
CTGSNA(l) ) CTGSNA(l)
NAME
CTGSNA  estimate reciprocal condition numbers for specified eigenvalues and/or eigenvec
tors 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 eigenvec
tors 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 eigenvec
tors (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 eigenpairs;
= '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 jth eigenvalue and/or
eigenvector, SELECT(j) must be set to .TRUE.. If HOWMNY = 'A', SELECT is not ref
erenced.
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 = 'E' or 'B', 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 = 'V', VL is not refer
enced.
LDVL (input) INTEGER
The leading dimension of the array VL. LDVL >= 1; and If JOB = 'E' or 'B', LDVL >=
N.
VR (input) COMPLEX array, dimension (LDVR,M)
IF JOB = 'E' or 'B', 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 = '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) REAL array, dimension (MM)
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) REAL array, dimension (MM)
If JOB = 'V' or 'B', 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 speci
fied by SELECT, DIF stores a Frobenius normbased 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 condi
tion numbers; for each selected eigenvalue one element is used. If HOWMNY = 'A', M
is set to N.
WORK (workspace/output) COMPLEX 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 ith argument had an illegal value
FURTHER DETAILS
The reciprocal of the condition number of the ith generalized 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 2norm 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 ith computed generalized
eigenvalue w and the corresponding exact eigenvalue 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 ) n1
1 n1 1 n1
Then the reciprocal condition number DIF(I) is
Difl[(a, b), (A22, B22)] = sigmamin( Zl )
where sigmamin(Zl) denotes the smallest singular value of
Zl = [ kron(a, In1) kron(1, A22) ]
[ kron(b, In1) kron(1, B22) ].
Here In1 is the identity matrix of size n1 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
CLATDF.
An approximate error bound for a computed eigenvector VL(i) or VR(i) is given by
EPS * norm(A, B) / DIF(i).
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
Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
M.S. Moonen et al (eds), Linear Algebra for Large Scale and
RealTime Applications, Kluwer Academic Publ. 1993, pp 195218.
[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,
S901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87.
To appear in Numerical Algorithms, 1996.
[3] B. Kagstrom and P. Poromaa, LAPACKStyle 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, S901 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.
LAPACK version 3.0 15 June 2000 CTGSNA(l) 
