CTGSNA - estimate reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a matrix pair (A, B)

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( * )

CTGSNA 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.

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 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.

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

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.

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

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[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.