + 1; k < blockIndex + diagIndex; k++) {
C += block(m_fT, blockIndex, k) * block(m_T, k, blockIndex+diagIndex);
C -= block(m_T, blockIndex, k) * block(m_fT, k, blockIndex+diagIndex);
}
block(m_fT, blockIndex, blockIndex+diagIndex) = solveTriangularSylvester(A, B, C);
}
}
}
/** \brief Solve a triangular Sylvester equation AX + XB = C
*
* \param[in] A the matrix A; should be square and upper triangular
* \param[in] B the matrix B; should be square and upper triangular
* \param[in] C the matrix C; should have correct size.
*
* \returns the solution X.
*
* If A is m-by-m and B is n-by-n, then both C and X are m-by-n.
* The (i,j)-th component of the Sylvester equation is
* \f[
* \sum_{k=i}^m A_{ik} X_{kj} + \sum_{k=1}^j X_{ik} B_{kj} = C_{ij}.
* \f]
* This can be re-arranged to yield:
* \f[
* X_{ij} = \frac{1}{A_{ii} + B_{jj}} \Bigl( C_{ij}
* - \sum_{k=i+1}^m A_{ik} X_{kj} - \sum_{k=1}^{j-1} X_{ik} B_{kj} \Bigr).
* \f]
* It is assumed that A and B are such that the numerator is never
* zero (otherwise the Sylvester equation does not have a unique
* solution). In that case, these equations can be evaluated in the
* order \f$ i=m,\ldots,1 \f$ and \f$ j=1,\ldots,n \f$.
*/
template <typename MatrixType, typename AtomicType>
typename MatrixFunction<MatrixType,AtomicType,1>::DynMatrixType MatrixFunction<MatrixType,AtomicType,1>::solveTriangularSylvester(
const DynMatrixType& A,
const DynMatrixType& B,
const DynMatrixType& C)
{
eigen_assert(A.rows() == A.cols());
eigen_assert(A.isUpperTriangular());
eigen_assert(B.rows() == B.cols());
eigen_assert(B.isUpperTriangular());
eigen_assert(C.rows() == A.rows());
eigen_assert(C.cols() == B.rows());
Index m = A.rows();
Index n = B.rows();
DynMatrixType X(m, n);
for (Index i = m - 1; i >= 0; --i) {
for (Index j = 0; j < n; ++j) {
// Compute AX = \sum_{k=i+1}^m A_{ik} X_{kj}
Scalar AX;
if (i == m - 1) {
AX = 0;
} else {
Matrix<Scalar,1,1> AXmatrix = A.row(i).tail(m-1-i) * X.col(j).tail(m-1-i);
AX = AXmatrix(0,0);
}
// Compute XB = \sum_{k=1}^{j-1} X_{ik} B_{kj}
Scalar XB;
if (j == 0) {
XB = 0;
} else {
Matrix<Scalar,1,1> XBmatrix = X.row(i).head(j) * B.col(j).head(j);
XB = XBmatrix(0,0);
}
X(i,j) = (C(i,j) - AX - XB) / (A(i,i) + B(j,j));
}
}
return X;
}
/** \ingroup MatrixFunctions_Module
*
* \brief Proxy for the matrix function of some matrix (expression).
*
* \tparam Derived Type of the argument to the matrix function.
*
* This class holds the argument to the matrix function until it is
* assigned or evaluated for some other reason (so the argument
* should not be changed in the meantime). It is the return type of
* matrixBase::matrixFunction() and related functions and most of the
* time this is the only way it is used.
*/
template<typename Derived> class MatrixFunctionReturnValue
: public ReturnByValue<MatrixFunctionReturnValue<Derived> >
{
public:
typedef typename Derived::Scalar Scalar;
typedef typename Derived::Index Index;
typedef typename internal::stem_function<Scalar>::type StemFunction;
/** \brief Constructor.
*
* \param[in] A %Matrix (expression) forming the argument of the
* matrix function.
* \param[in] f Stem function for matrix function under consideration.
*/
MatrixFunctionReturnValue(const Derived& A, StemFunction f) : m_A(A), m_f(f) { }
/** \brief Compute the matrix function.
*
* \param[out] result \p f applied to \p A, where \p f and \p A
* are as in the constructor.
*/
template <typename ResultType>
inline void evalTo(ResultType& result) const
{
typedef typename Derived::PlainObject PlainObject;
typedef internal::traits<PlainObject> Traits;
static const int RowsAtCompileTime = Traits::RowsAtCompileTime;
static const int ColsAtCompileTime = Traits::ColsAtCompileTime;
static const int Options = PlainObject::Options;
typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
typedef Matrix<ComplexScalar, Dynamic, Dynamic, Options, RowsAtCompileTime, ColsAtCompileTime> DynMatrixType;
typedef MatrixFunctionAtomic<DynMatrixType> AtomicType;
AtomicType atomic(m_f);
const PlainObject Aevaluated = m_A.eval();
MatrixFunction<PlainObject, AtomicType> mf(Aevaluated, atomic);
mf.compute(result);
}
Index rows() const { return m_A.rows(); }
Index cols() const { return m_A.cols(); }
private:
typename internal::nested<Derived>::type m_A;
StemFunction *m_f;
MatrixFunctionReturnValue& operator=(const MatrixFunctionReturnValue&);
};
namespace internal {
template<typename Derived>
struct traits<MatrixFunctionReturnValue<Derived> >
{
typedef typename Derived::PlainObject ReturnType;
};
}
/********** MatrixBase methods **********/
template <typename Derived>
const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::matrixFunction(typename internal::stem_function<typename internal::traits<Derived>::Scalar>::type f) const
{
eigen_assert(rows() == cols());
return MatrixFunctionReturnValue<Derived>(derived(), f);
}
template <typename Derived>
const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::sin() const
{
eigen_assert(rows() == cols());
typedef typename internal::stem_function<Scalar>::ComplexScalar ComplexScalar;
return MatrixFunctionReturnValue<Derived>(derived(), StdStemFunctions<ComplexScalar>::sin);
}
template <typename Derived>
const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::cos() const
{
eigen_assert(rows() == cols());
typedef typename internal::stem_function<Scalar>::ComplexScalar ComplexScalar;
return MatrixFunctionReturnValue<Derived>(derived(), StdStemFunctions<ComplexScalar>::cos);
}
template <typename Derived>
const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::sinh() const
{
eigen_assert(rows() == cols());
typedef typename internal::stem_function<Scalar>::ComplexScalar ComplexScalar;
return MatrixFunctionReturnValue<Derived>(derived(), StdStemFunctions<ComplexScalar>::sinh);
}
template <typename Derived>
const MatrixFunctionReturnValue<Derived> MatrixBase<Derived>::cosh() const
{
eigen_assert(rows() == cols());
typedef typename internal::stem_function<Scalar>::ComplexScalar ComplexScalar;
return MatrixFunctionReturnValue<Derived>(derived(), StdStemFunctions<ComplexScalar>::cosh);
}
} // end namespace Eigen
#endif // EIGEN_MATRIX_FUNCTION