dsbtrd.c

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/* dsbtrd.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 doublereal c_b9 = 0.;
static doublereal c_b10 = 1.;
static integer c__1 = 1;

/* Subroutine */ int dsbtrd_(char *vect, char *uplo, integer *n, integer *kd, 
        doublereal *ab, integer *ldab, doublereal *d__, doublereal *e, 
        doublereal *q, integer *ldq, doublereal *work, integer *info)
{
    /* System generated locals */
    integer ab_dim1, ab_offset, q_dim1, q_offset, i__1, i__2, i__3, i__4, 
            i__5;

    /* Local variables */
    integer i__, j, k, l, i2, j1, j2, nq, nr, kd1, ibl, iqb, kdn, jin, nrt, 
            kdm1, inca, jend, lend, jinc, incx, last;
    doublereal temp;
    extern /* Subroutine */ int drot_(integer *, doublereal *, integer *, 
            doublereal *, integer *, doublereal *, doublereal *);
    integer j1end, j1inc, iqend;
    extern logical lsame_(char *, char *);
    logical initq, wantq, upper;
    extern /* Subroutine */ int dlar2v_(integer *, doublereal *, doublereal *, 
             doublereal *, integer *, doublereal *, doublereal *, integer *);
    integer iqaend;
    extern /* Subroutine */ int dlaset_(char *, integer *, integer *, 
            doublereal *, doublereal *, doublereal *, integer *), 
            dlartg_(doublereal *, doublereal *, doublereal *, doublereal *, 
            doublereal *), xerbla_(char *, integer *), dlargv_(
            integer *, doublereal *, integer *, doublereal *, integer *, 
            doublereal *, integer *), dlartv_(integer *, doublereal *, 
            integer *, doublereal *, integer *, doublereal *, doublereal *, 
            integer *);


/*  -- LAPACK routine (version 3.2) -- */
/*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/*     November 2006 */

/*     .. Scalar Arguments .. */
/*     .. */
/*     .. Array Arguments .. */
/*     .. */

/*  Purpose */
/*  ======= */

/*  DSBTRD reduces a real symmetric band matrix A to symmetric */
/*  tridiagonal form T by an orthogonal similarity transformation: */
/*  Q**T * A * Q = T. */

/*  Arguments */
/*  ========= */

/*  VECT    (input) CHARACTER*1 */
/*          = 'N':  do not form Q; */
/*          = 'V':  form Q; */
/*          = 'U':  update a matrix X, by forming X*Q. */

/*  UPLO    (input) CHARACTER*1 */
/*          = 'U':  Upper triangle of A is stored; */
/*          = 'L':  Lower triangle of A is stored. */

/*  N       (input) INTEGER */
/*          The order of the matrix A.  N >= 0. */

/*  KD      (input) INTEGER */
/*          The number of superdiagonals of the matrix A if UPLO = 'U', */
/*          or the number of subdiagonals if UPLO = 'L'.  KD >= 0. */

/*  AB      (input/output) DOUBLE PRECISION array, dimension (LDAB,N) */
/*          On entry, the upper or lower triangle of the symmetric band */
/*          matrix A, stored in the first KD+1 rows of the array.  The */
/*          j-th column of A is stored in the j-th column of the array AB */
/*          as follows: */
/*          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; */
/*          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd). */
/*          On exit, the diagonal elements of AB are overwritten by the */
/*          diagonal elements of the tridiagonal matrix T; if KD > 0, the */
/*          elements on the first superdiagonal (if UPLO = 'U') or the */
/*          first subdiagonal (if UPLO = 'L') are overwritten by the */
/*          off-diagonal elements of T; the rest of AB is overwritten by */
/*          values generated during the reduction. */

/*  LDAB    (input) INTEGER */
/*          The leading dimension of the array AB.  LDAB >= KD+1. */

/*  D       (output) DOUBLE PRECISION array, dimension (N) */
/*          The diagonal elements of the tridiagonal matrix T. */

