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Rename Complex in ComplexSchur and ComplexEigenSolver to ComplexScalar

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for consistency with the RealScalar type; correct ComplexEigenSolver
docs to take non-diagonalizable matrices into account; refactor
ComplexEigenSolver::compute().
This commit is contained in:
Jitse Niesen 2010-03-21 21:57:34 +00:00
parent d91ffffc37
commit 8e5d2b6fc4
2 changed files with 68 additions and 62 deletions
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110
Eigen/src/Eigenvalues/ComplexEigenSolver.h
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@ -38,11 +38,12 @@
* instantiation of the Matrix class template. * instantiation of the Matrix class template.
* *
* The eigenvalues and eigenvectors of a matrix \f$ A \f$ are scalars * The eigenvalues and eigenvectors of a matrix \f$ A \f$ are scalars
* \f$ \lambda \f$ and vectors \f$ v \f$ such that \f$ Av = \lambda v \f$. * \f$ \lambda \f$ and vectors \f$ v \f$ such that \f$ Av = \lambda v
* The eigendecomposition of a matrix is \f$ A = V D V^{-1} \f$, * \f$. If \f$ D \f$ is a diagonal matrix with the eigenvalues on
* where \f$ D \f$ is a diagonal matrix. The entries on the diagonal * the diagonal, and \f$ V \f$ is a matrix with the eigenvectors as
* of \f$ D \f$ are the eigenvalues and the columns of \f$ V \f$ are * its columns, then \f$ A V = V D \f$. The matrix \f$ V \f$ is
* the eigenvectors. * almost always invertible, in which case we have \f$ A = V D V^{-1}
* \f$. This is called the eigendecomposition.
* *
* The main function in this class is compute(), which computes the * The main function in this class is compute(), which computes the
* eigenvalues and eigenvectors of a given function. The * eigenvalues and eigenvectors of a given function. The
@ -73,21 +74,21 @@ template<typename _MatrixType> class ComplexEigenSolver
* \c float or \c double) and just \c Scalar if #Scalar is * \c float or \c double) and just \c Scalar if #Scalar is
* complex. * complex.
*/ */
typedef std::complex<RealScalar> Complex; typedef std::complex<RealScalar> ComplexScalar;
/** \brief Type for vector of eigenvalues as returned by eigenvalues(). /** \brief Type for vector of eigenvalues as returned by eigenvalues().
* *
* This is a column vector with entries of type #Complex. * This is a column vector with entries of type #ComplexScalar.
* The length of the vector is the size of \p _MatrixType. * The length of the vector is the size of \p _MatrixType.
*/ */
typedef Matrix<Complex, ColsAtCompileTime, 1, Options, MaxColsAtCompileTime, 1> EigenvalueType; typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options, MaxColsAtCompileTime, 1> EigenvalueType;
/** \brief Type for matrix of eigenvectors as returned by eigenvectors(). /** \brief Type for matrix of eigenvectors as returned by eigenvectors().
* *
* This is a square matrix with entries of type #Complex. * This is a square matrix with entries of type #ComplexScalar.
* The size is the same as the size of \p _MatrixType. * The size is the same as the size of \p _MatrixType.
*/ */
typedef Matrix<Complex, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, ColsAtCompileTime> EigenvectorType; typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, ColsAtCompileTime> EigenvectorType;
/** \brief Default constructor. /** \brief Default constructor.
* *
@ -99,7 +100,7 @@ template<typename _MatrixType> class ComplexEigenSolver
/** \brief Constructor; computes eigendecomposition of given matrix. /** \brief Constructor; computes eigendecomposition of given matrix.
* *
* \param[in] matrix %Matrix whose eigendecomposition is to be computed. * \param[in] matrix Sqarae matrix whose eigendecomposition is to be computed.
* *
* This constructor calls compute() to compute the eigendecomposition. * This constructor calls compute() to compute the eigendecomposition.
