
SGEEVX(l) ) SGEEVX(l)
NAME
SGEEVX  compute for an NbyN real nonsymmetric matrix A, the eigenvalues and, option
ally, the left and/or right eigenvectors
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 NbyN real nonsymmetric matrix A, the eigenvalues and, optionally,
the left and/or right eigenvectors. Optionally also, it computes a balancing transforma
tion to improve the conditioning of the eigenvalues and eigenvectors (ILO, IHI, SCALE, and
ABNRM), reciprocal condition numbers for the eigenvalues (RCONDE), and reciprocal condi
tion 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, 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 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) REAL 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 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 imagi
nary 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 jth eigenvalue is real, then u(j) = VL(:,j), the jth column
of VL. If the jth and (j+1)st eigenvalues form a complex conjugate 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 jth eigenvalue is real, then v(j) = VR(:,j), the jth column
of VR. If the jth and (j+1)st eigenvalues form a complex conjugate 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 bal
anced. The balanced A(i,j) = 0 if I > J and J = 1,...,ILO1 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 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) REAL
The onenorm 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 jth eigenvalue.
RCONDV (output) REAL array, dimension (N)
RCONDV(j) is the reciprocal condition number of the jth 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*N2)
If SENSE = 'N' or 'E', not referenced.
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 WR and WI contain eigenvalues which have converged.
LAPACK version 3.0 15 June 2000 SGEEVX(l) 
