SGEESX
June 2010
Purpose
SGEESX 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 condition number for the average of the
selected eigenvalues (RCONDE); and computes a reciprocal condition
number for the right invariant subspace corresponding to the
selected eigenvalues (RCONDV). The leading columns of Z form an
orthonormal basis for this invariant subspace.
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 respectively).
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).
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 condition number for the average of the
selected eigenvalues (RCONDE); and computes a reciprocal condition
number for the right invariant subspace corresponding to the
selected eigenvalues (RCONDV). The leading columns of Z form an
orthonormal basis for this invariant subspace.
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 respectively).
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).
Arguments
| 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 |
(external procedure) LOGICAL FUNCTION of two REAL 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 computed; = '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) REAL array, dimension (LDA, N)
On entry, the N-by-N matrix A.
On exit, A is overwritten by its real Schur form T. |
| 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 sorting) for which SELECT is true. (Complex conjugate pairs for which SELECT is true for either eigenvalue count as 2.) |
| WR |
(output) REAL array, dimension (N)
|
| WI |
(output) REAL 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) REAL 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) REAL
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) REAL
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) REAL array, dimension (MAX(1,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 eigenvalues computed by this routine. Note that N+2*SDIM*(N-SDIM) <= N+N*N/2. Note also that an error is only returned if LWORK < max(1,3*N), but if SENSE = 'E' or 'V' or 'B' this may not be large enough. For good performance, LWORK must generally be larger. If LWORK = -1, then a workspace query is assumed; the routine only calculates upper bounds on the optimal sizes of the arrays WORK and IWORK, returns these values as the first entries of the WORK and IWORK arrays, and no error messages related to LWORK or LIWORK are issued by XERBLA. |
| IWORK |
(workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
|
| LIWORK |
(input) INTEGER
The dimension of the array IWORK.
LIWORK >= 1; if SENSE = 'V' or 'B', LIWORK >= SDIM*(N-SDIM). Note that SDIM*(N-SDIM) <= N*N/4. Note also that an error is only returned if LIWORK < 1, but if SENSE = 'V' or 'B' this may not be large enough. If LIWORK = -1, then a workspace query is assumed; the routine only calculates upper bounds on the optimal sizes of the arrays WORK and IWORK, returns these values as the first entries of the WORK and IWORK arrays, and no error messages related to LWORK or LIWORK are issued by XERBLA. |
| 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. |