*/ */
@ -117,11 +118,13 @@ template<typename _MatrixType> class ComplexEigenSolver
* ComplexEigenSolver(const MatrixType& matrix) or the member * ComplexEigenSolver(const MatrixType& matrix) or the member
* function compute(const MatrixType& matrix) has been called * function compute(const MatrixType& matrix) has been called
* before to compute the eigendecomposition of a matrix. This * before to compute the eigendecomposition of a matrix. This
* function returns the matrix \f$ V \f$ in the * function returns a matrix whose columns are the
* eigendecomposition \f$ A = V D V^{-1} \f$. The columns of \f$ * eigenvectors. Column \f$ k \f$ is an eigenvector
* V \f$ are the eigenvectors. The eigenvectors are normalized to * corresponding to eigenvalue number \f$ k \f$ as returned by
* have (Euclidean) norm equal to one, and are in the same order * eigenvalues(). The eigenvectors are normalized to have
* as the eigenvalues as returned by eigenvalues(). * (Euclidean) norm equal to one. The matrix returned by this
* function is the matrix \f$ V \f$ in the eigendecomposition \f$
* A = V D V^{-1} \f$, if it exists.
* *
* Example: \include ComplexEigenSolver_eigenvectors.cpp * Example: \include ComplexEigenSolver_eigenvectors.cpp
* Output: \verbinclude ComplexEigenSolver_eigenvectors.out * Output: \verbinclude ComplexEigenSolver_eigenvectors.out
@ -138,7 +141,10 @@ template<typename _MatrixType> class ComplexEigenSolver
* ComplexEigenSolver(const MatrixType& matrix) or the member * ComplexEigenSolver(const MatrixType& matrix) or the member
* function compute(const MatrixType& matrix) has been called * function compute(const MatrixType& matrix) has been called
* before to compute the eigendecomposition of a matrix. This * before to compute the eigendecomposition of a matrix. This
* function returns a column vector containing the eigenvalues. * function returns a column vector containing the
* eigenvalues. Eigenvalues are repeated according to their
* algebraic multiplicity, so there are as many eigenvalues as
* rows in the matrix.
* *
* Example: \include ComplexEigenSolver_eigenvalues.cpp * Example: \include ComplexEigenSolver_eigenvalues.cpp
* Output: \verbinclude ComplexEigenSolver_eigenvalues.out * Output: \verbinclude ComplexEigenSolver_eigenvalues.out
@ -151,7 +157,7 @@ template<typename _MatrixType> class ComplexEigenSolver
/** \brief Computes eigendecomposition of given matrix. /** \brief Computes eigendecomposition of given matrix.
* *
* \param[in] matrix %Matrix whose eigendecomposition is to be computed. * \param[in] matrix Square matrix whose eigendecomposition is to be computed.
* *
* This function computes the eigenvalues and eigenvectors of \p * This function computes the eigenvalues and eigenvectors of \p
* matrix. The eigenvalues() and eigenvectors() functions can be * matrix. The eigenvalues() and eigenvectors() functions can be
@ -182,56 +188,56 @@ void ComplexEigenSolver<MatrixType>::compute(const MatrixType& matrix)
{ {
// this code is inspired from Jampack // this code is inspired from Jampack
assert(matrix.cols() == matrix.rows()); assert(matrix.cols() == matrix.rows());
int n = matrix.cols(); const int n = matrix.cols();
m_eivalues.resize(n,1); const RealScalar matrixnorm = matrix.norm();
m_eivec.resize(n,n);
RealScalar eps = NumTraits<RealScalar>::epsilon(); // Step 1: Do a complex Schur decomposition, A = U T U^*
// The eigenvalues are on the diagonal of T.
// Reduce to complex Schur form
ComplexSchur<MatrixType> schur(matrix); ComplexSchur<MatrixType> schur(matrix);
m_eivalues = schur.matrixT().diagonal(); m_eivalues = schur.matrixT().diagonal();
m_eivec.setZero(); // Step 2: Compute X such that T = X D X^(-1), where D is the diagonal of T.
