Contents
stgsna - estimate reciprocal condition numbers for specified
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
SUBROUTINE STGSNA(JOB, HOWMNT, SELECT, N, A, LDA, B, LDB, VL, LDVL,
VR, LDVR, S, DIF, MM, M, WORK, LWORK, IWORK, INFO)
CHARACTER * 1 JOB, HOWMNT
INTEGER N, LDA, LDB, LDVL, LDVR, MM, M, LWORK, INFO
INTEGER IWORK(*)
LOGICAL SELECT(*)
REAL A(LDA,*), B(LDB,*), VL(LDVL,*), VR(LDVR,*), S(*),
DIF(*), WORK(*)
SUBROUTINE STGSNA_64(JOB, HOWMNT, SELECT, N, A, LDA, B, LDB, VL,
LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, IWORK, INFO)
CHARACTER * 1 JOB, HOWMNT
INTEGER*8 N, LDA, LDB, LDVL, LDVR, MM, M, LWORK, INFO
INTEGER*8 IWORK(*)
LOGICAL*8 SELECT(*)
REAL A(LDA,*), B(LDB,*), VL(LDVL,*), VR(LDVR,*), S(*),
DIF(*), WORK(*)
F95 INTERFACE
SUBROUTINE TGSNA(JOB, HOWMNT, SELECT, [N], A, [LDA], B, [LDB], VL,
[LDVL], VR, [LDVR], S, DIF, MM, M, [WORK], [LWORK], [IWORK],
[INFO])
CHARACTER(LEN=1) :: JOB, HOWMNT
INTEGER :: N, LDA, LDB, LDVL, LDVR, MM, M, LWORK, INFO
INTEGER, DIMENSION(:) :: IWORK
LOGICAL, DIMENSION(:) :: SELECT
REAL, DIMENSION(:) :: S, DIF, WORK
REAL, DIMENSION(:,:) :: A, B, VL, VR
SUBROUTINE TGSNA_64(JOB, HOWMNT, SELECT, [N], A, [LDA], B, [LDB], VL,
[LDVL], VR, [LDVR], S, DIF, MM, M, [WORK], [LWORK], [IWORK],
[INFO])
CHARACTER(LEN=1) :: JOB, HOWMNT
INTEGER(8) :: N, LDA, LDB, LDVL, LDVR, MM, M, LWORK, INFO
INTEGER(8), DIMENSION(:) :: IWORK
LOGICAL(8), DIMENSION(:) :: SELECT
REAL, DIMENSION(:) :: S, DIF, WORK
REAL, DIMENSION(:,:) :: A, B, VL, VR
C INTERFACE
#include <sunperf.h>
void stgsna(char job, char howmnt, int *select, int n, float
*a, int lda, float *b, int ldb, float *vl, int
ldvl, float *vr, int ldvr, float *s, float *dif,
int mm, int *m, int *info);
void stgsna_64(char job, char howmnt, long *select, long n,
float *a, long lda, float *b, long ldb, float *vl,
long ldvl, float *vr, long ldvr, float *s, float
*dif, long mm, long *m, long *info);
stgsna estimates reciprocal condition numbers for specified
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.
JOB (input)
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).
HOWMNT (input)
= 'A': compute condition numbers for all eigen-
pairs;
= 'S': compute condition numbers for selected
eigenpairs specified by the array SELECT.
SELECT (input)
If HOWMNT = 'S', SELECT specifies the eigenpairs
for which condition numbers are required. To
select condition numbers for the eigenpair
corresponding to a real eigenvalue w(j), SELECT(j)
must be set to .TRUE.. To select condition numbers
corresponding to a complex conjugate pair of
eigenvalues w(j) and w(j+1), either SELECT(j) or
SELECT(j+1) or both, must be set to .TRUE.. If
HOWMNT = 'A', SELECT is not referenced.
N (input) The order of the square matrix pair (A, B). N >=
0.
A (input) The upper quasi-triangular matrix A in the pair
(A,B).
LDA (input)
The leading dimension of the array A. LDA >=
max(1,N).
B (input) The upper triangular matrix B in the pair (A,B).
LDB (input)
The leading dimension of the array B. LDB >=
max(1,N).
VL (input)
If JOB = 'E' or 'B', VL must contain left eigen-
vectors of (A, B), corresponding to the eigenpairs
specified by HOWMNT and SELECT. The eigenvectors
must be stored in consecutive columns of VL, as
returned by STGEVC. If JOB = 'V', VL is not
referenced.
LDVL (input)
The leading dimension of the array VL. LDVL >= 1.
If JOB = 'E' or 'B', LDVL >= N.
VR (input)
If JOB = 'E' or 'B', VR must contain right eigen-
vectors of (A, B), corresponding to the eigenpairs
specified by HOWMNT and SELECT. The eigenvectors
must be stored in consecutive columns ov VR, as
returned by STGEVC. If JOB = 'V', VR is not
referenced.
LDVR (input)
The leading dimension of the array VR. LDVR >= 1.
If JOB = 'E' or 'B', LDVR >= N.
S (output)
If JOB = 'E' or 'B', the reciprocal condition
numbers of the selected eigenvalues, stored in
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
correspond to the same eigenpair (but not in gen-
eral the j-th eigenpair, unless all eigenpairs are
selected). If JOB = 'V', S is not referenced.
DIF (output)
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)
The number of elements in the arrays S and DIF. MM
>= M.
M (output)
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 HOWMNT =
'A', M is set to N.
WORK (workspace)
If JOB = 'E', WORK is not referenced. Otherwise,
on exit, if INFO = 0, WORK(1) returns the optimal
LWORK.
LWORK (input)
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)
dimension(N+6) If JOB = 'E', IWORK is not refer-
enced.
INFO (output)
=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
(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
corresponding exact eigenvalue lambda is
hord(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.
Note that if the default method for computing DIF(i) 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.
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.
and d2 is an upper bound on Difl((S11, T11), (S22, 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 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 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.
[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.