stgsna(3)

NAME

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' de

notes 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' de

notes the transpose of Z. (A, B) must be in generalized real

Schur form (as returned by SGGES), i.e. A is block upper triangu

lar with 1-by-1 and 2-by-2 diagonal blocks. B is upper triangu

lar.

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 eigenpair corresponding to a real eigenvalue

w(j), SELECT(j) must be set to .TRUE.. To select condition num

bers 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 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

referenced.

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

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 num

bers of the selected eigenvalues, stored in consecutive elements

of the array. For a complex conjugate pair of eigenvalues two

consecutive elements 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 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 consecu

tive elements of the array. For a complex eigenvector two consec

utive elements of DIF are set to the same value. If the eigenval

ues 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 any

way. 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 com

plex 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

FURTHER DETAILS

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 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 (=

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

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 gener

alized 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 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 LA

PACK 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

STGSNA(3)