Generic Decomposition Operations<T>.Symmetric Generalized Eigenvalue Decompose Method
Definition
Assembly: Numerics.NET.Generic (in Numerics.NET.Generic.dll) Version: 9.0.1
Overload List
Symmetric | Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. |
Symmetric | Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. |
Symmetric | Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. |
SymmetricGeneralizedEigenvalueDecompose(Int32, Char, MatrixTriangle, Int32, Span<T>, Int32, Span<T>, Int32, Span<T>, Int32)
Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x.
public override void SymmetricGeneralizedEigenvalueDecompose(
int itype,
char jobz,
MatrixTriangle uplo,
int n,
Span<T> a,
int lda,
Span<T> b,
int ldb,
Span<T> w,
out int info
)
Parameters
- itype Int32
-
C# 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
-
C# JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.
- uplo MatrixTriangle
-
C# UPLO is CHARACTER*1 = 'U': Upper triangles of A and B are stored; = 'L': Lower triangles of A and B are stored.
- n Int32
-
C# N is INTEGER The order of the matrices A and B. N >= 0.
- a Span<T>
-
C# A is DOUBLE PRECISION array, dimension (LDA, N) On entry, the symmetric 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**T*B*Z = I; if ITYPE = 3, Z**T*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
-
C# LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).
- b Span<T>
-
C# B is DOUBLE PRECISION array, dimension (LDB, N) On entry, the symmetric 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**T*U or B = L*L**T.
- ldb Int32
-
C# LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).
- w Span<T>
-
C# W is DOUBLE PRECISION array, dimension (N) If INFO = 0, the eigenvalues in ascending order.
- info Int32
-
C# INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: DPOTRF or DSYEVD 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.
Remarks
Here A and
B are assumed to be symmetric 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 DSYEVD 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