dlaebz.c

/* dlaebz.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 "clapack.h"


/* Subroutine */ int dlaebz_(integer *ijob, integer *nitmax, integer *n, 
	integer *mmax, integer *minp, integer *nbmin, doublereal *abstol, 
	doublereal *reltol, doublereal *pivmin, doublereal *d__, doublereal *
	e, doublereal *e2, integer *nval, doublereal *ab, doublereal *c__, 
	integer *mout, integer *nab, doublereal *work, integer *iwork, 
	integer *info)
{
    /* System generated locals */
    integer nab_dim1, nab_offset, ab_dim1, ab_offset, i__1, i__2, i__3, i__4, 
	    i__5, i__6;
    doublereal d__1, d__2, d__3, d__4;

    /* Local variables */
    integer j, kf, ji, kl, jp, jit;
    doublereal tmp1, tmp2;
    integer itmp1, itmp2, kfnew, klnew;


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

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

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

/*  DLAEBZ contains the iteration loops which compute and use the */
/*  function N(w), which is the count of eigenvalues of a symmetric */
/*  tridiagonal matrix T less than or equal to its argument  w.  It */
/*  performs a choice of two types of loops: */

/*  IJOB=1, followed by */
/*  IJOB=2: It takes as input a list of intervals and returns a list of */
/*          sufficiently small intervals whose union contains the same */
/*          eigenvalues as the union of the original intervals. */
/*          The input intervals are (AB(j,1),AB(j,2)], j=1,...,MINP. */
/*          The output interval (AB(j,1),AB(j,2)] will contain */
/*          eigenvalues NAB(j,1)+1,...,NAB(j,2), where 1 <= j <= MOUT. */

/*  IJOB=3: It performs a binary search in each input interval */
/*          (AB(j,1),AB(j,2)] for a point  w(j)  such that */
/*          N(w(j))=NVAL(j), and uses  C(j)  as the starting point of */
/*          the search.  If such a w(j) is found, then on output */
/*          AB(j,1)=AB(j,2)=w.  If no such w(j) is found, then on output */
/*          (AB(j,1),AB(j,2)] will be a small interval containing the */
/*          point where N(w) jumps through NVAL(j), unless that point */
/*          lies outside the initial interval. */

/*  Note that the intervals are in all cases half-open intervals, */
/*  i.e., of the form  (a,b] , which includes  b  but not  a . */

/*  To avoid underflow, the matrix should be scaled so that its largest */
/*  element is no greater than  overflow**(1/2) * underflow**(1/4) */
/*  in absolute value.  To assure the most accurate computation */
/*  of small eigenvalues, the matrix should be scaled to be */
/*  not much smaller than that, either. */

/*  See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal */
/*  Matrix", Report CS41, Computer Science Dept., Stanford */
/*  University, July 21, 1966 */

/*  Note: the arguments are, in general, *not* checked for unreasonable */
/*  values. */

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

/*  IJOB    (input) INTEGER */
/*          Specifies what is to be done: */
/*          = 1:  Compute NAB for the initial intervals. */
/*          = 2:  Perform bisection iteration to find eigenvalues of T. */
/*          = 3:  Perform bisection iteration to invert N(w), i.e., */
/*                to find a point which has a specified number of */
/*                eigenvalues of T to its left. */
/*          Other values will cause DLAEBZ to return with INFO=-1. */

/*  NITMAX  (input) INTEGER */
/*          The maximum number of "levels" of bisection to be */
/*          performed, i.e., an interval of width W will not be made */
/*          smaller than 2^(-NITMAX) * W.  If not all intervals */
/*          have converged after NITMAX iterations, then INFO is set */
/*          to the number of non-converged intervals. */

/*  N       (input) INTEGER */
/*          The dimension n of the tridiagonal matrix T.  It must be at */
/*          least 1. */

/*  MMAX    (input) INTEGER */
/*          The maximum number of intervals.  If more than MMAX intervals */
/*          are generated, then DLAEBZ will quit with INFO=MMAX+1. */

/*  MINP    (input) INTEGER */
/*          The initial number of intervals.  It may not be greater than */
/*          MMAX. */

/*  NBMIN   (input) INTEGER */
/*          The smallest number of intervals that should be processed */
/*          using a vector loop.  If zero, then only the scalar loop */
/*          will be used. */

/*  ABSTOL  (input) DOUBLE PRECISION */
/*          The minimum (absolute) width of an interval.  When an */
/*          interval is narrower than ABSTOL, or than RELTOL times the */
/*          larger (in magnitude) endpoint, then it is considered to be */
/*          sufficiently small, i.e., converged.  This must be at least */
/*          zero. */

