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LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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| subroutine dgeesx | ( | character | jobvs, |
| character | sort, | ||
| external | select, | ||
| character | sense, | ||
| integer | n, | ||
| double precision, dimension( lda, * ) | a, | ||
| integer | lda, | ||
| integer | sdim, | ||
| double precision, dimension( * ) | wr, | ||
| double precision, dimension( * ) | wi, | ||
| double precision, dimension( ldvs, * ) | vs, | ||
| integer | ldvs, | ||
| double precision | rconde, | ||
| double precision | rcondv, | ||
| double precision, dimension( * ) | work, | ||
| integer | lwork, | ||
| integer, dimension( * ) | iwork, | ||
| integer | liwork, | ||
| logical, dimension( * ) | bwork, | ||
| integer | info ) |
DGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
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!> !> 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 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). !>
| [in] | JOBVS | !> JOBVS is CHARACTER*1 !> = 'N': Schur vectors are not computed; !> = 'V': Schur vectors are computed. !> |
| [in] | SORT | !> SORT is 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). !> |
| [in] | SELECT | !> SELECT is a 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). !> |
| [in] | SENSE | !> SENSE is 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'. !> |
| [in] | N | !> N is INTEGER !> The order of the matrix A. N >= 0. !> |
| [in,out] | A | !> A is DOUBLE PRECISION array, dimension (LDA, N) !> On entry, the N-by-N matrix A. !> On exit, A is overwritten by its real Schur form T. !> |
| [in] | LDA | !> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,N). !> |
| [out] | SDIM | !> SDIM is 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.) !> |
| [out] | WR | !> WR is DOUBLE PRECISION array, dimension (N) !> |
| [out] | WI | !> WI is 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. !> |
| [out] | VS | !> VS is 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. !> |
| [in] | LDVS | !> LDVS is INTEGER !> The leading dimension of the array VS. LDVS >= 1, and if !> JOBVS = 'V', LDVS >= N. !> |
| [out] | RCONDE | !> RCONDE is 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'. !> |
| [out] | RCONDV | !> RCONDV is 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'. !> |
| [out] | WORK | !> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) !> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. !> |
| [in] | LWORK | !> LWORK is 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. !> |
| [out] | IWORK | !> IWORK is INTEGER array, dimension (MAX(1,LIWORK)) !> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. !> |
| [in] | LIWORK | !> LIWORK is 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. !> |
| [out] | BWORK | !> BWORK is LOGICAL array, dimension (N) !> Not referenced if SORT = 'N'. !> |
| [out] | INFO | !> INFO is 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. !> |
Definition at line 276 of file dgeesx.f.
1.11.0