/* dgeesx.f -- translated by f2c (version 20061008). You must link the resulting object file with libf2c: on Microsoft Windows system, link with libf2c.lib; on Linux or Unix systems, link with .../path/to/libf2c.a -lm or, if you install libf2c.a in a standard place, with -lf2c -lm -- in that order, at the end of the command line, as in cc *.o -lf2c -lm Source for libf2c is in /netlib/f2c/libf2c.zip, e.g., http://www.netlib.org/f2c/libf2c.zip */ #include "f2c.h" #include "blaswrap.h" /* Table of constant values */ static integer c__1 = 1; static integer c__0 = 0; static integer c_n1 = -1; /* Subroutine */ int dgeesx_(char *jobvs, char *sort, L_fp select, char * sense, integer *n, doublereal *a, integer *lda, integer *sdim, doublereal *wr, doublereal *wi, doublereal *vs, integer *ldvs, doublereal *rconde, doublereal *rcondv, doublereal *work, integer * lwork, integer *iwork, integer *liwork, logical *bwork, integer *info) { /* System generated locals */ integer a_dim1, a_offset, vs_dim1, vs_offset, i__1, i__2, i__3; /* Builtin functions */ double sqrt(doublereal); /* Local variables */ integer i__, i1, i2, ip, ihi, ilo; doublereal dum[1], eps; integer ibal; doublereal anrm; integer ierr, itau, iwrk, lwrk, inxt, icond, ieval; extern logical lsame_(char *, char *); extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *, doublereal *, integer *), dswap_(integer *, doublereal *, integer *, doublereal *, integer *); logical cursl; integer liwrk; extern /* Subroutine */ int dlabad_(doublereal *, doublereal *), dgebak_( char *, char *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, integer *), dgebal_(char *, integer *, doublereal *, integer *, integer *, integer *, doublereal *, integer *); logical lst2sl, scalea; extern doublereal dlamch_(char *); doublereal cscale; extern doublereal dlange_(char *, integer *, integer *, doublereal *, integer *, doublereal *); extern /* Subroutine */ int dgehrd_(integer *, integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *, integer *), dlascl_(char *, integer *, integer *, doublereal *, doublereal *, integer *, integer *, doublereal *, integer *, integer *), dlacpy_(char *, integer *, integer *, doublereal *, integer *, doublereal *, integer *), xerbla_(char *, integer *); extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *); doublereal bignum; extern /* Subroutine */ int dorghr_(integer *, integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *, integer *), dhseqr_(char *, char *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, integer *, doublereal *, integer *, integer *); logical wantsb; extern /* Subroutine */ int dtrsen_(char *, char *, logical *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, integer *, integer *, integer *, integer *); logical wantse, lastsl; integer minwrk, maxwrk; logical wantsn; doublereal smlnum; integer hswork; logical wantst, lquery, wantsv, wantvs; /* -- LAPACK driver routine (version 3.2) -- */ /* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */ /* November 2006 */ /* .. Scalar Arguments .. */ /* .. */ /* .. Array Arguments .. */ /* .. */ /* .. Function Arguments .. */ /* .. */ /* Purpose */ /* ======= */ /* 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). */ /* 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 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 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) 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. */ /* 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) 