
ZGEEVX(l) ) ZGEEVX(l)
NAME
ZGEEVX  compute for an NbyN complex nonsymmetric matrix A, the eigenvalues and, option
ally, the left and/or right eigenvectors
SYNOPSIS
SUBROUTINE ZGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, W, VL, LDVL, VR, LDVR, ILO,
IHI, SCALE, ABNRM, RCONDE, RCONDV, WORK, LWORK, RWORK, INFO )
CHARACTER BALANC, JOBVL, JOBVR, SENSE
INTEGER IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N
DOUBLE PRECISION ABNRM
DOUBLE PRECISION RCONDE( * ), RCONDV( * ), RWORK( * ), SCALE( * )
COMPLEX*16 A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ), W( * ), WORK( * )
PURPOSE
ZGEEVX computes for an NbyN complex nonsymmetric matrix A, the eigenvalues and, option
ally, the left and/or right eigenvectors. Optionally also, it computes a balancing trans
formation to improve the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues (RCONDE), and recipro
cal 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 tri
angular, 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 condition numbers of
its eigenvalues and eigenvectors smaller. The computed reciprocal condition numbers cor
respond 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 bal
ancing, see section 4.10.2 of the LAPACK Users' Guide.
ARGUMENTS
BALANC (input) CHARACTER*1
Indicates how the input matrix should be diagonally 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': Diagonally scale the matrix, ie. 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 arith
metic), 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 com
puted;
= 'E': Computed for eigenvalues only;
= 'V': Computed for right eigenvectors only;
= 'B': Computed for eigenvalues and right eigenvectors.
If SENSE = 'E' or 'B', both left and right eigenvectors must also be computed
(JOBVL = 'V' and JOBVR = 'V').
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the NbyN matrix A. On exit, A has been overwritten. If JOBVL = 'V'
or JOBVR = 'V', A contains the Schur form of the balanced version of the matrix A.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
W (output) COMPLEX*16 array, dimension (N)
W contains the computed eigenvalues.
VL (output) COMPLEX*16 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. u(j) = VL(:,j), the jth column of VL.
LDVL (input) INTEGER
The leading dimension of the array VL. LDVL >= 1; if JOBVL = 'V', LDVL >= N.
VR (output) COMPLEX*16 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. v(j) = VR(:,j), the jth column of VR.
LDVR (input) INTEGER
The leading dimension of the array VR. LDVR >= 1; if JOBVR = 'V', LDVR >= N.
ILO,IHI (output) INTEGER ILO and IHI are integer values determined when A was bal
anced. The balanced A(i,j) = 0 if I > J and J = 1,...,ILO1 or I = IHI+1,...,N.
SCALE (output) DOUBLE PRECISION 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 scaling factor applied to row and column j, then SCALE(J) = P(J), for J =
1,...,ILO1 = 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 ILO1.
ABNRM (output) DOUBLE PRECISION
The onenorm of the balanced matrix (the maximum of the sum of absolute values of
elements of any column).
RCONDE (output) DOUBLE PRECISION array, dimension (N)
RCONDE(j) is the reciprocal condition number of the jth eigenvalue.
RCONDV (output) DOUBLE PRECISION array, dimension (N)
RCONDV(j) is the reciprocal condition number of the jth right eigenvector.
WORK (workspace/output) COMPLEX*16 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 SENSE = 'V' or 'B', LWORK >= N*N+2*N. For good performance, LWORK must gener
ally 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.
RWORK (workspace) DOUBLE PRECISION array, dimension (2*N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = i, the ith argument had an illegal value.
> 0: if INFO = i, the QR algorithm failed to compute all the eigenvalues, and no
eigenvectors or condition numbers have been computed; elements 1:ILO1 and i+1:N
of W contain eigenvalues which have converged.
LAPACK version 3.0 15 June 2000 ZGEEVX(l) 
