sgeev(3)

NAME

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

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

tors

SYNOPSIS

SUBROUTINE SGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR, LDVR, WORK, LWORK, INFO )
    CHARACTER     JOBVL, JOBVR
    INTEGER       INFO, LDA, LDVL, LDVR, LWORK, N
    REAL           A( LDA, * ), VL( LDVL, * ), VR( LDVR, *
), WI( * ), WORK( * ), WR( * )

PURPOSE

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

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

tors. 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.

ARGUMENTS

JOBVL (input) CHARACTER*1

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

JOBVR (input) CHARACTER*1

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

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.

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 ap

pear consecutively with the eigenvalue having the positive imagi

nary 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;

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

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(1,3*N), and if JOBVL = 'V' or JOBVR = 'V', LWORK >= 4*N. 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.

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 have been computed;

elements i+1:N of WR and WI contain eigenvalues which have con

verged.

LAPACK version 3.0 15 June 2000

SGEEV(3)