/*  E       (output) DOUBLE PRECISION array, dimension (N-1) */
/*          The off-diagonal elements of the tridiagonal matrix T: */
/*          E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'. */

/*  Q       (input/output) DOUBLE PRECISION array, dimension (LDQ,N) */
/*          On entry, if VECT = 'U', then Q must contain an N-by-N */
/*          matrix X; if VECT = 'N' or 'V', then Q need not be set. */

/*          On exit: */
/*          if VECT = 'V', Q contains the N-by-N orthogonal matrix Q; */
/*          if VECT = 'U', Q contains the product X*Q; */
/*          if VECT = 'N', the array Q is not referenced. */

/*  LDQ     (input) INTEGER */
/*          The leading dimension of the array Q. */
/*          LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'. */

/*  WORK    (workspace) DOUBLE PRECISION array, dimension (N) */

/*  INFO    (output) INTEGER */
/*          = 0:  successful exit */
/*          < 0:  if INFO = -i, the i-th argument had an illegal value */

/*  Further Details */
/*  =============== */

/*  Modified by Linda Kaufman, Bell Labs. */

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

/*     .. Parameters .. */
/*     .. */
/*     .. Local Scalars .. */
/*     .. */
/*     .. External Subroutines .. */
/*     .. */
/*     .. Intrinsic Functions .. */
/*     .. */
/*     .. External Functions .. */
/*     .. */
/*     .. Executable Statements .. */

/*     Test the input parameters */

    /* Parameter adjustments */
    ab_dim1 = *ldab;
    ab_offset = 1 + ab_dim1;
    ab -= ab_offset;
    --d__;
    --e;
    q_dim1 = *ldq;
    q_offset = 1 + q_dim1;
    q -= q_offset;
    --work;

    /* Function Body */
    initq = lsame_(vect, "V");
    wantq = initq || lsame_(vect, "U");
    upper = lsame_(uplo, "U");
    kd1 = *kd + 1;
    kdm1 = *kd - 1;
    incx = *ldab - 1;
    iqend = 1;

    *info = 0;
    if (! wantq && ! lsame_(vect, "N")) {
        *info = -1;
    } else if (! upper && ! lsame_(uplo, "L")) {
        *info = -2;
    } else if (*n < 0) {
        *info = -3;
    } else if (*kd < 0) {
        *info = -4;
    } else if (*ldab < kd1) {
        *info = -6;
    } else if (*ldq < max(1,*n) && wantq) {
        *info = -10;
    }
    if (*info != 0) {
        i__1 = -(*info);
        xerbla_("DSBTRD", &i__1);
        return 0;
    }

/*     Quick return if possible */

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

/*     Initialize Q to the unit matrix, if needed */

    if (initq) {
        dlaset_("Full", n, n, &c_b9, &c_b10, &q[q_offset], ldq);
    }

/*     Wherever possible, plane rotations are generated and applied in */
/*     vector operations of length NR over the index set J1:J2:KD1. */

/*     The cosines and sines of the plane rotations are stored in the */
/*     arrays D and WORK. */

    inca = kd1 * *ldab;
/* Computing MIN */
    i__1 = *n - 1;
    kdn = min(i__1,*kd);
    if (upper) {

        if (*kd > 1) {

/*           Reduce to tridiagonal form, working with upper triangle */

            nr = 0;
            j1 = kdn + 2;
            j2 = 1;

            i__1 = *n - 2;
            for (i__ = 1; i__ <= i__1; ++i__) {

/*              Reduce i-th row of matrix to tridiagonal form */

                for (k = kdn + 1; k >= 2; --k) {
                    j1 += kdn;
                    j2 += kdn;

                    if (nr > 0) {

/*                    generate plane rotations to annihilate nonzero */
/*                    elements which have been created outside the band */

                        dlargv_(&nr, &ab[(j1 - 1) * ab_dim1 + 1], &inca, &
                                work[j1], &kd1, &d__[j1], &kd1);

