ztgsna(3)

NAME

ZTGSNA - estimate reciprocal condition numbers for speci

fied eigenvalues and/or eigenvectors of a matrix pair (A, B)

SYNOPSIS

SUBROUTINE ZTGSNA( 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( * )
    DOUBLE         PRECISION DIF( * ), S( * )
    COMPLEX*16      A(  LDA, * ), B( LDB, * ), VL( LDVL, *
), VR( LDVR, * ), WORK( * )

PURPOSE

ZTGSNA estimates reciprocal condition numbers for speci

fied 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 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

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*16 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*16 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*16 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 consecu

tive columns of VL, as returned by ZTGEVC. 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*16 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 consecu

tive columns of VR, as returned by ZTGEVC. 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) DOUBLE PRECISION array, dimension (MM)

If JOB = 'E' or 'B', the reciprocal condition num

bers of the selected eigenvalues, stored in consecutive elements

of the array. If JOB = 'V', S is not referenced.

DIF (output) DOUBLE PRECISION array, dimension (MM)

If JOB = 'V' or 'B', the estimated reciprocal con

dition numbers of the selected eigenvectors, stored in consecu

tive elements of the array. If the eigenvalues cannot be re

ordered to compute DIF(j), DIF(j) is set to 0; this can only oc

cur 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 refer

enced.

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*16 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 ille

gal value

FURTHER DETAILS

The reciprocal of the condition number of the i-th gener

alized 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 com

plex 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 eigen

vector 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 be

tween the matrices X and Y.

We approximate the smallest singular value of Zl with an

upper bound. This is done by ZLATDF.

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 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.

[3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms

and Software

for Solving the Generalized Sylvester Equation and Es

timating 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 LA

PACK Working
Note 75.
To appear in ACM Trans. on Math. Software, Vol 22, No

1, 1996.

LAPACK version 3.0 15 June 2000

ZTGSNA(3)