ztgsna (l) - Linux Manuals
ztgsna: estimates reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a matrix pair (A, B)
NAME
ZTGSNA - estimates reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a matrix pair (A, B)SYNOPSIS
- SUBROUTINE ZTGSNA(
- JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, IWORK, INFO )
- CHARACTER HOWMNY, JOB
- INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N
- LOGICAL SELECT( * )
- INTEGER IWORK( * )
- DOUBLE PRECISION DIF( * ), S( * )
- COMPLEX*16 A( LDA, * ), B( LDB, * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )
PURPOSE
ZTGSNA estimates reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a matrix pair (A, B). (A, B) must be in generalized Schur canonical form, that is, A and B are both upper triangular.ARGUMENTS
- JOB (input) CHARACTER*1
-
Specifies whether condition numbers are required for
eigenvalues (S) or eigenvectors (DIF):
= aqEaq: for eigenvalues only (S);
= aqVaq: for eigenvectors only (DIF);
= aqBaq: for both eigenvalues and eigenvectors (S and DIF). - HOWMNY (input) CHARACTER*1
-
= aqAaq: compute condition numbers for all eigenpairs;
= aqSaq: compute condition numbers for selected eigenpairs specified by the array SELECT. - SELECT (input) LOGICAL array, dimension (N)
- If HOWMNY = aqSaq, SELECT specifies the eigenpairs for which condition numbers are required. To select condition numbers for the corresponding j-th eigenvalue and/or eigenvector, SELECT(j) must be set to .TRUE.. If HOWMNY = aqAaq, SELECT is not referenced.
- N (input) INTEGER
- The order of the square matrix pair (A, B). N >= 0.
- A (input) COMPLEX*16 array, dimension (LDA,N)
- The upper triangular matrix A in the pair (A,B).
- LDA (input) INTEGER
- The leading dimension of the array A. LDA >= max(1,N).
- B (input) COMPLEX*16 array, dimension (LDB,N)
- The upper triangular matrix B in the pair (A, B).
- LDB (input) INTEGER
- The leading dimension of the array B. LDB >= max(1,N).
- VL (input) COMPLEX*16 array, dimension (LDVL,M)
- IF JOB = aqEaq or aqBaq, VL must contain left eigenvectors of (A, B), corresponding to the eigenpairs specified by HOWMNY and SELECT. The eigenvectors must be stored in consecutive columns of VL, as returned by ZTGEVC. If JOB = aqVaq, VL is not referenced.
- LDVL (input) INTEGER
- The leading dimension of the array VL. LDVL >= 1; and If JOB = aqEaq or aqBaq, LDVL >= N.
- VR (input) COMPLEX*16 array, dimension (LDVR,M)
- IF JOB = aqEaq or aqBaq, VR must contain right eigenvectors of (A, B), corresponding to the eigenpairs specified by HOWMNY and SELECT. The eigenvectors must be stored in consecutive columns of VR, as returned by ZTGEVC. If JOB = aqVaq, VR is not referenced.
- LDVR (input) INTEGER
- The leading dimension of the array VR. LDVR >= 1; If JOB = aqEaq or aqBaq, LDVR >= N.
- S (output) DOUBLE PRECISION array, dimension (MM)
- If JOB = aqEaq or aqBaq, the reciprocal condition numbers of the selected eigenvalues, stored in consecutive elements of the array. If JOB = aqVaq, S is not referenced.
- DIF (output) DOUBLE PRECISION array, dimension (MM)
- If JOB = aqVaq or aqBaq, the estimated reciprocal condition numbers of the selected eigenvectors, stored in consecutive elements of the array. If the eigenvalues cannot be reordered to compute DIF(j), DIF(j) is set to 0; this can only occur when the true value would be very small anyway. For each eigenvalue/vector specified by SELECT, DIF stores a Frobenius norm-based estimate of Difl. If JOB = aqEaq, DIF is not referenced.
- MM (input) INTEGER
- The number of elements in the arrays S and DIF. MM >= M.
- M (output) INTEGER
- The number of elements of the arrays S and DIF used to store the specified condition numbers; for each selected eigenvalue one element is used. If HOWMNY = aqAaq, M is set to N.
- WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
- On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
- LWORK (input) INTEGER
- The dimension of the array WORK. LWORK >= max(1,N). If JOB = aqVaq or aqBaq, LWORK >= max(1,2*N*N).
- IWORK (workspace) INTEGER array, dimension (N+2)
- If JOB = aqEaq, IWORK is not referenced.
- INFO (output) INTEGER
-
= 0: Successful exit
< 0: If INFO = -i, the i-th argument had an illegal value
FURTHER DETAILS
The reciprocal of the condition number of the i-th generalized eigenvalue w = (a, b) is defined asAn approximate error bound on the chordal distance between the i-th computed generalized eigenvalue w and the corresponding exact eigenvalue lambda is
where EPS is the machine precision.
The reciprocal of the condition number of the right eigenvector u and left eigenvector v corresponding to the generalized eigenvalue w is defined as follows. Suppose
Then the reciprocal condition number DIF(I) is
where sigma-min(Zl) denotes the smallest singular value of
Here In-1 is the identity matrix of size n-1 and Xaq is the conjugate transpose of X. kron(X, Y) is the Kronecker product between the matrices X and Y.
We approximate the smallest singular value of Zl with an upper bound. This is done by ZLATDF.
An approximate error bound for a computed eigenvector VL(i) or VR(i) is given by
See ref. [2-3] for more details and further references.
Based on contributions by
Bo Kagstrom and Peter Poromaa, Department of Computing Science,
Umea University, S-901 87 Umea, Sweden.
References
==========
[1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
[3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software