/*  RELTOL  (input) DOUBLE PRECISION */
/*          The minimum relative width of an interval.  When an interval */
/*          is narrower than ABSTOL, or than RELTOL times the larger (in */
/*          magnitude) endpoint, then it is considered to be */
/*          sufficiently small, i.e., converged.  Note: this should */
/*          always be at least radix*machine epsilon. */

/*  PIVMIN  (input) DOUBLE PRECISION */
/*          The minimum absolute value of a "pivot" in the Sturm */
/*          sequence loop.  This *must* be at least  max |e(j)**2| * */
/*          safe_min  and at least safe_min, where safe_min is at least */
/*          the smallest number that can divide one without overflow. */

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

/*  E       (input) DOUBLE PRECISION array, dimension (N) */
/*          The offdiagonal elements of the tridiagonal matrix T in */
/*          positions 1 through N-1.  E(N) is arbitrary. */

/*  E2      (input) DOUBLE PRECISION array, dimension (N) */
/*          The squares of the offdiagonal elements of the tridiagonal */
/*          matrix T.  E2(N) is ignored. */

/*  NVAL    (input/output) INTEGER array, dimension (MINP) */
/*          If IJOB=1 or 2, not referenced. */
/*          If IJOB=3, the desired values of N(w).  The elements of NVAL */
/*          will be reordered to correspond with the intervals in AB. */
/*          Thus, NVAL(j) on output will not, in general be the same as */
/*          NVAL(j) on input, but it will correspond with the interval */
/*          (AB(j,1),AB(j,2)] on output. */

/*  AB      (input/output) DOUBLE PRECISION array, dimension (MMAX,2) */
/*          The endpoints of the intervals.  AB(j,1) is  a(j), the left */
/*          endpoint of the j-th interval, and AB(j,2) is b(j), the */
/*          right endpoint of the j-th interval.  The input intervals */
/*          will, in general, be modified, split, and reordered by the */
/*          calculation. */

/*  C       (input/output) DOUBLE PRECISION array, dimension (MMAX) */
/*          If IJOB=1, ignored. */
/*          If IJOB=2, workspace. */
/*          If IJOB=3, then on input C(j) should be initialized to the */
/*          first search point in the binary search. */

/*  MOUT    (output) INTEGER */
/*          If IJOB=1, the number of eigenvalues in the intervals. */
/*          If IJOB=2 or 3, the number of intervals output. */
/*          If IJOB=3, MOUT will equal MINP. */

/*  NAB     (input/output) INTEGER array, dimension (MMAX,2) */
/*          If IJOB=1, then on output NAB(i,j) will be set to N(AB(i,j)). */
/*          If IJOB=2, then on input, NAB(i,j) should be set.  It must */
/*             satisfy the condition: */
/*             N(AB(i,1)) <= NAB(i,1) <= NAB(i,2) <= N(AB(i,2)), */
/*             which means that in interval i only eigenvalues */
/*             NAB(i,1)+1,...,NAB(i,2) will be considered.  Usually, */
/*             NAB(i,j)=N(AB(i,j)), from a previous call to DLAEBZ with */
/*             IJOB=1. */
/*             On output, NAB(i,j) will contain */
/*             max(na(k),min(nb(k),N(AB(i,j)))), where k is the index of */
/*             the input interval that the output interval */
/*             (AB(j,1),AB(j,2)] came from, and na(k) and nb(k) are the */
/*             the input values of NAB(k,1) and NAB(k,2). */
/*          If IJOB=3, then on output, NAB(i,j) contains N(AB(i,j)), */
/*             unless N(w) > NVAL(i) for all search points  w , in which */
/*             case NAB(i,1) will not be modified, i.e., the output */
/*             value will be the same as the input value (modulo */
/*             reorderings -- see NVAL and AB), or unless N(w) < NVAL(i) */
/*             for all search points  w , in which case NAB(i,2) will */
/*             not be modified.  Normally, NAB should be set to some */
/*             distinctive value(s) before DLAEBZ is called. */

/*  WORK    (workspace) DOUBLE PRECISION array, dimension (MMAX) */
/*          Workspace. */

/*  IWORK   (workspace) INTEGER array, dimension (MMAX) */
/*          Workspace. */

/*  INFO    (output) INTEGER */
/*          = 0:       All intervals converged. */
/*          = 1--MMAX: The last INFO intervals did not converge. */
/*          = MMAX+1:  More than MMAX intervals were generated. */

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

/*      This routine is intended to be called only by other LAPACK */
/*  routines, thus the interface is less user-friendly.  It is intended */
/*  for two purposes: */

