sgees.c

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/* sgees.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 sgees_(char *jobvs, char *sort, L_fp select, integer *n, 
        real *a, integer *lda, integer *sdim, real *wr, real *wi, real *vs, 
        integer *ldvs, real *work, integer *lwork, 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__;
    real s;
    integer i1, i2, ip, ihi, ilo;
    real dum[1], eps, sep;
    integer ibal;
    real anrm;
    integer idum[1], ierr, itau, iwrk, inxt, icond, ieval;
    extern logical lsame_(char *, char *);
    logical cursl;
    extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *, 
            integer *), sswap_(integer *, real *, integer *, real *, integer *
);
    logical lst2sl;
    extern /* Subroutine */ int slabad_(real *, real *);
    logical scalea;
    real cscale;
    extern /* Subroutine */ int sgebak_(char *, char *, integer *, integer *, 
            integer *, real *, integer *, real *, integer *, integer *), sgebal_(char *, integer *, real *, integer *, 
            integer *, integer *, real *, integer *);
    extern doublereal slamch_(char *), slange_(char *, integer *, 
            integer *, real *, integer *, real *);
    extern /* Subroutine */ int sgehrd_(integer *, integer *, integer *, real 
            *, integer *, real *, real *, integer *, integer *), xerbla_(char 
            *, integer *);
    extern integer ilaenv_(integer *, char *, char *, integer *, integer *, 
            integer *, integer *);
    real bignum;
    extern /* Subroutine */ int slascl_(char *, integer *, integer *, real *, 
            real *, integer *, integer *, real *, integer *, integer *), slacpy_(char *, integer *, integer *, real *, integer *, 
            real *, integer *);
    logical lastsl;
    extern /* Subroutine */ int sorghr_(integer *, integer *, integer *, real 
            *, integer *, real *, real *, integer *, integer *), shseqr_(char 
            *, char *, integer *, integer *, integer *, real *, integer *, 
            real *, real *, real *, integer *, real *, integer *, integer *);
    integer minwrk, maxwrk;
    real smlnum;
    integer hswork;
    extern /* Subroutine */ int strsen_(char *, char *, logical *, integer *, 
            real *, integer *, real *, integer *, real *, real *, integer *, 
            real *, real *, real *, integer *, integer *, integer *, integer *
);
    logical wantst, lquery, 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 */
/*  ======= */

/*  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 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. */
/*  The leading columns of Z then form an orthonormal 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  (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 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 referenced. */

/*  LDVS    (input) INTEGER */
/*          The leading dimension of the array VS.  LDVS >= 1; if */
/*          JOBVS = 'V', LDVS >= N. */

/*  WORK    (workspace/output) REAL array, dimension (MAX(1,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 generally be larger. */

/*          If LWORK = -1, then a workspace query is assumed; the routine */
/*          only calculates the optimal size of the WORK array, returns */
/*          this value as the first entry of the WORK array, and no error */
/*          message related to LWORK is 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 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. */

/*  ===================================================================== */

/*     .. 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;
    --bwork;

    /* Function Body */
    *info = 0;
    lquery = *lwork == -1;
    wantvs = lsame_(jobvs, "V");
    wantst = lsame_(sort, "S");
    if (! wantvs && ! lsame_(jobvs, "N")) {
        *info = -1;
    } else if (! wantst && ! lsame_(sort, "N")) {
        *info = -2;
    } else if (*n < 0) {
        *info = -4;
    } else if (*lda < max(1,*n)) {
        *info = -6;
    } else if (*ldvs < 1 || wantvs && *ldvs < *n) {
        *info = -11;
    }

/*     Compute workspace */
/*      (Note: Comments in the code beginning "Workspace:" describe the */
/*       minimal amount of workspace needed at that point in the code, */
/*       as well as the preferred amount for good performance. */
/*       NB refers to the optimal block size for the immediately */
/*       following subroutine, as returned by ILAENV. */
/*       HSWORK refers to the workspace preferred by SHSEQR, as */
/*       calculated below. HSWORK is computed assuming ILO=1 and IHI=N, */
/*       the worst case.) */

    if (*info == 0) {
        if (*n == 0) {
            minwrk = 1;
            maxwrk = 1;
        } else {
            maxwrk = (*n << 1) + *n * ilaenv_(&c__1, "SGEHRD", " ", n, &c__1, 
                    n, &c__0);
            minwrk = *n * 3;

            shseqr_("S", jobvs, n, &c__1, n, &a[a_offset], lda, &wr[1], &wi[1]
, &vs[vs_offset], ldvs, &work[1], &c_n1, &ieval);
            hswork = 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, 
                        "SORGHR", " ", 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);
            }
        }
        work[1] = (real) maxwrk;

        if (*lwork < minwrk && ! lquery) {
            *info = -13;
        }
    }

    if (*info != 0) {
        i__1 = -(*info);
        xerbla_("SGEES ", &i__1);
        return 0;
    } else if (lquery) {
        return 0;
    }