// The matrix X is unit triangular.
Complex d2, z; EigenvectorType X = EigenvectorType::Zero(n, n);
RealScalar norm = matrix.norm();
// compute the (normalized) eigenvectors
for(int k=n-1 ; k>=0 ; k--) for(int k=n-1 ; k>=0 ; k--)
{ {
d2 = schur.matrixT().coeff(k,k); X.coeffRef(k,k) = ComplexScalar(1.0,0.0);
m_eivec.coeffRef(k,k) = Complex(1.0,0.0); // Compute X(i,k) using the (i,k) entry of the equation X T = D X
for(int i=k-1 ; i>=0 ; i--) for(int i=k-1 ; i>=0 ; i--)
{ {
m_eivec.coeffRef(i,k) = -schur.matrixT().coeff(i,k); X.coeffRef(i,k) = -schur.matrixT().coeff(i,k);
if(k-i-1>0) if(k-i-1>0)
m_eivec.coeffRef(i,k) -= (schur.matrixT().row(i).segment(i+1,k-i-1) * m_eivec.col(k).segment(i+1,k-i-1)).value(); X.coeffRef(i,k) -= (schur.matrixT().row(i).segment(i+1,k-i-1) * X.col(k).segment(i+1,k-i-1)).value();
z = schur.matrixT().coeff(i,i) - d2; ComplexScalar z = schur.matrixT().coeff(i,i) - schur.matrixT().coeff(k,k);
if(z==Complex(0)) if(z==ComplexScalar(0))
ei_real_ref(z) = eps * norm; {
m_eivec.coeffRef(i,k) = m_eivec.coeff(i,k) / z; // If the i-th and k-th eigenvalue are equal, then z equals 0.
// Use a small value instead, to prevent division by zero.
ei_real_ref(z) = NumTraits<RealScalar>::epsilon() * matrixnorm;
}
X.coeffRef(i,k) = X.coeff(i,k) / z;
} }
m_eivec.col(k).normalize();
} }
m_eivec = schur.matrixU() * m_eivec; // Step 3: Compute V as V = U X; now A = U T U^* = U X D X^(-1) U^* = V D V^(-1)
m_eivec = schur.matrixU() * X;
// .. and normalize the eigenvectors
for(int k=0 ; k<n ; k++)
{
m_eivec.col(k).normalize();
}
m_isInitialized = true; m_isInitialized = true;
// sort the eigenvalues // Step 4: Sort the eigenvalues
for (int i=0; i<n; i++)
{ {
for (int i=0; i<n; i++) int k;
m_eivalues.cwiseAbs().tail(n-i).minCoeff(&k);
if (k != 0)
{ {
int k; k += i;
m_eivalues.cwiseAbs().tail(n-i).minCoeff(&k); std::swap(m_eivalues[k],m_eivalues[i]);
if (k != 0) m_eivec.col(i).swap(m_eivec.col(k));
{
k += i;
std::swap(m_eivalues[k],m_eivalues[i]);
m_eivec.col(i).swap(m_eivec.col(k));
}
} }
} }
} }

20
Eigen/src/Eigenvalues/ComplexSchur.h
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@ -54,8 +54,8 @@ template<typename _MatrixType> class ComplexSchur
}; };
typedef typename MatrixType::Scalar Scalar; typedef typename MatrixType::Scalar Scalar;
typedef typename NumTraits<Scalar>::Real RealScalar; typedef typename NumTraits<Scalar>::Real RealScalar;
typedef std::complex<RealScalar> Complex; typedef std::complex<RealScalar> ComplexScalar;
typedef Matrix<Complex, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> ComplexMatrixType; typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> ComplexMatrixType;
enum { enum {
Size = MatrixType::RowsAtCompileTime Size = MatrixType::RowsAtCompileTime
}; };
@ -158,8 +158,8 @@ void ComplexSchur<MatrixType>::compute(const MatrixType& matrix, bool skipU)
// TODO skip Q if skipU = true // TODO skip Q if skipU = true
HessenbergDecomposition<MatrixType> hess(matrix); HessenbergDecomposition<MatrixType> hess(matrix);
m_matT = hess.matrixH().template cast<Complex>(); m_matT = hess.matrixH().template cast<ComplexScalar>();
if(!skipU) m_matU = hess.matrixQ().template cast<Complex>(); if(!skipU) m_matU = hess.matrixQ().template cast<ComplexScalar>();
// Reduce the Hessenberg matrix m_matT to triangular form by QR iteration. // Reduce the Hessenberg matrix m_matT to triangular form by QR iteration.
