cggevx(3)

NAME

CGGEVX - compute for a pair of N-by-N complex nonsymmetric

matrices (A,B) the generalized eigenvalues, and optionally, the

left and/or right generalized eigenvectors

SYNOPSIS

SUBROUTINE CGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA,
B,  LDB,  ALPHA,  BETA,  VL,  LDVL,  VR,  LDVR, ILO, IHI, LSCALE,
RSCALE, ABNRM, BBNRM, RCONDE, RCONDV, WORK, LWORK, RWORK,  IWORK,
BWORK, INFO )
    CHARACTER      BALANC, JOBVL, JOBVR, SENSE
    INTEGER         IHI,  ILO, INFO, LDA, LDB, LDVL, LDVR,
LWORK, N
    REAL           ABNRM, BBNRM
    LOGICAL        BWORK( * )
    INTEGER        IWORK( * )
    REAL           LSCALE( * ), RCONDE( * ), RCONDV( *  ),
RSCALE( * ), RWORK( * )
    COMPLEX         A(  LDA, * ), ALPHA( * ), B( LDB, * ),
BETA( * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )

PURPOSE

CGGEVX computes for a pair of N-by-N complex nonsymmetric

matrices (A,B) the generalized eigenvalues, and optionally, the

left and/or right generalized eigenvectors. Optionally, it also

computes a balancing transformation to improve the conditioning

of the eigenvalues and eigenvectors (ILO, IHI, LSCALE, RSCALE,

ABNRM, and BBNRM), reciprocal condition numbers for the eigenval

ues (RCONDE), and reciprocal condition numbers for the right

eigenvectors (RCONDV).

A generalized eigenvalue for a pair of matrices (A,B) is a

scalar lambda or a ratio alpha/beta = lambda, such that A - lamb

da*B is singular. It is usually represented as the pair (al

pha,beta), as there is a reasonable interpretation for beta=0,

and even for both being zero.

The right eigenvector v(j) corresponding to the eigenvalue

lambda(j) of (A,B) satisfies

A * v(j) = lambda(j) * B * v(j) .

The left eigenvector u(j) corresponding to the eigenvalue

lambda(j) of (A,B) satisfies

u(j)**H * A = lambda(j) * u(j)**H * B.

where u(j)**H is the conjugate-transpose of u(j).

ARGUMENTS

BALANC (input) CHARACTER*1

Specifies the balance option to be performed:
= 'N': do not diagonally scale or permute;
= 'P': permute only;
= 'S': scale only;
= 'B': both permute and scale. Computed recipro

cal condition numbers will be for the matrices after permuting

and/or balancing. Permuting does not change condition numbers (in

exact arithmetic), but balancing does.

JOBVL (input) CHARACTER*1

= 'N': do not compute the left generalized eigen

vectors;
= 'V': compute the left generalized eigenvectors.

JOBVR (input) CHARACTER*1

= 'N': do not compute the right generalized

eigenvectors;
= 'V': compute the right generalized eigenvec

tors.

SENSE (input) CHARACTER*1

Determines which reciprocal condition numbers are

computed. = 'N': none are computed;
= 'E': computed for eigenvalues only;
= 'V': computed for eigenvectors only;
= 'B': computed for eigenvalues and eigenvectors.

N (input) INTEGER

The order of the matrices A, B, VL, and VR. N >=

0.

A (input/output) COMPLEX array, dimension (LDA, N)

On entry, the matrix A in the pair (A,B). On ex

it, A has been overwritten. If JOBVL='V' or JOBVR='V' or both,

then A contains the first part of the complex Schur form of the

"balanced" versions of the input A and B.

LDA (input) INTEGER

The leading dimension of A. LDA >= max(1,N).

B (input/output) COMPLEX array, dimension (LDB, N)

On entry, the matrix B in the pair (A,B). On ex

it, B has been overwritten. If JOBVL='V' or JOBVR='V' or both,

then B contains the second part of the complex Schur form of the

"balanced" versions of the input A and B.

LDB (input) INTEGER

The leading dimension of B. LDB >= max(1,N).

ALPHA (output) COMPLEX array, dimension (N)

BETA (output) COMPLEX array, dimension (N) On

exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigen

values.

Note: the quotient ALPHA(j)/BETA(j) ) may easily

over- or underflow, and BETA(j) may even be zero. Thus, the user

should avoid naively computing the ratio ALPHA/BETA. However,

ALPHA will be always less than and usually comparable with

norm(A) in magnitude, and BETA always less than and usually com

parable with norm(B).