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 (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. */ /* ===================================================================== */ /* .. Parameters .. */ /* .. */ /* .. Local Scalars .. */ /* .. */ /* .. Local Arrays .. */ /* .. */ /* .. External Subroutines .. */ /* .. */ /* .. External Functions .. */ /* .. */ /* .. Intrinsic Functions .. */ /* .. */ /* .. Executable Statements .. */ /* Test the input arguments */ /* Parameter adjustments */ a_dim1 = *lda; a_offset = 1 + a_dim1; a -= a_offset; --wr; --wi; vs_dim1 = *ldvs; vs_offset = 1 + vs_dim1; vs -= vs_offset; --work; --iwork; --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"); lquery = *lwork == -1 || *liwork == -1; 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 = -12; } /* Compute workspace */ /* (Note: Comments in the code beginning "RWorkspace:" describe the */ /* minimal amount of real workspace needed at that point in the */ /* code, as well as the preferred amount for good performance. */ /* IWorkspace refers to integer workspace. */ /* NB refers to the optimal block size for the immediately */ /* following subroutine, as returned by ILAENV. */ /* HSWORK refers to the workspace preferred by DHSEQR, 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 DTRSEN later */ /* in the code.) */ if (*info == 0) { liwrk = 1; if (*n == 0) { minwrk = 1; lwrk = 1; } else { maxwrk = (*n << 1) + *n * ilaenv_(&c__1, "DGEHRD", " ", n, &c__1, n, &c__0); minwrk = *n * 3; dhseqr_("S", jobvs, n, &c__1, n, &a[a_offset], lda, &wr[1], &wi[1] , &vs[vs_offset], ldvs, &work[1], &c_n1, &ieval); hswork = (integer) work[1]; if (! wantvs) { /* Computing MAX */ i__1 = maxwrk, i__2 = *n + hswork; maxwrk = max(i__1,i__2); } else { /* Computing MAX */ i__1 = maxwrk, i__2 = (*n << 1) + (*n - 1) * ilaenv_(&c__1, "DORGHR", " ", n, &c__1, n, &c_n1); maxwrk = max(i__1,i__2); /* Computing MAX */ i__1 = maxwrk, i__2 = *n + hswork; maxwrk = max(i__1,i__2); } lwrk = maxwrk; if (! wantsn) { /* Computing MAX */ i__1 = lwrk, i__2 = *n + *n * *n / 2; lwrk = max(i__1,i__2); } if (wantsv || wantsb) { liwrk = *n * *n / 4; } } iwork[1] = liwrk; work[1] = (doublereal) lwrk; if (*lwork < minwrk && ! lquery) { *info = -16; } else if (*liwork < 1 && ! lquery) { *info = -18; } } if (*info != 0) { i__1 = -(*info); xerbla_("DGEESX", &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 = dlange_("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) { dlascl_("G", &c__0, &c__0, &anrm, &cscale, n, n, &a[a_offset], lda, & ierr); } /* Permute the matrix to make it more nearly triangular */ /* (RWorkspace: need N) */ ibal = 1; dgebal_("P", n, &a[a_offset], lda, &ilo, &ihi, &work[ibal], &ierr); /* Reduce to upper Hessenberg form */ /* (RWorkspace: need 3*N, prefer 2*N+N*NB) */ itau = *n + ibal; iwrk = *n + itau; i__1 = *lwork - iwrk + 1; dgehrd_(n, &ilo, &ihi, &a[a_offset], lda, &work[itau], &work[iwrk], &i__1, &ierr); if (wantvs) { /* Copy Householder vectors to VS */ dlacpy_("L", n, n, &a[a_offset], lda, &vs[vs_offset], ldvs) ; /* Generate orthogonal matrix in VS */ /* (RWorkspace: need 3*N-1, prefer 2*N+(N-1)*NB) */ i__1 = *lwork - iwrk + 1; dorghr_(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 */ /* (RWorkspace: need N+1, prefer N+HSWORK (see comments) ) */ iwrk = itau; i__1 = *lwork - iwrk + 1; dhseqr_("S", jobvs, n, &ilo, &ihi, &a[a_offset], lda, &wr[1], &wi[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) { dlascl_("G", &c__0, &c__0, &cscale, &anrm, n, &c__1, &wr[1], n, & ierr); dlascl_("G", &c__0, &c__0, &cscale, &anrm, n, &c__1, &wi[1], n, & ierr); } i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { bwork[i__] = (*select)(&wr[i__], &wi[i__]); /* L10: */ } /* Reorder eigenvalues, transform Schur vectors, and compute */ /* reciprocal condition numbers */ /* (RWorkspace: if SENSE is not 'N', need N+2*SDIM*(N-SDIM) */ /* otherwise, need N ) */ /* (IWorkspace: if SENSE is 'V' or 'B', need SDIM*(N-SDIM) */ /* otherwise, need 0 ) */ i__1 = *lwork - iwrk + 1; dtrsen_(sense, jobvs, &bwork[1], n, &a[a_offset], lda, &vs[vs_offset], ldvs, &wr[1], &wi[1], sdim, rconde, rcondv, &work[iwrk], & i__1, &iwork[1], liwork, &icond); if (! wantsn) { /* Computing MAX */ i__1 = maxwrk, i__2 = *n + (*sdim << 1) * (*n - *sdim); maxwrk = max(i__1,i__2); } if (icond == -15) { /* Not enough real workspace */ *info = -16; } else if (icond == -17) { /* Not enough integer workspace */ *info = -18; } else if (icond > 0) { /* DTRSEN failed to reorder or to restore standard Schur form */ *info = icond + *n; } } if (wantvs) { /* Undo balancing */ /* (RWorkspace: need N) */ dgebak_("P", "R", n, &ilo, &ihi, &work[ibal], n, &vs[vs_offset], ldvs, &ierr); } if (scalea) { /* Undo scaling for the Schur form of A */ dlascl_("H", &c__0, &c__0, &cscale, &anrm, n, n, &a[a_offset], lda, & ierr); i__1 = *lda + 1; dcopy_(n, &a[a_offset], &i__1, &wr[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]; } if (cscale == smlnum) { /* If scaling back towards underflow, adjust WI if an */ /* offdiagonal element of a 2-by-2 block in the Schur form */ /* underflows. */ if (ieval > 0) { i1 = ieval + 1; i2 = ihi - 1; i__1 = ilo - 1; dlascl_("G", &c__0, &c__0, &cscale, &anrm, &i__1, &c__1, &wi[ 1], n, &ierr); } else if (wantst) { i1 = 1; i2 = *n - 1; } else { i1 = ilo; i2 = ihi - 1; } inxt = i1 - 1; i__1 = i2; for (i__ = i1; i__ <= i__1; ++i__) { if (i__ < inxt) { goto L20; } if (wi[i__] == 0.) { inxt = i__ + 1; } else { if (a[i__ + 1 + i__ * a_dim1] == 0.) { wi[i__] = 0.; wi[i__ + 1] = 0.; } else if (a[i__ + 1 + i__ * a_dim1] != 0. && a[i__ + ( i__ + 1) * a_dim1] == 0.) { wi[i__] = 0.; wi[i__ + 1] = 0.; if (i__ > 1) { i__2 = i__ - 1; dswap_(&i__2, &a[i__ * a_dim1 + 1], &c__1, &a[( i__ + 1) * a_dim1 + 1], &c__1); } if (*n > i__ + 1) { i__2 = *n - i__ - 1; dswap_(&i__2, &a[i__ + (i__ + 2) * a_dim1], lda, & a[i__ + 1 + (i__ + 2) * a_dim1], lda); } dswap_(n, &vs[i__ * vs_dim1 + 1], &c__1, &vs[(i__ + 1) * vs_dim1 + 1], &c__1); a[i__ + (i__ + 1) * a_dim1] = a[i__ + 1 + i__ * a_dim1]; a[i__ + 1 + i__ * a_dim1] = 0.; } inxt = i__ + 2; } L20: ; } } i__1 = *n - ieval; /* Computing MAX */ i__3 = *n - ieval; i__2 = max(i__3,1); dlascl_("G", &c__0, &c__0, &cscale, &anrm, &i__1, &c__1, &wi[ieval + 1], &i__2, &ierr); } if (wantst && *info == 0) { /* Check if reordering successful */ lastsl = TRUE_; lst2sl = TRUE_; *sdim = 0; ip = 0; i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { cursl = (*select)(&wr[i__], &wi[i__]); if (wi[i__] == 0.) { if (cursl) { ++(*sdim); } ip = 0; if (cursl && ! lastsl) { *info = *n + 2; } } else { if (ip == 1) { /* Last eigenvalue of conjugate pair */ cursl = cursl || lastsl; lastsl = cursl; if (cursl) { *sdim += 2; } ip = -1; if (cursl && ! lst2sl) { *info = *n + 2; } } else { /* First eigenvalue of conjugate pair */ ip = 1; } } lst2sl = lastsl; lastsl = cursl; /* L30: */ } } work[1] = (doublereal) maxwrk; if (wantsv || wantsb) { /* Computing MAX */ i__1 = 1, i__2 = *sdim * (*n - *sdim); iwork[1] = max(i__1,i__2); } else { iwork[1] = 1; } return 0; /* End of DGEESX */ } /* dgeesx_ */