/*  (a) finding eigenvalues.  In this case, DLAEBZ should have one or */
/*      more initial intervals set up in AB, and DLAEBZ should be called */
/*      with IJOB=1.  This sets up NAB, and also counts the eigenvalues. */
/*      Intervals with no eigenvalues would usually be thrown out at */
/*      this point.  Also, if not all the eigenvalues in an interval i */
/*      are desired, NAB(i,1) can be increased or NAB(i,2) decreased. */
/*      For example, set NAB(i,1)=NAB(i,2)-1 to get the largest */
/*      eigenvalue.  DLAEBZ is then called with IJOB=2 and MMAX */
/*      no smaller than the value of MOUT returned by the call with */
/*      IJOB=1.  After this (IJOB=2) call, eigenvalues NAB(i,1)+1 */
/*      through NAB(i,2) are approximately AB(i,1) (or AB(i,2)) to the */
/*      tolerance specified by ABSTOL and RELTOL. */

/*  (b) finding an interval (a',b'] containing eigenvalues w(f),...,w(l). */
/*      In this case, start with a Gershgorin interval  (a,b).  Set up */
/*      AB to contain 2 search intervals, both initially (a,b).  One */
/*      NVAL element should contain  f-1  and the other should contain  l */
/*      , while C should contain a and b, resp.  NAB(i,1) should be -1 */
/*      and NAB(i,2) should be N+1, to flag an error if the desired */
/*      interval does not lie in (a,b).  DLAEBZ is then called with */
/*      IJOB=3.  On exit, if w(f-1) < w(f), then one of the intervals -- */
/*      j -- will have AB(j,1)=AB(j,2) and NAB(j,1)=NAB(j,2)=f-1, while */
/*      if, to the specified tolerance, w(f-k)=...=w(f+r), k > 0 and r */
/*      >= 0, then the interval will have  N(AB(j,1))=NAB(j,1)=f-k and */
/*      N(AB(j,2))=NAB(j,2)=f+r.  The cases w(l) < w(l+1) and */
/*      w(l-r)=...=w(l+k) are handled similarly. */

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

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

/*     Check for Errors */

    /* Parameter adjustments */
    nab_dim1 = *mmax;
    nab_offset = 1 + nab_dim1;
    nab -= nab_offset;
    ab_dim1 = *mmax;
    ab_offset = 1 + ab_dim1;
    ab -= ab_offset;
    --d__;
    --e;
    --e2;
    --nval;
    --c__;
    --work;
    --iwork;

    /* Function Body */
    *info = 0;
    if (*ijob < 1 || *ijob > 3) {
	*info = -1;
	return 0;
    }

/*     Initialize NAB */

    if (*ijob == 1) {

/*        Compute the number of eigenvalues in the initial intervals. */

	*mout = 0;
/* DIR$ NOVECTOR */
	i__1 = *minp;
	for (ji = 1; ji <= i__1; ++ji) {
	    for (jp = 1; jp <= 2; ++jp) {
		tmp1 = d__[1] - ab[ji + jp * ab_dim1];
		if (abs(tmp1) < *pivmin) {
		    tmp1 = -(*pivmin);
		}
		nab[ji + jp * nab_dim1] = 0;
		if (tmp1 <= 0.) {
		    nab[ji + jp * nab_dim1] = 1;
		}

		i__2 = *n;
		for (j = 2; j <= i__2; ++j) {
		    tmp1 = d__[j] - e2[j - 1] / tmp1 - ab[ji + jp * ab_dim1];
		    if (abs(tmp1) < *pivmin) {
			tmp1 = -(*pivmin);
		    }
		    if (tmp1 <= 0.) {
			++nab[ji + jp * nab_dim1];
		    }
/* L10: */
		}
/* L20: */
	    }
	    *mout = *mout + nab[ji + (nab_dim1 << 1)] - nab[ji + nab_dim1];
/* L30: */
	}
	return 0;
    }

/*     Initialize for loop */

/*     KF and KL have the following meaning: */
/*        Intervals 1,...,KF-1 have converged. */
/*        Intervals KF,...,KL  still need to be refined. */

    kf = 1;
    kl = *minp;

/*     If IJOB=2, initialize C. */
/*     If IJOB=3, use the user-supplied starting point. */

    if (*ijob == 2) {
	i__1 = *minp;
	for (ji = 1; ji <= i__1; ++ji) {
	    c__[ji] = (ab[ji + ab_dim1] + ab[ji + (ab_dim1 << 1)]) * .5;
/* L40: */
	}
    }