@ -172,7 +172,7 @@ void ComplexSchur<MatrixType>::compute(const MatrixType& matrix, bool skipU)
int iu = m_matT.cols() - 1; int iu = m_matT.cols() - 1;
int il; int il;
RealScalar d,sd,sf; RealScalar d,sd,sf;
Complex c,b,disc,r1,r2,kappa; ComplexScalar c,b,disc,r1,r2,kappa;
RealScalar eps = NumTraits<RealScalar>::epsilon(); RealScalar eps = NumTraits<RealScalar>::epsilon();
@ -188,7 +188,7 @@ void ComplexSchur<MatrixType>::compute(const MatrixType& matrix, bool skipU)
if(!ei_isMuchSmallerThan(sd,d,eps)) if(!ei_isMuchSmallerThan(sd,d,eps))
break; break;
m_matT.coeffRef(iu,iu-1) = Complex(0); m_matT.coeffRef(iu,iu-1) = ComplexScalar(0);
iter = 0; iter = 0;
--iu; --iu;
} }
@ -216,12 +216,12 @@ void ComplexSchur<MatrixType>::compute(const MatrixType& matrix, bool skipU)
--il; --il;
} }
if( il != 0 ) m_matT.coeffRef(il,il-1) = Complex(0); if( il != 0 ) m_matT.coeffRef(il,il-1) = ComplexScalar(0);
// compute the shift kappa as one of the eigenvalues of the 2x2 // compute the shift kappa as one of the eigenvalues of the 2x2
// diagonal block on the bottom of the active submatrix // diagonal block on the bottom of the active submatrix
Matrix<Complex,2,2> t = m_matT.template block<2,2>(iu-1,iu-1); Matrix<ComplexScalar,2,2> t = m_matT.template block<2,2>(iu-1,iu-1);
sf = t.cwiseAbs().sum(); sf = t.cwiseAbs().sum();
t /= sf; // the normalization by sf is to avoid under/overflow t /= sf; // the normalization by sf is to avoid under/overflow
@ -251,7 +251,7 @@ void ComplexSchur<MatrixType>::compute(const MatrixType& matrix, bool skipU)
} }
// perform the QR step using Givens rotations // perform the QR step using Givens rotations
PlanarRotation<Complex> rot; PlanarRotation<ComplexScalar> rot;
rot.makeGivens(m_matT.coeff(il,il) - kappa, m_matT.coeff(il+1,il)); rot.makeGivens(m_matT.coeff(il,il) - kappa, m_matT.coeff(il+1,il));
for(int i=il ; i<iu ; i++) for(int i=il ; i<iu ; i++)
@ -266,7 +266,7 @@ void ComplexSchur<MatrixType>::compute(const MatrixType& matrix, bool skipU)
int i2 = i+2; int i2 = i+2;
rot.makeGivens(m_matT.coeffRef(i1,i), m_matT.coeffRef(i2,i), &m_matT.coeffRef(i1,i)); rot.makeGivens(m_matT.coeffRef(i1,i), m_matT.coeffRef(i2,i), &m_matT.coeffRef(i1,i));
m_matT.coeffRef(i2,i) = Complex(0); m_matT.coeffRef(i2,i) = ComplexScalar(0);
} }
} }
} }