// -*- tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 2 -*-
// vi: set et ts=4 sw=2 sts=2:
#ifndef DUNE_FMATRIXEIGENVALUES_HH
#define DUNE_FMATRIXEIGENVALUES_HH
/** \file
* \brief Eigenvalue computations for the FieldMatrix class
*/
#include <iostream>
#include <cmath>
#include <cassert>
#include <dune/common/exceptions.hh>
#include <dune/common/fvector.hh>
#include <dune/common/fmatrix.hh>
namespace Dune {
/**
@addtogroup DenseMatVec
@{
*/
namespace FMatrixHelp {
// defined in fmatrixev.cc
extern void eigenValuesLapackCall(
const char* jobz, const char* uplo, const long
int* n, double* a, const long int* lda, double* w,
double* work, const long int* lwork, long int* info);
extern void eigenValuesNonsymLapackCall(
const char* jobvl, const char* jobvr, const long
int* n, double* a, const long int* lda, double* wr, double* wi, double* vl,
const long int* ldvl, double* vr, const long int* ldvr, double* work,
const long int* lwork, const long int* info);
/** \brief calculates the eigenvalues of a symmetric field matrix
\param[in] matrix matrix eigenvalues are calculated for
\param[out] eigenvalues FieldVector that contains eigenvalues in
ascending order
*/
template <typename K>
static void eigenValues(const FieldMatrix<K, 1, 1>& matrix,
FieldVector<K, 1>& eigenvalues)
{
eigenvalues[0] = matrix[0][0];
}
/** \brief calculates the eigenvalues of a symmetric field matrix
\param[in] matrix matrix eigenvalues are calculated for
\param[out] eigenvalues FieldVector that contains eigenvalues in
ascending order
*/
template <typename K>
static void eigenValues(const FieldMatrix<K, 2, 2>& matrix,
FieldVector<K, 2>& eigenvalues)
{
const K detM = matrix[0][0] * matrix[1][1] - matrix[1][0] * matrix[0][1];
const K p = 0.5 * (matrix[0][0] + matrix [1][1]);
K q = p * p - detM;
if( q < 0 && q > -1e-14 ) q = 0;
if (p < 0 || q < 0)
{
std::cout << p << " p | q " << q << "\n";
std::cout << matrix << std::endl;
std::cout << "something went wrong in Eigenvalues for matrix!" << std::endl;
assert(false);
abort();
}
// get square root
q = std :: sqrt(q);
// store eigenvalues in ascending order
eigenvalues[0] = p - q;
eigenvalues[1] = p + q;
}
/** \brief Calculates the eigenvalues of a symmetric 3x3 field matrix
\param[in] matrix matrix eigenvalues are calculated for
\param[out] eigenvalues Eigenvalues in ascending order
\note If the input matrix is not symmetric the behavior of this method is undefined.
This implementation was adapted from the pseudo-code (Python?) implementation found on
http://en.wikipedia.org/wiki/Eigenvalue_algorithm (retrieved late August 2014).
Wikipedia claims to have taken it from
Smith, Oliver K. (April 1961), Eigenvalues of a symmetric 3 × 3 matrix.,
Communications of the ACM 4 (4): 168, doi:10.1145/355578.366316
*/
template <typename K>
static void eigenValues(const FieldMatrix<K, 3, 3>& matrix,
FieldVector<K, 3>& eigenvalues)
{
K p1 = matrix[0][1]*matrix[0][1] + matrix[0][2]*matrix[0][2] + matrix[1][2]*matrix[1][2];
if (p1 <= 1e-8)
{
// A is diagonal.
eigenvalues[0] = matrix[0][0];
eigenvalues[1] = matrix[1][1];
eigenvalues[2] = matrix[2][2];
}
else
{
// q = trace(A)/3
K q = 0;
for (int i=0; i<3; i++)
q += matrix[i][i]/3.0;
K p2 = (matrix[0][0] - q)*(matrix[0][0] - q) + (matrix[1][1] - q)*(matrix[1][1] - q) + (matrix[2][2] - q)*(matrix[2][2] - q) + 2 * p1;
K p = std::sqrt(p2 / 6);
// B = (1 / p) * (A - q * I); // I is the identity matrix
FieldMatrix<K,3,3> B;
for (int i=0; i<3; i++)
for (int j=0; j<3; j++)
B[i][j] = (1/p) * (matrix[i][j] - q*(i==j));
K r = B.determinant() / 2.0;
// In exact arithmetic for a symmetric matrix -1 <= r <= 1
// but computation error can leave it slightly outside this range.
