#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int zgeesx_(char *jobvs, char *sort, L_fp select, char * sense, integer *n, doublecomplex *a, integer *lda, integer *sdim, doublecomplex *w, doublecomplex *vs, integer *ldvs, doublereal * rconde, doublereal *rcondv, doublecomplex *work, integer *lwork, doublereal *rwork, logical *bwork, integer *info) { /* -- LAPACK driver routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 Purpose ======= ZGEESX computes for an N-by-N complex nonsymmetric matrix A, the eigenvalues, the Schur form T, and, optionally, the matrix of Schur vectors Z. This gives the Schur factorization A = Z*T*(Z**H). Optionally, it also orders the eigenvalues on the diagonal of the 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 complex matrix is in Schur form if it is upper triangular. 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 one COMPLEX*16 argument SELECT must be declared EXTERNAL in the calling subroutine. If SORT = 'S', SELECT is used to select eigenvalues to order to the top left of the Schur form. If SORT = 'N', SELECT is not referenced. An eigenvalue W(j) is selected if SELECT(W(j)) is true. 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) COMPLEX*16 array, dimension (LDA, N) On entry, the N-by-N matrix A. On exit, A is overwritten by its 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 for which SELECT is true. W (output) COMPLEX*16 array, dimension (N) W contains the computed eigenvalues, in the same order that they appear on the diagonal of the output Schur form T. VS (output) COMPLEX*16 array, dimension (LDVS,N) If JOBVS = 'V', VS contains the unitary 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) COMPLEX*16 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(1,2*N). Also, if SENSE = 'E' or 'V' or 'B', LWORK >= 2*SDIM*(N-SDIM), where SDIM is the number of selected eigenvalues computed by this routine. Note that 2*SDIM*(N-SDIM) <= N*N/2. Note also that an error is only returned if LWORK < max(1,2*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 bound on the optimal size of the array WORK, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. RWORK (workspace) DOUBLE PRECISION array, dimension (N) 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 W 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. ===================================================================== Test the input arguments Parameter adjustments */ /* Table of constant values */ static integer c__1 = 1; static integer c__0 = 0; static integer c_n1 = -1; /* System generated locals */ integer a_dim1, a_offset, vs_dim1, vs_offset, i__1, i__2; /* Builtin functions */ double sqrt(doublereal); /* Local variables */ static integer i__, ihi, ilo; static doublereal dum[1], eps; static integer ibal; static doublereal anrm; static integer ierr, itau, iwrk, lwrk, icond, ieval; extern logical lsame_(char *, char *); extern /* Subroutine */ int zcopy_(integer *, doublecomplex *, integer *, doublecomplex *, integer *), dlabad_(doublereal *, doublereal *); static logical scalea; extern doublereal dlamch_(char *); static doublereal cscale; extern /* Subroutine */ int dlascl_(char *, integer *, integer *, doublereal *, doublereal *, integer *, integer *, doublereal *, integer *, integer *), zgebak_(char *, char *, integer *, integer *, integer *, doublereal *, integer *, doublecomplex *, integer *, integer *), zgebal_(char *, integer *, doublecomplex *, integer *, integer *, integer *, doublereal *, integer *), xerbla_(char *, integer *); extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *, ftnlen, ftnlen); extern doublereal zlange_(char *, integer *, integer *, doublecomplex *, integer *, doublereal *); static doublereal bignum; extern /* Subroutine */ int zgehrd_(integer *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, doublecomplex *, integer *, integer *), zlascl_(char *, integer *, integer *, doublereal *, doublereal *, integer *, integer *, doublecomplex *, integer *, integer *); static logical wantsb, wantse; extern /* Subroutine */ int zlacpy_(char *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, integer *); static integer minwrk, maxwrk; static logical wantsn; static doublereal smlnum; extern /* Subroutine */ int zhseqr_(char *, char *, integer *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, doublecomplex *, integer *, doublecomplex *, integer *, integer *); static integer hswork; extern /* Subroutine */ int zunghr_(integer *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, doublecomplex *, integer *, integer *); static logical wantst, wantsv, wantvs; extern /* Subroutine */ int ztrsen_(char *, char *, logical *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, doublereal *, doublereal *, doublecomplex *, integer *, integer *); a_dim1 = *lda; a_offset = 1 + a_dim1; a -= a_offset; --w; vs_dim1 = *ldvs; vs_offset = 1 + vs_dim1; vs -= vs_offset; --work; --rwork; --bwork; /* Function Body */ *info = 0; wantvs = lsame_(jobvs, "V"); wantst = lsame_(sort, "S"); wantsn = lsame_(sense, "N"); wantse = lsame_(sense, "E"); wantsv = lsame_(sense, "V"); wantsb = lsame_(sense, "B"); if (! wantvs && ! lsame_(jobvs, "N")) { *info = -1; } else if (! wantst && ! lsame_(sort, "N")) { *info = -2; } else if (! (wantsn || wantse || wantsv || wantsb) || ! wantst && ! wantsn) { *info = -4; } else if (*n < 0) { *info = -5; } else if (*lda < max(1,*n)) { *info = -7; } else if (*ldvs < 1 || wantvs && *ldvs < *n) { *info = -11; } /* Compute workspace (Note: Comments in the code beginning "Workspace:" describe the minimal amount of real workspace needed at that point in the code, as well as the preferred amount for good performance. CWorkspace refers to complex workspace, and RWorkspace to real workspace. NB refers to the optimal block size for the immediately following subroutine, as returned by ILAENV. HSWORK refers to the workspace preferred by ZHSEQR, as calculated below. HSWORK is computed assuming ILO=1 and IHI=N, the worst case. If SENSE = 'E', 'V' or 'B', then the amount of workspace needed depends on SDIM, which is computed by the routine ZTRSEN later in the code.) */ if (*info == 0) { if (*n == 0) { minwrk = 1; lwrk = 1; } else { maxwrk = *n + *n * ilaenv_(&c__1, "ZGEHRD", " ", n, &c__1, n, & c__0, (ftnlen)6, (ftnlen)1); minwrk = *n << 1; zhseqr_("S", jobvs, n, &c__1, n, &a[a_offset], lda, &w[1], &vs[ vs_offset], ldvs, &work[1], &c_n1, &ieval); hswork = (integer) work[1].r; if (! wantvs) { maxwrk = max(maxwrk,hswork); } else { /* Computing MAX */ i__1 = maxwrk, i__2 = *n + (*n - 1) * ilaenv_(&c__1, "ZUNGHR", " ", n, &c__1, n, &c_n1, (ftnlen)6, (ftnlen)1); maxwrk = max(i__1,i__2); maxwrk = max(maxwrk,hswork); } lwrk = maxwrk; if (! wantsn) { /* Computing MAX */ i__1 = lwrk, i__2 = *n * *n / 2; lwrk = max(i__1,i__2); } } work[1].r = (doublereal) lwrk, work[1].i = 0.; if (*lwork < minwrk) { *info = -15; } } if (*info != 0) { i__1 = -(*info); xerbla_("ZGEESX", &i__1); return 0; } /* Quick return if possible */ if (*n == 0) { *sdim = 0; return 0; } /* Get machine constants */ eps = dlamch_("P"); smlnum = dlamch_("S"); bignum = 1. / smlnum; dlabad_(&smlnum, &bignum); smlnum = sqrt(smlnum) / eps; bignum = 1. / smlnum; /* Scale A if max element outside range [SMLNUM,BIGNUM] */ anrm = zlange_("M", n, n, &a[a_offset], lda, dum); scalea = FALSE_; if (anrm > 0. && anrm < smlnum) { scalea = TRUE_; cscale = smlnum; } else if (anrm > bignum) { scalea = TRUE_; cscale = bignum; } if (scalea) { zlascl_("G", &c__0, &c__0, &anrm, &cscale, n, n, &a[a_offset], lda, & ierr); } /* Permute the matrix to make it more nearly triangular (CWorkspace: none) (RWorkspace: need N) */ ibal = 1; zgebal_("P", n, &a[a_offset], lda, &ilo, &ihi, &rwork[ibal], &ierr); /* Reduce to upper Hessenberg form (CWorkspace: need 2*N, prefer N+N*NB) (RWorkspace: none) */ itau = 1; iwrk = *n + itau; i__1 = *lwork - iwrk + 1; zgehrd_(n, &ilo, &ihi, &a[a_offset], lda, &work[itau], &work[iwrk], &i__1, &ierr); if (wantvs) { /* Copy Householder vectors to VS */ zlacpy_("L", n, n, &a[a_offset], lda, &vs[vs_offset], ldvs) ; /* Generate unitary matrix in VS (CWorkspace: need 2*N-1, prefer N+(N-1)*NB) (RWorkspace: none) */ i__1 = *lwork - iwrk + 1; zunghr_(n, &ilo, &ihi, &vs[vs_offset], ldvs, &work[itau], &work[iwrk], &i__1, &ierr); } *sdim = 0; /* Perform QR iteration, accumulating Schur vectors in VS if desired (CWorkspace: need 1, prefer HSWORK (see comments) ) (RWorkspace: none) */ iwrk = itau; i__1 = *lwork - iwrk + 1; zhseqr_("S", jobvs, n, &ilo, &ihi, &a[a_offset], lda, &w[1], &vs[ vs_offset], ldvs, &work[iwrk], &i__1, &ieval); if (ieval > 0) { *info = ieval; } /* Sort eigenvalues if desired */ if (wantst && *info == 0) { if (scalea) { zlascl_("G", &c__0, &c__0, &cscale, &anrm, n, &c__1, &w[1], n, & ierr); } i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { bwork[i__] = (*select)(&w[i__]); /* L10: */ } /* Reorder eigenvalues, transform Schur vectors, and compute reciprocal condition numbers (CWorkspace: if SENSE is not 'N', need 2*SDIM*(N-SDIM) otherwise, need none ) (RWorkspace: none) */ i__1 = *lwork - iwrk + 1; ztrsen_(sense, jobvs, &bwork[1], n, &a[a_offset], lda, &vs[vs_offset], ldvs, &w[1], sdim, rconde, rcondv, &work[iwrk], &i__1, & icond); if (! wantsn) { /* Computing MAX */ i__1 = maxwrk, i__2 = (*sdim << 1) * (*n - *sdim); maxwrk = max(i__1,i__2); } if (icond == -14) { /* Not enough complex workspace */ *info = -15; } } if (wantvs) { /* Undo balancing (CWorkspace: none) (RWorkspace: need N) */ zgebak_("P", "R", n, &ilo, &ihi, &rwork[ibal], n, &vs[vs_offset], ldvs, &ierr); } if (scalea) { /* Undo scaling for the Schur form of A */ zlascl_("U", &c__0, &c__0, &cscale, &anrm, n, n, &a[a_offset], lda, & ierr); i__1 = *lda + 1; zcopy_(n, &a[a_offset], &i__1, &w[1], &c__1); if ((wantsv || wantsb) && *info == 0) { dum[0] = *rcondv; dlascl_("G", &c__0, &c__0, &cscale, &anrm, &c__1, &c__1, dum, & c__1, &ierr); *rcondv = dum[0]; } } work[1].r = (doublereal) maxwrk, work[1].i = 0.; return 0; /* End of ZGEESX */ } /* zgeesx_ */