ManagedLapack.HermitianGeneralizedEigenvalueDecompose Method

public override void HermitianGeneralizedEigenvalueDecompose(
	int itype,
	char jobz,
	MatrixTriangle uplo,
	int n,
	Span<Complex<double>> a,
	int lda,
	Span<Complex<double>> b,
	int ldb,
	Span<double> w,
	out int info
)

Public Overrides Sub HermitianGeneralizedEigenvalueDecompose ( 
	itype As Integer,
	jobz As Char,
	uplo As MatrixTriangle,
	n As Integer,
	a As Span(Of Complex(Of Double)),
	lda As Integer,
	b As Span(Of Complex(Of Double)),
	ldb As Integer,
	w As Span(Of Double),
	<OutAttribute> ByRef info As Integer
)

public:
virtual void HermitianGeneralizedEigenvalueDecompose(
	int itype, 
	wchar_t jobz, 
	MatrixTriangle uplo, 
	int n, 
	Span<Complex<double>> a, 
	int lda, 
	Span<Complex<double>> b, 
	int ldb, 
	Span<double> w, 
	[OutAttribute] int% info
) override

abstract HermitianGeneralizedEigenvalueDecompose : 
        itype : int * 
        jobz : char * 
        uplo : MatrixTriangle * 
        n : int * 
        a : Span<Complex<float>> * 
        lda : int * 
        b : Span<Complex<float>> * 
        ldb : int * 
        w : Span<float> * 
        info : int byref -> unit 
override HermitianGeneralizedEigenvalueDecompose : 
        itype : int * 
        jobz : char * 
        uplo : MatrixTriangle * 
        n : int * 
        a : Span<Complex<float>> * 
        lda : int * 
        b : Span<Complex<float>> * 
        ldb : int * 
        w : Span<float> * 
        info : int byref -> unit 

Parameters

itype  Int32

ITYPE is INTEGER Specifies the problem type to be solved: = 1: A*x = (lambda)*B*x = 2: A*B*x = (lambda)*x = 3: B*A*x = (lambda)*x

jobz  Char

JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.

uplo  MatrixTriangle

UPLO is CHARACTER*1 = 'U': Upper triangles of A and B are stored; = 'L': Lower triangles of A and B are stored.

n  Int32

N is INTEGER The order of the matrices A and B. N >= 0.

a  Span<Complex<Double>>

A is COMPLEX*16 array, dimension (LDA, N) On entry, the Hermitian matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A. On exit, if JOBZ = 'V', then if INFO = 0, A contains the matrix Z of eigenvectors. The eigenvectors are normalized as follows: if ITYPE = 1 or 2, Z**H*B*Z = I; if ITYPE = 3, Z**H*inv(B)*Z = I. If JOBZ = 'N', then on exit the upper triangle (if UPLO='U') or the lower triangle (if UPLO='L') of A, including the diagonal, is destroyed.

lda  Int32

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

b  Span<Complex<Double>>

B is COMPLEX*16 array, dimension (LDB, N) On entry, the Hermitian matrix B. If UPLO = 'U', the leading N-by-N upper triangular part of B contains the upper triangular part of the matrix B. If UPLO = 'L', the leading N-by-N lower triangular part of B contains the lower triangular part of the matrix B. On exit, if INFO <= N, the part of B containing the matrix is overwritten by the triangular factor U or L from the Cholesky factorization B = U**H*U or B = L*L**H.

ldb  Int32

LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

w  Span<Double>

W is DOUBLE PRECISION array, dimension (N) If INFO = 0, the eigenvalues in ascending order.

info  Int32

INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: ZPOTRF or ZHEEVD returned an error code: <= N: if INFO = i and JOBZ = 'N', then the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero; if INFO = i and JOBZ = 'V', then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns INFO/(N+1) through mod(INFO,N+1); > N: if INFO = N + i, for 1 <= i <= N, then the leading minor of order i of B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.

Here A and B are assumed to be Hermitian and B is also positive definite. If eigenvectors are desired, it uses a divide and conquer algorithm. The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.

Further Details:

Modified so that no backsubstitution is performed if ZHEEVD fails to converge (NEIG in old code could be greater than N causing out of bounds reference to A - reported by Ralf Meyer). Also corrected the description of INFO and the test on ITYPE. Sven, 16 Feb 05.

Contributors:

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Authors: Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver, NAG Ltd.

Date: November 2015

Reference