sggevx(3)

NAME

SGGEVX - compute for a pair of N-by-N real nonsymmetric

matrices (A,B)

SYNOPSIS

SUBROUTINE SGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA,
B, LDB, ALPHAR, ALPHAI, BETA,  VL,  LDVL,  VR,  LDVR,  ILO,  IHI,
LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, RCONDV, WORK, LWORK, 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            A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
B( LDB, * ), BETA( * ), LSCALE( * ), RCONDE( * ),  RCONDV(  *  ),
RSCALE( * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )

PURPOSE

SGGEVX computes for a pair of N-by-N real nonsymmetric ma

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

left and/or right generalized eigenvectors.

Optionally also, it computes a balancing transformation to

improve the conditioning of the eigenvalues and eigenvectors

(ILO, IHI, LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condi

tion numbers for the eigenvalues (RCONDE), and reciprocal condi

tion 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) REAL 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 real 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) REAL 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 real Schur form of the

"balanced" versions of the input A and B.

LDB (input) INTEGER

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

ALPHAR (output) REAL array, dimension (N)

ALPHAI (output) REAL array, dimension (N) BETA

(output) REAL array, dimension (N) On exit, (ALPHAR(j) + AL

PHAI(j)*i)/BETA(j), j=1,...,N, will be the generalized eigenval

ues. If ALPHAI(j) is zero, then the j-th eigenvalue is real; if

positive, then the j-th and (j+1)-st eigenvalues are a complex

conjugate pair, with ALPHAI(j+1) negative.

Note: the quotients ALPHAR(j)/BETA(j) and AL

PHAI(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, ALPHAR and ALPHAI will be always less

than and usually comparable with norm(A) in magnitude, and BETA

always less than and usually comparable with norm(B).

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

If JOBVL = 'V', the left eigenvectors u(j) are

stored one after another in the columns of VL, in the same order

as their eigenvalues. If the j-th eigenvalue is real, then u(j) =

VL(:,j), the j-th column of VL. If the j-th and (j+1)-th eigen

values form a complex conjugate pair, then u(j) =

VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1). Each

eigenvector will be scaled so the largest component 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) REAL array, dimension (LDVR,N)

If JOBVR = 'V', the right eigenvectors v(j) are

stored one after another in the columns of VR, in the same order

as their eigenvalues. If the j-th eigenvalue is real, then v(j) =

VR(:,j), the j-th column of VR. If the j-th and (j+1)-th eigen

values form a complex conjugate pair, then v(j) =

VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1). Each

eigenvector will be scaled so the largest component 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. For a complex conjugate pair of eigenvalues

two consecutive elements of RCONDE are set to the same value.

Thus RCONDE(j), RCONDV(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 SENSE = 'V',

RCONDE is not referenced.

RCONDV (output) REAL array, dimension (N)

If SENSE = 'V' or 'B', the estimated reciprocal

condition numbers of the selected eigenvectors, stored in consec

utive elements of the array. For a complex eigenvector two con

secutive elements of RCONDV are set to the same value. If the

eigenvalues cannot be reordered to compute RCONDV(j), RCONDV(j)

is set to 0; this can only occur when the true value would be

very small anyway. If SENSE = 'E', RCONDV is not referenced.

WORK (workspace/output) REAL 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(1,6*N). If SENSE = 'E', LWORK >= 12*N. If SENSE = 'V' or

'B', LWORK >= 2*N*N+12*N+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 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 ALPHAR(j), ALPHAI(j), and BETA(j)

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

iteration failed in SHGEQZ.
=N+2: error return from STGEVC.

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

SGGEVX(3)