/*                    apply rotations from the right */


/*                    Dependent on the the number of diagonals either */
/*                    DLARTV or DROT is used */

                        if (nr >= (*kd << 1) - 1) {
                            i__2 = *kd - 1;
                            for (l = 1; l <= i__2; ++l) {
                                dlartv_(&nr, &ab[l + 1 + (j1 - 1) * ab_dim1], 
                                        &inca, &ab[l + j1 * ab_dim1], &inca, &
                                        d__[j1], &work[j1], &kd1);
/* L10: */
                            }

                        } else {
                            jend = j1 + (nr - 1) * kd1;
                            i__2 = jend;
                            i__3 = kd1;
                            for (jinc = j1; i__3 < 0 ? jinc >= i__2 : jinc <= 
                                    i__2; jinc += i__3) {
                                drot_(&kdm1, &ab[(jinc - 1) * ab_dim1 + 2], &
                                        c__1, &ab[jinc * ab_dim1 + 1], &c__1, 
                                        &d__[jinc], &work[jinc]);
/* L20: */
                            }
                        }
                    }


                    if (k > 2) {
                        if (k <= *n - i__ + 1) {

/*                       generate plane rotation to annihilate a(i,i+k-1) */
/*                       within the band */

                            dlartg_(&ab[*kd - k + 3 + (i__ + k - 2) * ab_dim1]
, &ab[*kd - k + 2 + (i__ + k - 1) * 
                                    ab_dim1], &d__[i__ + k - 1], &work[i__ + 
                                    k - 1], &temp);
                            ab[*kd - k + 3 + (i__ + k - 2) * ab_dim1] = temp;

/*                       apply rotation from the right */

                            i__3 = k - 3;
                            drot_(&i__3, &ab[*kd - k + 4 + (i__ + k - 2) * 
                                    ab_dim1], &c__1, &ab[*kd - k + 3 + (i__ + 
                                    k - 1) * ab_dim1], &c__1, &d__[i__ + k - 
                                    1], &work[i__ + k - 1]);
                        }
                        ++nr;
                        j1 = j1 - kdn - 1;
                    }

/*                 apply plane rotations from both sides to diagonal */
/*                 blocks */

                    if (nr > 0) {
                        dlar2v_(&nr, &ab[kd1 + (j1 - 1) * ab_dim1], &ab[kd1 + 
                                j1 * ab_dim1], &ab[*kd + j1 * ab_dim1], &inca, 
                                 &d__[j1], &work[j1], &kd1);
                    }

/*                 apply plane rotations from the left */

                    if (nr > 0) {
                        if ((*kd << 1) - 1 < nr) {

/*                    Dependent on the the number of diagonals either */
/*                    DLARTV or DROT is used */

                            i__3 = *kd - 1;
                            for (l = 1; l <= i__3; ++l) {
                                if (j2 + l > *n) {
                                    nrt = nr - 1;
                                } else {
                                    nrt = nr;
                                }
                                if (nrt > 0) {
                                    dlartv_(&nrt, &ab[*kd - l + (j1 + l) * 
                                            ab_dim1], &inca, &ab[*kd - l + 1 
                                            + (j1 + l) * ab_dim1], &inca, &
                                            d__[j1], &work[j1], &kd1);
                                }
/* L30: */
                            }
                        } else {
                            j1end = j1 + kd1 * (nr - 2);
                            if (j1end >= j1) {
                                i__3 = j1end;
                                i__2 = kd1;
                                for (jin = j1; i__2 < 0 ? jin >= i__3 : jin <=
                                         i__3; jin += i__2) {
                                    i__4 = *kd - 1;
                                    drot_(&i__4, &ab[*kd - 1 + (jin + 1) * 
                                            ab_dim1], &incx, &ab[*kd + (jin + 
                                            1) * ab_dim1], &incx, &d__[jin], &
                                            work[jin]);
/* L40: */
                                }
                            }
/* Computing MIN */
                            i__2 = kdm1, i__3 = *n - j2;
                            lend = min(i__2,i__3);
                            last = j1end + kd1;
                            if (lend > 0) {
                                drot_(&lend, &ab[*kd - 1 + (last + 1) * 
                                        ab_dim1], &incx, &ab[*kd + (last + 1) 
                                        * ab_dim1], &incx, &d__[last], &work[
                                        last]);
                            }
                        }
                    }

                    if (wantq) {

/*                    accumulate product of plane rotations in Q */

                        if (initq) {

/*                 take advantage of the fact that Q was */
/*                 initially the Identity matrix */

