ztgsna(3P) Sun Performance Library ztgsna(3P)NAMEztgsna - estimate reciprocal condition numbers for specified eigenval‐
ues and/or eigenvectors of a matrix pair (A, B)
SYNOPSIS
SUBROUTINE ZTGSNA(JOB, HOWMNT, SELECT, N, A, LDA, B, LDB, VL, LDVL,
VR, LDVR, S, DIF, MM, M, WORK, LWORK, IWORK, INFO)
CHARACTER * 1 JOB, HOWMNT
DOUBLE COMPLEX A(LDA,*), B(LDB,*), VL(LDVL,*), VR(LDVR,*), WORK(*)
INTEGER N, LDA, LDB, LDVL, LDVR, MM, M, LWORK, INFO
INTEGER IWORK(*)
LOGICAL SELECT(*)
DOUBLE PRECISION S(*), DIF(*)
SUBROUTINE ZTGSNA_64(JOB, HOWMNT, SELECT, N, A, LDA, B, LDB, VL,
LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, IWORK, INFO)
CHARACTER * 1 JOB, HOWMNT
DOUBLE COMPLEX A(LDA,*), B(LDB,*), VL(LDVL,*), VR(LDVR,*), WORK(*)
INTEGER*8 N, LDA, LDB, LDVL, LDVR, MM, M, LWORK, INFO
INTEGER*8 IWORK(*)
LOGICAL*8 SELECT(*)
DOUBLE PRECISION S(*), DIF(*)
F95 INTERFACE
SUBROUTINE TGSNA(JOB, HOWMNT, SELECT, [N], A, [LDA], B, [LDB], VL,
[LDVL], VR, [LDVR], S, DIF, MM, M, [WORK], [LWORK], [IWORK],
[INFO])
CHARACTER(LEN=1) :: JOB, HOWMNT
COMPLEX(8), DIMENSION(:) :: WORK
COMPLEX(8), DIMENSION(:,:) :: A, B, VL, VR
INTEGER :: N, LDA, LDB, LDVL, LDVR, MM, M, LWORK, INFO
INTEGER, DIMENSION(:) :: IWORK
LOGICAL, DIMENSION(:) :: SELECT
REAL(8), DIMENSION(:) :: S, DIF
SUBROUTINE TGSNA_64(JOB, HOWMNT, SELECT, [N], A, [LDA], B, [LDB], VL,
[LDVL], VR, [LDVR], S, DIF, MM, M, [WORK], [LWORK], [IWORK],
[INFO])
CHARACTER(LEN=1) :: JOB, HOWMNT
COMPLEX(8), DIMENSION(:) :: WORK
COMPLEX(8), DIMENSION(:,:) :: A, B, VL, VR
INTEGER(8) :: N, LDA, LDB, LDVL, LDVR, MM, M, LWORK, INFO
INTEGER(8), DIMENSION(:) :: IWORK
LOGICAL(8), DIMENSION(:) :: SELECT
REAL(8), DIMENSION(:) :: S, DIF
C INTERFACE
#include <sunperf.h>
void ztgsna(char job, char howmnt, int *select, int n, doublecomplex
*a, int lda, doublecomplex *b, int ldb, doublecomplex *vl,
int ldvl, doublecomplex *vr, int ldvr, double *s, double
*dif, int mm, int *m, int *info);
void ztgsna_64(char job, char howmnt, long *select, long n, doublecom‐
plex *a, long lda, doublecomplex *b, long ldb, doublecomplex
*vl, long ldvl, doublecomplex *vr, long ldvr, double *s, dou‐
ble *dif, long mm, long *m, long *info);
PURPOSEztgsna 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)
Specifies whether condition numbers are required for eigen‐
values (S) or eigenvectors (DIF):
= 'E': for eigenvalues only (S);
= 'V': for eigenvectors only (DIF);
= 'B': for both eigenvalues and eigenvectors (S and DIF).
HOWMNT (input)
= 'A': compute condition numbers for all eigenpairs;
= 'S': compute condition numbers for selected eigenpairs
specified by the array SELECT.
SELECT (input)
If HOWMNT = 'S', 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 HOWMNT = 'A', SELECT is
not referenced.
N (input) The order of the square matrix pair (A, B). N >= 0.
A (input) The upper triangular matrix A in the pair (A,B).
