STGSNA - estimate reciprocal condition numbers for speci
fied eigenvalues and/or eigenvectors of a matrix pair (A,
B) in generalized real Schur canonical form (or of any
matrix pair (Q*A*Z', Q*B*Z') with orthogonal matrices Q
and Z, where Z' denotes the transpose of Z
SYNOPSIS
SUBROUTINE STGSNA( 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 A( LDA, * ), B( LDB, * ), DIF( * ), S(
* ), VL( LDVL, * ), VR( LDVR, * ),
WORK( * )
PURPOSE
STGSNA estimates reciprocal condition numbers for speci
fied eigenvalues and/or eigenvectors of a matrix pair (A,
B) in generalized real Schur canonical form (or of any
matrix pair (Q*A*Z', Q*B*Z') with orthogonal matrices Q
and Z, where Z' denotes the transpose of Z. (A, B) must
be in generalized real Schur form (as returned by SGGES),
i.e. A is block upper triangular with 1-by-1 and 2-by-2
diagonal blocks. B is 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
select condition numbers for the eigenpair corre
sponding to a real eigenvalue w(j), SELECT(j) must
be set to .TRUE.. To select condition numbers cor
responding to a complex conjugate pair of eigen
values w(j) and w(j+1), either SELECT(j) or
SELECT(j+1) or both, 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) REAL array, dimension (LDA,N)
The upper quasi-triangular matrix A in the pair
(A,B).
LDA (input) INTEGER
The leading dimension of the array A. LDA >=
max(1,N).
B (input) REAL 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) REAL 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 STGEVC. If JOB = 'V', VL is not ref
erenced.
LDVL (input) INTEGER
The leading dimension of the array VL. LDVL >= 1.
If JOB = 'E' or 'B', LDVL >= N.
VR (input) REAL 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 ov VR, as
returned by STGEVC. 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
consecutive elements of the array. For a complex
conjugate pair of eigenvalues two consecutive ele
ments of S are set to the same value. Thus S(j),
DIF(j), and the j-th columns of VL and VR all cor
respond to the same eigenpair (but not in general
the j-th eigenpair, unless all eigenpairs are
selected). If JOB = 'V', S is not referenced.
DIF (output) REAL 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. For a
complex eigenvector two consecutive elements of
DIF are set to the same value. 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. 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 real eigenvalue one element is used,
and for each selected complex conjugate pair of
eigenvalues, two elements are used. If HOWMNY =
'A', M is set to N.
WORK (workspace/output) REAL 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 >= N. If
JOB = 'V' or 'B' LWORK >= 2*N*(N+2)+16.
If LWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of
the WORK array, returns this value as the first
entry of the WORK array, and no error message
related to LWORK is issued by XERBLA.
IWORK (workspace) INTEGER array, dimension (N + 6)
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
The reciprocal of the condition number of a generalized
eigenvalue w = (a, b) is defined as
S(w) = (|u'Av|**2 + |u'Bv|**2)**(1/2) /
(norm(u)*norm(v))
where u and v are the left and right 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 (=
u'Av/u'Bv) 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)
where EPS is the machine precision.
The reciprocal of the condition number DIF(i) of right
eigenvector u and left eigenvector v corresponding to the
generalized eigenvalue w is defined as follows:
a) If the i-th eigenvalue w = (a,b) is real
Suppose U and V are orthogonal transformations such
that
U'*(A, B)*V = (S, T) = ( a * ) ( b * )
1
( 0 S22 ),( 0 T22 )
n-1
1 n-1 1 n-1
Then the reciprocal condition number DIF(i) is
Difl((a, b), (S22, T22)) = sigma-min( Zl ),
where sigma-min(Zl) denotes the smallest singular value
of the
2(n-1)-by-2(n-1) matrix
Zl = [ kron(a, In-1) -kron(1, S22) ]
[ kron(b, In-1) -kron(1, T22) ] .
Here In-1 is the identity matrix of size n-1. kron(X,
Y) is the
Kronecker product between the matrices X and Y.
wanted
(see SLATDF), then the parameter DIFDRI (see below)
should be
changed from 3 to 4 (routine SLATDF(IJOB = 2 will be
used)).
See STGSYL for more details.
b) If the i-th and (i+1)-th eigenvalues are complex conju
gate pair,
Suppose U and V are orthogonal transformations such
that
U'*(A, B)*V = (S, T) = ( S11 * ) ( T11 *
) 2
( 0 S22 ),( 0
T22) n-2
2 n-2 2
n-2
and (S11, T11) corresponds to the complex conjugate
eigenvalue
pair (w, conjg(w)). There exist unitary matrices U1 and
V1 such
that
U1'*S11*V1 = ( s11 s12 ) and U1'*T11*V1 = ( t11
t12 )
( 0 s22 ) ( 0
t22 )
where the generalized eigenvalues w = s11/t11 and
conjg(w) = s22/t22.
Then the reciprocal condition number DIF(i) is bounded
by
min( d1, max( 1, |real(s11)/real(s22)| )*d2 )
where, d1 = Difl((s11, t11), (s22, t22)) = sigma-
min(Z1), where
Z1 is the complex 2-by-2 matrix
Z1 = [ s11 -s22 ]
[ t11 -t22 ],
This is done by computing (using real arithmetic) the
roots of the characteristical polynomial det(Z1' * Z1 -
lambda I),
where Z1' denotes the conjugate transpose of Z1 and
det(X) denotes
the determinant of X.
T22)), i.e. an
upper bound on sigma-min(Z2), where Z2 is
(2n-2)-by-(2n-2)
Z2 = [ kron(S11', In-2) -kron(I2, S22) ]
[ kron(T11', In-2) -kron(I2, T22) ]
Note that if the default method for computing DIF is
wanted (see
SLATDF), then the parameter DIFDRI (see below) should
be changed
from 3 to 4 (routine SLATDF(IJOB = 2 will be used)).
See STGSYL
for more details.
For each eigenvalue/vector specified by SELECT, DIF stores
a Frobenius norm-based estimate of Difl.
An approximate error bound for the i-th computed eigenvec
tor 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
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.
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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