/*     Quick return if possible */

    if (*n == 0) {
        *sdim = 0;
        return 0;
    }

/*     Get machine constants */

    eps = slamch_("P");
    smlnum = slamch_("S");
    bignum = 1.f / smlnum;
    slabad_(&smlnum, &bignum);
    smlnum = sqrt(smlnum) / eps;
    bignum = 1.f / smlnum;

/*     Scale A if max element outside range [SMLNUM,BIGNUM] */

    anrm = slange_("M", n, n, &a[a_offset], lda, dum);
    scalea = FALSE_;
    if (anrm > 0.f && anrm < smlnum) {
        scalea = TRUE_;
        cscale = smlnum;
    } else if (anrm > bignum) {
        scalea = TRUE_;
        cscale = bignum;
    }
    if (scalea) {
        slascl_("G", &c__0, &c__0, &anrm, &cscale, n, n, &a[a_offset], lda, &
                ierr);
    }

/*     Permute the matrix to make it more nearly triangular */
/*     (Workspace: need N) */

    ibal = 1;
    sgebal_("P", n, &a[a_offset], lda, &ilo, &ihi, &work[ibal], &ierr);

/*     Reduce to upper Hessenberg form */
/*     (Workspace: need 3*N, prefer 2*N+N*NB) */

    itau = *n + ibal;
    iwrk = *n + itau;
    i__1 = *lwork - iwrk + 1;
    sgehrd_(n, &ilo, &ihi, &a[a_offset], lda, &work[itau], &work[iwrk], &i__1, 
             &ierr);

    if (wantvs) {

/*        Copy Householder vectors to VS */

        slacpy_("L", n, n, &a[a_offset], lda, &vs[vs_offset], ldvs)
                ;

/*        Generate orthogonal matrix in VS */
/*        (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB) */

        i__1 = *lwork - iwrk + 1;
        sorghr_(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 */
/*     (Workspace: need N+1, prefer N+HSWORK (see comments) ) */

    iwrk = itau;
    i__1 = *lwork - iwrk + 1;
    shseqr_("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) {
            slascl_("G", &c__0, &c__0, &cscale, &anrm, n, &c__1, &wr[1], n, &
                    ierr);
            slascl_("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 and transform Schur vectors */
/*        (Workspace: none needed) */

        i__1 = *lwork - iwrk + 1;
        strsen_("N", jobvs, &bwork[1], n, &a[a_offset], lda, &vs[vs_offset], 
                ldvs, &wr[1], &wi[1], sdim, &s, &sep, &work[iwrk], &i__1, 
                idum, &c__1, &icond);
        if (icond > 0) {
            *info = *n + icond;
        }
    }

    if (wantvs) {

/*        Undo balancing */
/*        (Workspace: need N) */

        sgebak_("P", "R", n, &ilo, &ihi, &work[ibal], n, &vs[vs_offset], ldvs, 
                 &ierr);
    }

    if (scalea) {

/*        Undo scaling for the Schur form of A */

        slascl_("H", &c__0, &c__0, &cscale, &anrm, n, n, &a[a_offset], lda, &
                ierr);
        i__1 = *lda + 1;
        scopy_(n, &a[a_offset], &i__1, &wr[1], &c__1);
        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;
/* Computing MAX */
                i__3 = ilo - 1;
                i__2 = max(i__3,1);
                slascl_("G", &c__0, &c__0, &cscale, &anrm, &i__1, &c__1, &wi[
                        1], &i__2, &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.f) {
                    inxt = i__ + 1;
                } else {
                    if (a[i__ + 1 + i__ * a_dim1] == 0.f) {
                        wi[i__] = 0.f;
                        wi[i__ + 1] = 0.f;
                    } else if (a[i__ + 1 + i__ * a_dim1] != 0.f && a[i__ + (
                            i__ + 1) * a_dim1] == 0.f) {
                        wi[i__] = 0.f;
                        wi[i__ + 1] = 0.f;
                        if (i__ > 1) {
                            i__2 = i__ - 1;
                            sswap_(&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;
                            sswap_(&i__2, &a[i__ + (i__ + 2) * a_dim1], lda, &
                                    a[i__ + 1 + (i__ + 2) * a_dim1], lda);
                        }
                        if (wantvs) {
                            sswap_(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.f;
                    }
                    inxt = i__ + 2;
                }
L20:
                ;
            }
        }

/*        Undo scaling for the imaginary part of the eigenvalues */

        i__1 = *n - ieval;
/* Computing MAX */
        i__3 = *n - ieval;
        i__2 = max(i__3,1);
        slascl_("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.f) {
                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] = (real) maxwrk;
    return 0;

/*     End of SGEES */

} /* sgees_ */