DGEESX(l) ) DGEESX(l)
DGEESX - compute for an N-by-N real nonsymmetric matrix A, the eigenvalues, the real Schur
form T, and, optionally, the matrix of Schur vectors Z
SUBROUTINE DGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, WR, WI, VS, LDVS, RCONDE,
RCONDV, WORK, LWORK, IWORK, LIWORK, BWORK, INFO )
CHARACTER JOBVS, SENSE, SORT
INTEGER INFO, LDA, LDVS, LIWORK, LWORK, N, SDIM
DOUBLE PRECISION RCONDE, RCONDV
LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ), WR( * )
DGEESX computes for an N-by-N real nonsymmetric matrix A, the eigenvalues, the real Schur
form T, and, optionally, the matrix of Schur vectors Z. This gives the Schur factorization
A = Z*T*(Z**T). Optionally, it also orders the eigenvalues on the diagonal of the real
Schur form so that selected eigenvalues are at the top left; computes a reciprocal condi-
tion number for the average of the selected eigenvalues (RCONDE); and computes a recipro-
cal condition number for the right invariant subspace corresponding to the selected eigen-
values (RCONDV). The leading columns of Z form an orthonormal basis for this invariant
For further explanation of the reciprocal condition numbers RCONDE and RCONDV, see Section
4.10 of the LAPACK Users' Guide (where these quantities are called s and sep respec-
A real matrix is in real Schur form if it is upper quasi-triangular with 1-by-1 and 2-by-2
blocks. 2-by-2 blocks will be standardized in the form
[ a b ]
[ c a ]
where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).
JOBVS (input) CHARACTER*1
= 'N': Schur vectors are not computed;
= 'V': Schur vectors are computed.
SORT (input) CHARACTER*1
Specifies whether or not to order the eigenvalues on the diagonal of the Schur
form. = 'N': Eigenvalues are not ordered;
= 'S': Eigenvalues are ordered (see SELECT).
SELECT (input) LOGICAL FUNCTION of two DOUBLE PRECISION arguments
SELECT must be declared EXTERNAL in the calling subroutine. If SORT = 'S', SELECT
is used to select eigenvalues to sort to the top left of the Schur form. If SORT
= 'N', SELECT is not referenced. An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected
if SELECT(WR(j),WI(j)) is true; i.e., if either one of a complex conjugate pair of
eigenvalues is selected, then both are. Note that a selected complex eigenvalue
may no longer satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering, since ordering
may change the value of complex eigenvalues (especially if the eigenvalue is ill-
conditioned); in this case INFO may be set to N+3 (see INFO below).
SENSE (input) CHARACTER*1
Determines which reciprocal condition numbers are computed. = 'N': None are com-
= 'E': Computed for average of selected eigenvalues only;
= 'V': Computed for selected right invariant subspace only;
= 'B': Computed for both. If SENSE = 'E', 'V' or 'B', SORT must equal 'S'.
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 is overwritten by its real Schur form
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
SDIM (output) INTEGER
If SORT = 'N', SDIM = 0. If SORT = 'S', SDIM = number of eigenvalues (after sort-
ing) for which SELECT is true. (Complex conjugate pairs for which SELECT is true
for either eigenvalue count as 2.)
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, in the same order
that they appear on the diagonal of the output Schur form T. Complex conjugate
pairs of eigenvalues appear consecutively with the eigenvalue having the positive
imaginary part first.
VS (output) DOUBLE PRECISION array, dimension (LDVS,N)
If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur vectors. If JOBVS =
'N', VS is not referenced.
LDVS (input) INTEGER
The leading dimension of the array VS. LDVS >= 1, and if JOBVS = 'V', LDVS >= N.
RCONDE (output) DOUBLE PRECISION
If SENSE = 'E' or 'B', RCONDE contains the reciprocal condition number for the
average of the selected eigenvalues. Not referenced if SENSE = 'N' or 'V'.
RCONDV (output) DOUBLE PRECISION
If SENSE = 'V' or 'B', RCONDV contains the reciprocal condition number for the
selected right invariant subspace. Not referenced if SENSE = 'N' or 'E'.
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. LWORK >= max(1,3*N). Also, if SENSE = 'E' or
'V' or 'B', LWORK >= N+2*SDIM*(N-SDIM), where SDIM is the number of selected ei-
genvalues computed by this routine. Note that N+2*SDIM*(N-SDIM) <= N+N*N/2. For
good performance, LWORK must generally be larger.
IWORK (workspace/output) INTEGER array, dimension (LIWORK)
Not referenced if SENSE = 'N' or 'E'. On exit, if INFO = 0, IWORK(1) returns the
LIWORK (input) INTEGER
The dimension of the array IWORK. LIWORK >= 1; if SENSE = 'V' or 'B', LIWORK >=
BWORK (workspace) LOGICAL array, dimension (N)
Not referenced if SORT = 'N'.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = i, and i is
<= N: the QR algorithm failed to compute all the
eigenvalues; elements 1:ILO-1 and i+1:N of WR and WI contain those eigenvalues
which have converged; if JOBVS = 'V', VS contains the transformation which reduces
A to its partially converged Schur form. = N+1: the eigenvalues could not be
reordered because some eigenvalues were too close to separate (the problem is very
ill-conditioned); = N+2: after reordering, roundoff changed values of some complex
eigenvalues so that leading eigenvalues in the Schur form no longer satisfy
SELECT=.TRUE. This could also be caused by underflow due to scaling.
LAPACK version 3.0 15 June 2000 DGEESX(l)