CTGSNA(3S)CTGSNA(3S)NAMECTGSNA - estimate 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( * )
IMPLEMENTATION
These routines are part of the SCSL Scientific Library and can be loaded
using either the -lscs or the -lscs_mp option. The -lscs_mp option
directs the linker to use the multi-processor version of the library.
When linking to SCSL with -lscs or -lscs_mp, the default integer size is
4 bytes (32 bits). Another version of SCSL is available in which integers
are 8 bytes (64 bits). This version allows the user access to larger
memory sizes and helps when porting legacy Cray codes. It can be loaded
by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
only one of the two versions; 4-byte integer and 8-byte integer library
calls cannot be mixed.
PURPOSECTGSNA 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):
= '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
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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.
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 referenced.
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
referenced.
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.
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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 (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 illegal value
FURTHER DETAILS
The reciprocal of the condition number of the i-th 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 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 eigenvalue
lambda is
chord(w, lambda) <= EPS * norm(A, B) / S(I),
where EPS is the machine precision.
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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 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. [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
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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.
SEE ALSOINTRO_LAPACK(3S), INTRO_SCSL(3S)
This man page is available only online.
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