dtgsna - 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 DTGSNA( 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(*) DOUBLE PRECISION A(LDA,*), B(LDB,*), VL(LDVL,*), VR(LDVR,*), S(*), DIF(*), WORK(*)
SUBROUTINE DTGSNA_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(*) DOUBLE PRECISION A(LDA,*), B(LDB,*), VL(LDVL,*), VR(LDVR,*), S(*), DIF(*), WORK(*)
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(8), DIMENSION(:) :: S, DIF, WORK REAL(8), 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(8), DIMENSION(:) :: S, DIF, WORK REAL(8), DIMENSION(:,:) :: A, B, VL, VR
#include <sunperf.h>
void dtgsna(char job, char howmnt, logical *select, int n, double *a, int lda, double *b, int ldb, double *vl, int ldvl, double *vr, int ldvr, double *s, double *dif, int mm, int *m, int *info);
void dtgsna_64(char job, char howmnt, logical *select, long n, double *a, long lda, double *b, long ldb, double *vl, long ldvl, double *vr, long ldvr, double *s, double *dif, long mm, long *m, long *info);
dtgsna 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.
= 'E': for eigenvalues only (S);
= 'V': for eigenvectors only (DIF);
= 'B': for both eigenvalues and eigenvectors (S and DIF).
= 'A': compute condition numbers for all eigenpairs;
= 'S': compute condition numbers for selected eigenpairs specified by the array SELECT.
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.
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.
WORK(1) returns the optimal LWORK.
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.
dimension(N+6)
If JOB = 'E', IWORK is not referenced.
=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 conjugate 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 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.