ctgsna (l)  Linux Man Pages
ctgsna: estimates reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a matrix pair (A, B)
NAME
CTGSNA  estimates reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a matrix pair (A, B)SYNOPSIS
 SUBROUTINE CTGSNA(
 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( * )
 REAL DIF( * ), S( * )
 COMPLEX A( LDA, * ), B( LDB, * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )
PURPOSE
CTGSNA 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 jth 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 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 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 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 CTGEVC. 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 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 CTGEVC. 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) REAL 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) REAL 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 normbased 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 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 ith argument had an illegal value
FURTHER DETAILS
The reciprocal of the condition number of the ith generalized eigenvalue w = (a, b) is defined asAn approximate error bound on the chordal distance between the ith 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 sigmamin(Zl) denotes the smallest singular value of
Here In1 is the identity matrix of size n1 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 CLATDF.
An approximate error bound for a computed eigenvector VL(i) or VR(i) is given by
See ref. [23] for more details and further references.
Based on contributions by
Bo Kagstrom and Peter Poromaa, Department of Computing Science,
Umea University, S901 87 Umea, Sweden.
References
==========
[1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
[3] B. Kagstrom and P. Poromaa, LAPACKStyle Algorithms and Software