                            iqend = max(iqend,j2);
/* Computing MAX */
                            i__2 = 0, i__3 = k - 3;
                            i2 = max(i__2,i__3);
                            iqaend = i__ * *kd + 1;
                            if (k == 2) {
                                iqaend += *kd;
                            }
                            iqaend = min(iqaend,iqend);
                            i__2 = j2;
                            i__3 = kd1;
                            for (j = j1; i__3 < 0 ? j >= i__2 : j <= i__2; j 
                                    += i__3) {
                                ibl = i__ - i2 / kdm1;
                                ++i2;
/* Computing MAX */
                                i__4 = 1, i__5 = j - ibl;
                                iqb = max(i__4,i__5);
                                nq = iqaend + 1 - iqb;
/* Computing MIN */
                                i__4 = iqaend + *kd;
                                iqaend = min(i__4,iqend);
                                drot_(&nq, &q[iqb + (j - 1) * q_dim1], &c__1, 
                                        &q[iqb + j * q_dim1], &c__1, &d__[j], 
                                        &work[j]);
/* L50: */
                            }
                        } else {

                            i__3 = j2;
                            i__2 = kd1;
                            for (j = j1; i__2 < 0 ? j >= i__3 : j <= i__3; j 
                                    += i__2) {
                                drot_(n, &q[(j - 1) * q_dim1 + 1], &c__1, &q[
                                        j * q_dim1 + 1], &c__1, &d__[j], &
                                        work[j]);
/* L60: */
                            }
                        }

                    }

                    if (j2 + kdn > *n) {

/*                    adjust J2 to keep within the bounds of the matrix */

                        --nr;
                        j2 = j2 - kdn - 1;
                    }

                    i__2 = j2;
                    i__3 = kd1;
                    for (j = j1; i__3 < 0 ? j >= i__2 : j <= i__2; j += i__3) 
                            {

/*                    create nonzero element a(j-1,j+kd) outside the band */
/*                    and store it in WORK */

                        work[j + *kd] = work[j] * ab[(j + *kd) * ab_dim1 + 1];
                        ab[(j + *kd) * ab_dim1 + 1] = d__[j] * ab[(j + *kd) * 
                                ab_dim1 + 1];
/* L70: */
                    }
/* L80: */
                }
/* L90: */
            }
        }

        if (*kd > 0) {

/*           copy off-diagonal elements to E */

            i__1 = *n - 1;
            for (i__ = 1; i__ <= i__1; ++i__) {
                e[i__] = ab[*kd + (i__ + 1) * ab_dim1];
/* L100: */
            }
        } else {

/*           set E to zero if original matrix was diagonal */

            i__1 = *n - 1;
            for (i__ = 1; i__ <= i__1; ++i__) {
                e[i__] = 0.;
/* L110: */
            }
        }

/*        copy diagonal elements to D */

        i__1 = *n;
        for (i__ = 1; i__ <= i__1; ++i__) {
            d__[i__] = ab[kd1 + i__ * ab_dim1];
/* L120: */
        }

    } else {

        if (*kd > 1) {

/*           Reduce to tridiagonal form, working with lower triangle */

            nr = 0;
            j1 = kdn + 2;
            j2 = 1;

            i__1 = *n - 2;
            for (i__ = 1; i__ <= i__1; ++i__) {

/*              Reduce i-th column of matrix to tridiagonal form */

                for (k = kdn + 1; k >= 2; --k) {
                    j1 += kdn;
                    j2 += kdn;

                    if (nr > 0) {

/*                    generate plane rotations to annihilate nonzero */
/*                    elements which have been created outside the band */