/*     Iteration loop */

    i__1 = *nitmax;
    for (jit = 1; jit <= i__1; ++jit) {

/*        Loop over intervals */

	if (kl - kf + 1 >= *nbmin && *nbmin > 0) {

/*           Begin of Parallel Version of the loop */

	    i__2 = kl;
	    for (ji = kf; ji <= i__2; ++ji) {

/*              Compute N(c), the number of eigenvalues less than c */

		work[ji] = d__[1] - c__[ji];
		iwork[ji] = 0;
		if (work[ji] <= *pivmin) {
		    iwork[ji] = 1;
/* Computing MIN */
		    d__1 = work[ji], d__2 = -(*pivmin);
		    work[ji] = min(d__1,d__2);
		}

		i__3 = *n;
		for (j = 2; j <= i__3; ++j) {
		    work[ji] = d__[j] - e2[j - 1] / work[ji] - c__[ji];
		    if (work[ji] <= *pivmin) {
			++iwork[ji];
/* Computing MIN */
			d__1 = work[ji], d__2 = -(*pivmin);
			work[ji] = min(d__1,d__2);
		    }
/* L50: */
		}
/* L60: */
	    }

	    if (*ijob <= 2) {

/*              IJOB=2: Choose all intervals containing eigenvalues. */

		klnew = kl;
		i__2 = kl;
		for (ji = kf; ji <= i__2; ++ji) {

/*                 Insure that N(w) is monotone */

/* Computing MIN */
/* Computing MAX */
		    i__5 = nab[ji + nab_dim1], i__6 = iwork[ji];
		    i__3 = nab[ji + (nab_dim1 << 1)], i__4 = max(i__5,i__6);
		    iwork[ji] = min(i__3,i__4);

/*                 Update the Queue -- add intervals if both halves */
/*                 contain eigenvalues. */

		    if (iwork[ji] == nab[ji + (nab_dim1 << 1)]) {

/*                    No eigenvalue in the upper interval: */
/*                    just use the lower interval. */

			ab[ji + (ab_dim1 << 1)] = c__[ji];

		    } else if (iwork[ji] == nab[ji + nab_dim1]) {

/*                    No eigenvalue in the lower interval: */
/*                    just use the upper interval. */

			ab[ji + ab_dim1] = c__[ji];
		    } else {
			++klnew;
			if (klnew <= *mmax) {

/*                       Eigenvalue in both intervals -- add upper to */
/*                       queue. */

			    ab[klnew + (ab_dim1 << 1)] = ab[ji + (ab_dim1 << 
				    1)];
			    nab[klnew + (nab_dim1 << 1)] = nab[ji + (nab_dim1 
				    << 1)];
			    ab[klnew + ab_dim1] = c__[ji];
			    nab[klnew + nab_dim1] = iwork[ji];
			    ab[ji + (ab_dim1 << 1)] = c__[ji];
			    nab[ji + (nab_dim1 << 1)] = iwork[ji];
			} else {
			    *info = *mmax + 1;
			}
		    }
/* L70: */
		}
		if (*info != 0) {
		    return 0;
		}
		kl = klnew;
	    } else {

/*              IJOB=3: Binary search.  Keep only the interval containing */
/*                      w   s.t. N(w) = NVAL */

		i__2 = kl;
		for (ji = kf; ji <= i__2; ++ji) {
		    if (iwork[ji] <= nval[ji]) {
			ab[ji + ab_dim1] = c__[ji];
			nab[ji + nab_dim1] = iwork[ji];
		    }
		    if (iwork[ji] >= nval[ji]) {
			ab[ji + (ab_dim1 << 1)] = c__[ji];
			nab[ji + (nab_dim1 << 1)] = iwork[ji];
		    }
/* L80: */
		}
	    }

	} else {

/*           End of Parallel Version of the loop */

/*           Begin of Serial Version of the loop */

	    klnew = kl;
	    i__2 = kl;
	    for (ji = kf; ji <= i__2; ++ji) {

/*              Compute N(w), the number of eigenvalues less than w */

		tmp1 = c__[ji];
		tmp2 = d__[1] - tmp1;
		itmp1 = 0;
		if (tmp2 <= *pivmin) {
		    itmp1 = 1;
/* Computing MIN */
		    d__1 = tmp2, d__2 = -(*pivmin);
		    tmp2 = min(d__1,d__2);
		}

/*              A series of compiler directives to defeat vectorization */
/*              for the next loop */