LDA (input)
The leading dimension of the array A. LDA >= max(1,N).
B (input) The upper triangular matrix B in the pair (A, B).
LDB (input)
The leading dimension of the array B. LDB >= max(1,N).
VL (input)
If JOB = 'E' or 'B', VL must contain left eigenvectors of (A,
B), corresponding to the eigenpairs specified by HOWMNT and
SELECT. The eigenvectors must be stored in consecutive col‐
umns of VL, as returned by ZTGEVC. If JOB = 'V', VL is not
referenced.
LDVL (input)
The leading dimension of the array VL. LDVL >= 1; and If JOB
= 'E' or 'B', LDVL >= N.
VR (input)
If JOB = 'E' or 'B', VR must contain right eigenvectors of
(A, B), corresponding to the eigenpairs specified by HOWMNT
and SELECT. The eigenvectors must be stored in consecutive
columns of VR, as returned by ZTGEVC. If JOB = 'V', VR is
not referenced.
LDVR (input)
The leading dimension of the array VR. LDVR >= 1; If JOB =
'E' or 'B', LDVR >= N.
S (output)
If JOB = 'E' or 'B', the reciprocal condition numbers of the
selected eigenvalues, stored in consecutive elements of the
array. If JOB = 'V', S is not referenced.
DIF (output)
If JOB = 'V' or 'B', the estimated reciprocal condition num‐
bers of the selected eigenvectors, stored in consecutive ele‐
ments 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 ei‐
genvalue/vector specified by SELECT, DIF stores a Frobenius
norm-based estimate of Difl. If JOB = 'E', DIF is not refer‐
enced.
MM (input)
The number of elements in the arrays S and DIF. MM >= M.
M (output)
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 HOWMNT = 'A', M is set to N.
WORK (workspace)
If JOB = 'E', WORK is not referenced. Otherwise, on exit, if
INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input)
The dimension of the array WORK. LWORK >= max(1, N). If JOB
= 'V' or 'B', LWORK >= max(1, 2*N*N).
IWORK (workspace)
dimension(N+2) If JOB = 'E', IWORK is not referenced.
INFO (output)
= 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 eigen‐
value w = (a, b) is defined as
S(I) = (|v'Au|**2 + |v'Bu|**2)**(1/2) / (norm(u)*norm(v))
where u and v are the right and left eigenvectors of (A, B) correspond‐
ing to w; |z| denotes the absolute value of the complex number, and
norm(u) denotes the 2-norm of the vector u. The pair (a, b) corresponds
to an eigenvalue w = a/b (= v'Au/v'Bu) of the matrix pair (A, B). If
both a and b equal zero, then (A,B) is singular and S(I) = -1 is
returned.
An approximate error bound on the chordal distance between the i-th
computed generalized eigenvalue w and the corresponding exact eigenval‐
ue lambda is
chord(w, lambda) <= EPS * norm(A, B) / S(I),
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
(A, B) = ( a * ) ( b * ) 1
( 0 A22 ),( 0 B22 ) n-1
1 n-1 1 n-1
Then the reciprocal condition number DIF(I) is
Difl[(a, b), (A22, B22)] = sigma-min( Zl )
where sigma-min(Zl) denotes the smallest singular value of
Zl = [ kron(a, In-1) -kron(1, A22) ]
[ kron(b, In-1) -kron(1, B22) ].
Here In-1 is the identity matrix of size n-1 and X' is the conjugate
transpose of X. kron(X, Y) is the Kronecker product between the matri‐
ces 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
EPS * norm(A, B) / DIF(i).
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
Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
M.S. Moonen et al (eds), Linear Algebra for Large Scale and
Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
[2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
Eigenvalues of a Regular Matrix Pair (A, B) and Condition
Estimation: Theory, Algorithms and Software, Report
UMINF - 94.04, Department of Computing Science, Umea University,
S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87.
To appear in Numerical Algorithms, 1996.
[3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
for Solving the Generalized Sylvester Equation and Estimating the
Separation between Regular Matrix Pairs, Report UMINF - 93.23,
Department of Computing Science, Umea University, S-901 87 Umea,
Sweden, December 1993, Revised April 1994, Also as LAPACK Working
Note 75.
To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.
6 Mar 2009 ztgsna(3P)