                        dlargv_(&nr, &ab[kd1 + (j1 - kd1) * ab_dim1], &inca, &
                                work[j1], &kd1, &d__[j1], &kd1);

/*                    apply plane rotations from one side */


/*                    Dependent on the the number of diagonals either */
/*                    DLARTV or DROT is used */

                        if (nr > (*kd << 1) - 1) {
                            i__3 = *kd - 1;
                            for (l = 1; l <= i__3; ++l) {
                                dlartv_(&nr, &ab[kd1 - l + (j1 - kd1 + l) * 
                                        ab_dim1], &inca, &ab[kd1 - l + 1 + (
                                        j1 - kd1 + l) * ab_dim1], &inca, &d__[
                                        j1], &work[j1], &kd1);
/* L130: */
                            }
                        } else {
                            jend = j1 + kd1 * (nr - 1);
                            i__3 = jend;
                            i__2 = kd1;
                            for (jinc = j1; i__2 < 0 ? jinc >= i__3 : jinc <= 
                                    i__3; jinc += i__2) {
                                drot_(&kdm1, &ab[*kd + (jinc - *kd) * ab_dim1]
, &incx, &ab[kd1 + (jinc - *kd) * 
                                        ab_dim1], &incx, &d__[jinc], &work[
                                        jinc]);
/* L140: */
                            }
                        }

                    }

                    if (k > 2) {
                        if (k <= *n - i__ + 1) {

/*                       generate plane rotation to annihilate a(i+k-1,i) */
/*                       within the band */

                            dlartg_(&ab[k - 1 + i__ * ab_dim1], &ab[k + i__ * 
                                    ab_dim1], &d__[i__ + k - 1], &work[i__ + 
                                    k - 1], &temp);
                            ab[k - 1 + i__ * ab_dim1] = temp;

/*                       apply rotation from the left */

                            i__2 = k - 3;
                            i__3 = *ldab - 1;
                            i__4 = *ldab - 1;
                            drot_(&i__2, &ab[k - 2 + (i__ + 1) * ab_dim1], &
                                    i__3, &ab[k - 1 + (i__ + 1) * ab_dim1], &
                                    i__4, &d__[i__ + k - 1], &work[i__ + k - 
                                    1]);
                        }
                        ++nr;
                        j1 = j1 - kdn - 1;
                    }

/*                 apply plane rotations from both sides to diagonal */
/*                 blocks */

                    if (nr > 0) {
                        dlar2v_(&nr, &ab[(j1 - 1) * ab_dim1 + 1], &ab[j1 * 
                                ab_dim1 + 1], &ab[(j1 - 1) * ab_dim1 + 2], &
                                inca, &d__[j1], &work[j1], &kd1);
                    }

/*                 apply plane rotations from the right */


/*                    Dependent on the the number of diagonals either */
/*                    DLARTV or DROT is used */

                    if (nr > 0) {
                        if (nr > (*kd << 1) - 1) {
                            i__2 = *kd - 1;
                            for (l = 1; l <= i__2; ++l) {
                                if (j2 + l > *n) {
                                    nrt = nr - 1;
                                } else {
                                    nrt = nr;
                                }
                                if (nrt > 0) {
                                    dlartv_(&nrt, &ab[l + 2 + (j1 - 1) * 
                                            ab_dim1], &inca, &ab[l + 1 + j1 * 
                                            ab_dim1], &inca, &d__[j1], &work[
                                            j1], &kd1);
                                }
/* L150: */
                            }
                        } else {
                            j1end = j1 + kd1 * (nr - 2);
                            if (j1end >= j1) {
                                i__2 = j1end;
                                i__3 = kd1;
                                for (j1inc = j1; i__3 < 0 ? j1inc >= i__2 : 
                                        j1inc <= i__2; j1inc += i__3) {
                                    drot_(&kdm1, &ab[(j1inc - 1) * ab_dim1 + 
                                            3], &c__1, &ab[j1inc * ab_dim1 + 
                                            2], &c__1, &d__[j1inc], &work[
                                            j1inc]);
/* L160: */
                                }
                            }
/* Computing MIN */
                            i__3 = kdm1, i__2 = *n - j2;
                            lend = min(i__3,i__2);
                            last = j1end + kd1;
                            if (lend > 0) {
                                drot_(&lend, &ab[(last - 1) * ab_dim1 + 3], &
                                        c__1, &ab[last * ab_dim1 + 2], &c__1, 
                                        &d__[last], &work[last]);
                            }
                        }
                    }



