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 eigen
values and eigenvectors (ILO, IHI, SCALE, and ABNRM),
reciprocal condition numbers for the eigenvalues (RCONDE),
and reciprocal 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 diag
onal 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 correspond to the balanced matrix. Permuting rows
and columns will not change the condition numbers (in
exact arithmetic) but diagonal scaling will. For further
Users' Guide.
ARGUMENTS
BALANC (input) CHARACTER*1
Indicates how the input matrix should be diago
nally scaled and/or permuted to improve the condi
tioning of its eigenvalues. = 'N': Do not diago
nally 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. Per
muting 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 eigen
vectors.
If SENSE = 'E' or 'B', both left and right eigen
vectors 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 contains the real Schur form of the balanced
version of the input matrix A.
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 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 bal
anced 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 col
umn j, and D(j) is the scaling factor applied to
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 eigenvalue.
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 ille
gal 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.
Man(1) output converted with
man2html