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NAME SGEES - 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 SYNOPSIS SUBROUTINE SGEES( JOBVS, SORT, SELECT, N, A, LDA, SDIM, WR, WI, VS, LDVS, WORK, LWORK, BWORK, INFO ) CHARACTER JOBVS, SORT INTEGER INFO, LDA, LDVS, LWORK, N, SDIM LOGICAL BWORK( * ) REAL A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ), WR( * ) LOGICAL SELECT EXTERNAL SELECT PURPOSE SGEES 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 factoriza- tion 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. The leading columns of Z then form an ortho- normal basis for the invariant subspace corresponding to the selected eigenvalues. A 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 (input) LOGICAL FUNCTION of two REAL variables SELECT must be declared EXTERNAL in the calling sub- routine. 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 complex eigenvalues are selected. 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 is set to N+2 (see INFO below). 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 has been 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 will 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 refer- enced. LDVS (input) INTEGER The leading dimension of the array VS. LDVS >= 1; if JOBVS = 'V', LDVS >= N. WORK (workspace/output) REAL array, dimension (LWORK) On exit, if INFO = 0, WORK(1) contains the optimal LWORK. LWORK (input) INTEGER The dimension of the array WORK. LWORK >= max(1,3*N). For good performance, LWORK must gen- erally be larger. 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 matrix 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.