K phi;
if (r <= -1)
phi = M_PI / 3.0;
else if (r >= 1)
phi = 0;
else
phi = std::acos(r) / 3;
// the eigenvalues satisfy eig[2] <= eig[1] <= eig[0]
eigenvalues[2] = q + 2 * p * cos(phi);
eigenvalues[0] = q + 2 * p * cos(phi + (2*M_PI/3));
eigenvalues[1] = 3 * q - eigenvalues[0] - eigenvalues[2]; // since trace(matrix) = eig1 + eig2 + eig3
}
}
/** \brief calculates the eigenvalues of a symmetric field matrix
\param[in] matrix matrix eigenvalues are calculated for
\param[out] eigenvalues FieldVector that contains eigenvalues in
ascending order
\note LAPACK::dsyev is used to calculate the eigenvalues
*/
template <int dim, typename K>
static void eigenValues(const FieldMatrix<K, dim, dim>& matrix,
FieldVector<K, dim>& eigenvalues)
{
{
const long int N = dim ;
const char jobz = 'n'; // only calculate eigenvalues
const char uplo = 'u'; // use upper triangular matrix
// length of matrix vector
const long int w = N * N ;
// matrix to put into dsyev
double matrixVector[dim * dim];
// copy matrix
int row = 0;
for(int i=0; i<dim; ++i)
{
for(int j=0; j<dim; ++j, ++row)
{
matrixVector[ row ] = matrix[ i ][ j ];
}
}
// working memory
double workSpace[dim * dim];
// return value information
long int info = 0;
// call LAPACK routine (see fmatrixev.cc)
eigenValuesLapackCall(&jobz, &uplo, &N, &matrixVector[0], &N,
&eigenvalues[0], &workSpace[0], &w, &info);
if( info != 0 )
{
std::cerr << "For matrix " << matrix << " eigenvalue calculation failed! " << std::endl;
DUNE_THROW(InvalidStateException,"eigenValues: Eigenvalue calculation failed!");
}
}
}
/** \brief calculates the eigenvalues of a symmetric field matrix
\param[in] matrix matrix eigenvalues are calculated for
\param[out] eigenValues FieldVector that contains eigenvalues in
ascending order
\note LAPACK::dgeev is used to calculate the eigen values
*/
template <int dim, typename K, class C>
static void eigenValuesNonSym(const FieldMatrix<K, dim, dim>& matrix,
FieldVector<C, dim>& eigenValues)
{
{
const long int N = dim ;
const char jobvl = 'n';
const char jobvr = 'n';
// matrix to put into dgeev
double matrixVector[dim * dim];
// copy matrix
int row = 0;
for(int i=0; i<dim; ++i)
{
for(int j=0; j<dim; ++j, ++row)
{
matrixVector[ row ] = matrix[ i ][ j ];
}
}
// working memory
double eigenR[dim];
double eigenI[dim];
double work[3*dim];
// return value information
long int info = 0;
long int lwork = 3*dim;
// call LAPACK routine (see fmatrixev_ext.cc)
eigenValuesNonsymLapackCall(&jobvl, &jobvr, &N, &matrixVector[0], &N,
&eigenR[0], &eigenI[0], 0, &N, 0, &N, &work[0],
&lwork, &info);
if( info != 0 )
{
std::cerr << "For matrix " << matrix << " eigenvalue calculation failed! " << std::endl;
DUNE_THROW(InvalidStateException,"eigenValues: Eigenvalue calculation failed!");
}
for (int i=0; i<N; ++i) {
eigenValues[i].real = eigenR[i];
eigenValues[i].imag = eigenI[i];
}
}
}
} // end namespace FMatrixHelp
/** @} end documentation */
} // end namespace Dune
#endif