VL (output) COMPLEX array, dimension (LDVL,N)

If JOBVL = 'V', the left generalized eigenvectors

u(j) are stored one after another in the columns of VL, in the

same order as their eigenvalues. Each eigenvector will be scaled

so the largest component will have abs(real part) + abs(imag.

part) = 1. Not referenced if JOBVL = 'N'.

LDVL (input) INTEGER

The leading dimension of the matrix VL. LDVL >= 1,

and if JOBVL = 'V', LDVL >= N.

VR (output) COMPLEX array, dimension (LDVR,N)

If JOBVR = 'V', the right generalized eigenvectors

v(j) are stored one after another in the columns of VR, in the

same order as their eigenvalues. Each eigenvector will be scaled

so the largest component will have abs(real part) + abs(imag.

part) = 1. Not referenced if JOBVR = 'N'.

LDVR (input) INTEGER

The leading dimension of the matrix VR. LDVR >= 1,

and if JOBVR = 'V', LDVR >= N.

ILO,IHI (output) INTEGER ILO and IHI are integer

values such that on exit A(i,j) = 0 and B(i,j) = 0 if i > j and j

= 1,...,ILO-1 or i = IHI+1,...,N. If BALANC = 'N' or 'S', ILO =

1 and IHI = N.

LSCALE (output) REAL array, dimension (N)

Details of the permutations and scaling factors

applied to the left side of A and B. If PL(j) is the index of

the row interchanged with row j, and DL(j) is the scaling factor

applied to row j, then LSCALE(j) = PL(j) for j = 1,...,ILO-1 =

DL(j) for j = ILO,...,IHI = PL(j) for j = IHI+1,...,N. The or

der in which the interchanges are made is N to IHI+1, then 1 to

ILO-1.

RSCALE (output) REAL array, dimension (N)

Details of the permutations and scaling factors

applied to the right side of A and B. If PR(j) is the index of

the column interchanged with column j, and DR(j) is the scaling

factor applied to column j, then RSCALE(j) = PR(j) for j =

1,...,ILO-1 = DR(j) for j = ILO,...,IHI = PR(j) for j =

IHI+1,...,N The order in which the interchanges are made is N to

IHI+1, then 1 to ILO-1.

ABNRM (output) REAL

The one-norm of the balanced matrix A.

BBNRM (output) REAL

The one-norm of the balanced matrix B.

RCONDE (output) REAL array, dimension (N)

If SENSE = 'E' or 'B', the reciprocal condition

numbers of the selected eigenvalues, stored in consecutive ele

ments of the array. If SENSE = 'V', RCONDE is not referenced.

RCONDV (output) REAL array, dimension (N)

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 RCONDV(j), RCONDV(j) is set to 0; this can on

ly occur when the true value would be very small anyway. If

SENSE = 'E', RCONDV is not referenced. Not referenced if JOB =

'E'.

WORK (workspace/output) COMPLEX array, dimension

(LWORK)

On exit, if INFO = 0, WORK(1) returns the optimal

LWORK.

LWORK (input) INTEGER

The dimension of the array WORK. LWORK >=

max(1,2*N). If SENSE = 'N' or 'E', LWORK >= 2*N. If SENSE = 'V'

or 'B', LWORK >= 2*N*N+2*N.

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.

RWORK (workspace) REAL array, dimension (6*N)

Real workspace.

IWORK (workspace) INTEGER array, dimension (N+2)

If SENSE = 'E', IWORK is not referenced.

BWORK (workspace) LOGICAL array, dimension (N)

If SENSE = 'N', BWORK is not referenced.

INFO (output) INTEGER

= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille

gal value.
= 1,...,N: The QZ iteration failed. No eigenvec

tors have been calculated, but ALPHA(j) and BETA(j) should be

correct for j=INFO+1,...,N. > N: =N+1: other than QZ iteration

failed in CHGEQZ.
=N+2: error return from CTGEVC.

FURTHER DETAILS

Balancing a matrix pair (A,B) includes, first, permuting

rows and columns to isolate eigenvalues, second, applying diago

nal similarity transformation to the rows and columns to make the

rows and columns as close in norm as possible. The computed re

ciprocal condition numbers correspond to the balanced matrix.

Permuting rows and columns will not change the condition numbers

(in exact arithmetic) but diagonal scaling will. For further ex

planation of balancing, see section 4.11.1.2 of LAPACK Users'

Guide.

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(ABNRM, BBNRM) /

RCONDE(I)

An approximate error bound for the angle between the i-th

computed eigenvector VL(i) or VR(i) is given by

EPS * norm(ABNRM, BBNRM) / DIF(i).

For further explanation of the reciprocal condition num

bers RCONDE and RCONDV, see section 4.11 of LAPACK User's Guide.

LAPACK version 3.0 15 June 2000

CGGEVX(3)