sgeevx(3)

NAME

SGEEVX - compute for an N-by-N real nonsymmetric matrix A,

the eigenvalues and, optionally, the left and/or right eigenvec

tors

SYNOPSIS

SUBROUTINE SGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA,
WR,  WI,  VL,  LDVL,  VR,  LDVR,  ILO, IHI, SCALE, ABNRM, RCONDE,
RCONDV, WORK, LWORK, IWORK, INFO )
    CHARACTER      BALANC, JOBVL, JOBVR, SENSE
    INTEGER        IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK,
N
    REAL           ABNRM
    INTEGER        IWORK( * )
    REAL            A( LDA, * ), RCONDE( * ), RCONDV( * ),
SCALE( * ), VL( LDVL, * ), VR( LDVR, * ), WI( * ), WORK( * ), WR(
* )

PURPOSE

SGEEVX computes for an N-by-N real nonsymmetric matrix A,

the eigenvalues and, optionally, the left and/or right eigenvec

tors. Optionally also, it computes a balancing transformation to

improve the conditioning of the eigenvalues and eigenvectors

(ILO, IHI, SCALE, and ABNRM), reciprocal condition numbers for

the eigenvalues (RCONDE), and reciprocal condition numbers for

the right
eigenvectors (RCONDV).

The right eigenvector v(j) of A satisfies

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

where lambda(j) is its eigenvalue.
The left eigenvector u(j) of A satisfies

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

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

The computed eigenvectors are normalized to have Euclidean

norm equal to 1 and largest component real.

Balancing a matrix means permuting the rows and columns to

make it more nearly upper triangular, and applying a diagonal

similarity transformation D * A * D**(-1), where D is a diagonal

matrix, to make its rows and columns closer in norm and the con

dition numbers of its eigenvalues and eigenvectors smaller. The

computed reciprocal 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 explanation of balancing, see section 4.10.2 of the LA

PACK Users' Guide.

ARGUMENTS

BALANC (input) CHARACTER*1

Indicates how the input matrix should be diagonal

ly scaled and/or permuted to improve the conditioning of its

eigenvalues. = 'N': Do not diagonally scale or permute;
= 'P': Perform permutations to make the matrix

more nearly upper triangular. Do not diagonally scale; = 'S': Di

agonally scale the matrix, i.e. replace A by D*A*D**(-1), where D

is a diagonal matrix chosen to make the rows and columns of A

more equal in norm. Do not permute; = 'B': Both diagonally scale

and permute A.

Computed reciprocal condition numbers will be for

the matrix after balancing and/or permuting. Permuting does not

change condition numbers (in exact arithmetic), but balancing

does.

JOBVL (input) CHARACTER*1

= 'N': left eigenvectors of A are not computed;
= 'V': left eigenvectors of A are computed. If

SENSE = 'E' or 'B', JOBVL must = 'V'.

JOBVR (input) CHARACTER*1

= 'N': right eigenvectors of A are not computed;
= 'V': right eigenvectors of A are computed. If

SENSE = 'E' or 'B', JOBVR must = 'V'.

SENSE (input) CHARACTER*1

Determines which reciprocal condition numbers are

computed. = 'N': None are computed;
= 'E': Computed for eigenvalues only;
= 'V': Computed for right eigenvectors only;
= 'B': Computed for eigenvalues and right eigen

vectors.

If SENSE = 'E' or 'B', both left and right eigen

vectors must also be computed (JOBVL = 'V' and JOBVR = 'V').

N (input) INTEGER

The order of the matrix A. N >= 0.

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

On entry, the N-by-N matrix A. On exit, A has

been overwritten. If JOBVL = 'V' or JOBVR = 'V', A contains the

real Schur form of the balanced version of the input matrix A.

LDA (input) INTEGER

The leading dimension of the array A. LDA >=

max(1,N).

WR (output) REAL array, dimension (N)

WI (output) REAL array, dimension (N) WR and

WI contain the real and imaginary parts, respectively, of the

computed eigenvalues. Complex conjugate pairs of eigenvalues

will appear consecutively with the eigenvalue having the positive

imaginary part first.

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 JOBVL = 'N', VL is not referenced. 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)-st eigenvalues form a complex con

jugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and
u(j+1) = VL(:,j) - i*VL(:,j+1).

LDVL (input) INTEGER

The leading dimension of the array VL. LDVL >= 1;

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 JOBVR = 'N', VR is not referenced. 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)-st eigenvalues form a complex con

jugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and
v(j+1) = VR(:,j) - i*VR(:,j+1).

LDVR (input) INTEGER

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

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

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

values determined when A was balanced. The balanced A(i,j) = 0

if I > J and J = 1,...,ILO-1 or I = IHI+1,...,N.

SCALE (output) REAL array, dimension (N)

Details of the permutations and scaling factors

applied when balancing A. If P(j) is the index of the row and

column interchanged with row and column j, and D(j) is the scal

ing factor applied to row and column j, then SCALE(J) = P(J),

for J = 1,...,ILO-1 = D(J), for J = ILO,...,IHI = P(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 (the maximum

of the sum of absolute values of elements of any column).

RCONDE (output) REAL array, dimension (N)

RCONDE(j) is the reciprocal condition number of

the j-th eigenvalue.

RCONDV (output) REAL array, dimension (N)

RCONDV(j) is the reciprocal condition number of

the j-th right eigenvector.

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. If SENSE = 'N'

or 'E', LWORK >= max(1,2*N), and if JOBVL = 'V' or JOBVR = 'V',

LWORK >= 3*N. If SENSE = 'V' or 'B', LWORK >= N*(N+6). For good

performance, LWORK must generally be larger.

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 (2*N-2)

If SENSE = 'N' or 'E', not referenced.

INFO (output) INTEGER

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

gal value.
> 0: if INFO = i, the QR algorithm failed to com

pute all the eigenvalues, and no eigenvectors or condition num

bers have been computed; elements 1:ILO-1 and i+1:N of WR and WI

contain eigenvalues which have converged.

LAPACK version 3.0 15 June 2000

SGEEVX(3)