ZTGSNA - estimate reciprocal condition numbers for speci
fied 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 speci
fied 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):
= 'E': for eigenvalues only (S);
= 'V': for eigenvectors only (DIF);
= 'B': for both eigenvalues and eigenvectors (S
and DIF).
HOWMNY (input) CHARACTER*1
= 'A': compute condition numbers for all eigen
pairs;
= 'S': compute condition numbers for selected
eigenpairs specified by the array SELECT.
SELECT (input) LOGICAL array, dimension (N)
If HOWMNY = '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 HOWMNY = 'A', SELECT is not
referenced.
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 = 'E' or 'B', VL must contain left eigen
vectors 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 = 'V', VL is not ref
erenced.
LDVL (input) INTEGER
The leading dimension of the array VL. LDVL >= 1;
and If JOB = 'E' or 'B', LDVL >= N.
VR (input) COMPLEX*16 array, dimension (LDVR,M)
IF JOB = 'E' or 'B', VR must contain right eigen
vectors 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 = 'V', VR is not ref
erenced.
LDVR (input) INTEGER
The leading dimension of the array VR. LDVR >= 1;
If JOB = 'E' or 'B', LDVR >= N.
S (output) DOUBLE PRECISION array, dimension (MM)
If JOB = 'E' or 'B', the reciprocal condition num
bers of the selected eigenvalues, stored in con
secutive elements of the array. If JOB = 'V', S
is not referenced.
DIF (output) DOUBLE PRECISION array, dimension (MM)
If JOB = 'V' or 'B', the estimated reciprocal con
dition 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
For each eigenvalue/vector specified by SELECT,
DIF stores a Frobenius norm-based estimate of
Difl. If JOB = 'E', 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 = 'A', M is set to N.
WORK (workspace/output) COMPLEX*16 array, dimension
(LWORK)
If JOB = 'E', WORK is not referenced. Otherwise,
on exit, if INFO = 0, WORK(1) returns the optimal
LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= 1. If
JOB = 'V' or 'B', LWORK >= 2*N*N.
IWORK (workspace) INTEGER array, dimension (N+2)
If JOB = 'E', IWORK is not referenced.
INFO (output) INTEGER
= 0: Successful exit
< 0: If INFO = -i, the i-th argument had an ille
gal value
FURTHER DETAILS
The reciprocal of the condition number of the i-th gener
alized eigenvalue 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) corresponding 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 corre
sponding exact eigenvalue lambda is
chord(w, lambda) <= EPS * norm(A, B) / S(I),
The reciprocal of the condition number of the right eigen
vector u and left eigenvector v corresponding to the gen
eralized 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 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
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 Eigenval
ues 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 Condi
tion
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.
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