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NAME
DGEEVX - compute for an N-by-N real nonsymmetric matrix A,
the eigenvalues 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 eigenvalues and, optionally, the left and/or right
eigenvectors.
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 left eigenvectors of A are the same as the right eigen-
vectors of A**T. If u(j) and v(j) are the left and right
eigenvectors, respectively, corresponding to the eigenvalue
lambda(j), then (u(j)**T)*A = lambda(j)*(u(j)**T) and A*v(j)
= lambda(j) * v(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 diago-
nal 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 eigenvec-
tors 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 explana-
tion 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. Permut-
ing 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 eigenvec-
tors.
If SENSE = 'E' or 'B', both left and right eigenvec-
tors 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
overwritten. If JOBVL = 'V' or JOBVR = 'V', A con-
tains 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 consecu-
tively with the eigenvalue having the positive ima-
ginary 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 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
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
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 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 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,...,ILO-1 = D(J), for J = ILO,...,IHI = P(J)
for J = IHI+1,...,N. The order in which the inter-
changes 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 entries of any
column).
RCONDE (output) DOUBLE PRECISION array, dimension (N)
RCONDE(j) is the reciprocal condition number of the
j-th eigenvalue.
RCONDV (output) DOUBLE PRECISION array, dimension (N)
RCONDV(j) is the reciprocal condition number of the
j-th right eigenvector.
(LWORK)
WORK (workspace/output) DOUBLE PRECISION array, dimension
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.
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 com-
pute 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 eigenvalues
which have converged.