DGEEVX(l) ) DGEEVX(l)NAME
DGEEVX - compute for an N-by-N real nonsymmetric matrix A, the eigen‐
values and, optionally, the left and/or right eigenvectors
SYNOPSIS
SUBROUTINE DGEEVX( 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
DOUBLE PRECISION ABNRM
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), RCONDE( * ), RCONDV( * ),
SCALE( * ), VL( LDVL, * ), VR( LDVR, * ), WI( * ),
WORK( * ), WR( * )
PURPOSE
DGEEVX computes for an N-by-N real nonsymmetric matrix A, the eigenval‐
ues and, optionally, the left and/or right eigenvectors. Optionally
also, it computes a balancing transformation to improve the condition‐
ing 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 triangular, and applying a diagonal similarity transforma‐
tion 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
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
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 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 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) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the N-by-N matrix A. On exit, A has been overwrit‐
ten. 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) DOUBLE PRECISION array, dimension (N)
WI (output) DOUBLE PRECISION 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 posi‐
tive imaginary part first.
VL (output) DOUBLE PRECISION 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 eigen‐
values. If JOBVL = 'N', VL is not referenced. If the j-th ei‐
genvalue 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 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) DOUBLE PRECISION 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 conju‐
gate 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 deter‐
mined 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) 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 inter‐
changed 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,...,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) DOUBLE PRECISION
The one-norm 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 j-th eigen‐
value.
RCONDV (output) DOUBLE PRECISION array, dimension (N)
RCONDV(j) is the reciprocal condition number of the j-th right
eigenvector.
WORK (workspace/output) DOUBLE PRECISION 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 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:ILO-1 and i+1:N of WR and WI contain ei‐
genvalues which have converged.
LAPACK version 3.0 15 June 2000 DGEEVX(l)
