CTGSNA(1) LAPACK routine (version 3.2) CTGSNA(1)NAME
CTGSNA - estimates reciprocal condition numbers for specified eigenval‐
ues 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 gener‐
alized Schur canonical form, that is, A and B are both upper triangu‐
lar.
ARGUMENTS
JOB (input) CHARACTER*1
Specifies whether condition numbers are required for eigenval‐
ues (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 eigenpairs;
= 'S': compute condition numbers for selected eigenpairs speci‐
fied by the array SELECT.
SELECT (input) LOGICAL array, dimension (N)
If HOWMNY = 'S', SELECT specifies the eigenpairs for which con‐
dition 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 refer‐
enced.
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 refer‐
enced.
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
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 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 =
'V' or 'B', LWORK >= max(1,2*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 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) 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 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 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. [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.
LAPACK routine (version 3.2) November 2008 CTGSNA(1)