                    if (wantq) {

/*                    accumulate product of plane rotations in Q */

                        if (initq) {

/*                 take advantage of the fact that Q was */
/*                 initially the Identity matrix */

                            iqend = max(iqend,j2);
/* Computing MAX */
                            i__3 = 0, i__2 = k - 3;
                            i2 = max(i__3,i__2);
                            iqaend = i__ * *kd + 1;
                            if (k == 2) {
                                iqaend += *kd;
                            }
                            iqaend = min(iqaend,iqend);
                            i__3 = j2;
                            i__2 = kd1;
                            for (j = j1; i__2 < 0 ? j >= i__3 : j <= i__3; j 
                                    += i__2) {
                                ibl = i__ - i2 / kdm1;
                                ++i2;
/* Computing MAX */
                                i__4 = 1, i__5 = j - ibl;
                                iqb = max(i__4,i__5);
                                nq = iqaend + 1 - iqb;
/* Computing MIN */
                                i__4 = iqaend + *kd;
                                iqaend = min(i__4,iqend);
                                drot_(&nq, &q[iqb + (j - 1) * q_dim1], &c__1, 
                                        &q[iqb + j * q_dim1], &c__1, &d__[j], 
                                        &work[j]);
/* L170: */
                            }
                        } else {

                            i__2 = j2;
                            i__3 = kd1;
                            for (j = j1; i__3 < 0 ? j >= i__2 : j <= i__2; j 
                                    += i__3) {
                                drot_(n, &q[(j - 1) * q_dim1 + 1], &c__1, &q[
                                        j * q_dim1 + 1], &c__1, &d__[j], &
                                        work[j]);
/* L180: */
                            }
                        }
                    }

                    if (j2 + kdn > *n) {

/*                    adjust J2 to keep within the bounds of the matrix */

                        --nr;
                        j2 = j2 - kdn - 1;
                    }

                    i__3 = j2;
                    i__2 = kd1;
                    for (j = j1; i__2 < 0 ? j >= i__3 : j <= i__3; j += i__2) 
                            {

/*                    create nonzero element a(j+kd,j-1) outside the */
/*                    band and store it in WORK */

                        work[j + *kd] = work[j] * ab[kd1 + j * ab_dim1];
                        ab[kd1 + j * ab_dim1] = d__[j] * ab[kd1 + j * ab_dim1]
                                ;
/* L190: */
                    }
/* L200: */
                }
/* L210: */
            }
        }

        if (*kd > 0) {

/*           copy off-diagonal elements to E */

            i__1 = *n - 1;
            for (i__ = 1; i__ <= i__1; ++i__) {
                e[i__] = ab[i__ * ab_dim1 + 2];
/* L220: */
            }
        } else {

/*           set E to zero if original matrix was diagonal */

            i__1 = *n - 1;
            for (i__ = 1; i__ <= i__1; ++i__) {
                e[i__] = 0.;
/* L230: */
            }
        }

/*        copy diagonal elements to D */

        i__1 = *n;
        for (i__ = 1; i__ <= i__1; ++i__) {
            d__[i__] = ab[i__ * ab_dim1 + 1];
/* L240: */
        }
    }

    return 0;

/*     End of DSBTRD */

} /* dsbtrd_ */