/* $PL$ CMCHAR=' ' */
/* DIR$          NEXTSCALAR */
/* $DIR          SCALAR */
/* DIR$          NEXT SCALAR */
/* VD$L          NOVECTOR */
/* DEC$          NOVECTOR */
/* VD$           NOVECTOR */
/* VDIR          NOVECTOR */
/* VOCL          LOOP,SCALAR */
/* IBM           PREFER SCALAR */
/* $PL$ CMCHAR='*' */

		i__3 = *n;
		for (j = 2; j <= i__3; ++j) {
		    tmp2 = d__[j] - e2[j - 1] / tmp2 - tmp1;
		    if (tmp2 <= *pivmin) {
			++itmp1;
/* Computing MIN */
			d__1 = tmp2, d__2 = -(*pivmin);
			tmp2 = min(d__1,d__2);
		    }
/* L90: */
		}

		if (*ijob <= 2) {

/*                 IJOB=2: Choose all intervals containing eigenvalues. */

/*                 Insure that N(w) is monotone */

/* Computing MIN */
/* Computing MAX */
		    i__5 = nab[ji + nab_dim1];
		    i__3 = nab[ji + (nab_dim1 << 1)], i__4 = max(i__5,itmp1);
		    itmp1 = min(i__3,i__4);

/*                 Update the Queue -- add intervals if both halves */
/*                 contain eigenvalues. */

		    if (itmp1 == nab[ji + (nab_dim1 << 1)]) {

/*                    No eigenvalue in the upper interval: */
/*                    just use the lower interval. */

			ab[ji + (ab_dim1 << 1)] = tmp1;

		    } else if (itmp1 == nab[ji + nab_dim1]) {

/*                    No eigenvalue in the lower interval: */
/*                    just use the upper interval. */

			ab[ji + ab_dim1] = tmp1;
		    } else if (klnew < *mmax) {

/*                    Eigenvalue in both intervals -- add upper to queue. */

			++klnew;
			ab[klnew + (ab_dim1 << 1)] = ab[ji + (ab_dim1 << 1)];
			nab[klnew + (nab_dim1 << 1)] = nab[ji + (nab_dim1 << 
				1)];
			ab[klnew + ab_dim1] = tmp1;
			nab[klnew + nab_dim1] = itmp1;
			ab[ji + (ab_dim1 << 1)] = tmp1;
			nab[ji + (nab_dim1 << 1)] = itmp1;
		    } else {
			*info = *mmax + 1;
			return 0;
		    }
		} else {

/*                 IJOB=3: Binary search.  Keep only the interval */
/*                         containing  w  s.t. N(w) = NVAL */

		    if (itmp1 <= nval[ji]) {
			ab[ji + ab_dim1] = tmp1;
			nab[ji + nab_dim1] = itmp1;
		    }
		    if (itmp1 >= nval[ji]) {
			ab[ji + (ab_dim1 << 1)] = tmp1;
			nab[ji + (nab_dim1 << 1)] = itmp1;
		    }
		}
/* L100: */
	    }
	    kl = klnew;

/*           End of Serial Version of the loop */

	}

/*        Check for convergence */

	kfnew = kf;
	i__2 = kl;
	for (ji = kf; ji <= i__2; ++ji) {
	    tmp1 = (d__1 = ab[ji + (ab_dim1 << 1)] - ab[ji + ab_dim1], abs(
		    d__1));
/* Computing MAX */
	    d__3 = (d__1 = ab[ji + (ab_dim1 << 1)], abs(d__1)), d__4 = (d__2 =
		     ab[ji + ab_dim1], abs(d__2));
	    tmp2 = max(d__3,d__4);
/* Computing MAX */
	    d__1 = max(*abstol,*pivmin), d__2 = *reltol * tmp2;
	    if (tmp1 < max(d__1,d__2) || nab[ji + nab_dim1] >= nab[ji + (
		    nab_dim1 << 1)]) {

/*              Converged -- Swap with position KFNEW, */
/*                           then increment KFNEW */

		if (ji > kfnew) {
		    tmp1 = ab[ji + ab_dim1];
		    tmp2 = ab[ji + (ab_dim1 << 1)];
		    itmp1 = nab[ji + nab_dim1];
		    itmp2 = nab[ji + (nab_dim1 << 1)];
		    ab[ji + ab_dim1] = ab[kfnew + ab_dim1];
		    ab[ji + (ab_dim1 << 1)] = ab[kfnew + (ab_dim1 << 1)];
		    nab[ji + nab_dim1] = nab[kfnew + nab_dim1];
		    nab[ji + (nab_dim1 << 1)] = nab[kfnew + (nab_dim1 << 1)];
		    ab[kfnew + ab_dim1] = tmp1;
		    ab[kfnew + (ab_dim1 << 1)] = tmp2;
		    nab[kfnew + nab_dim1] = itmp1;
		    nab[kfnew + (nab_dim1 << 1)] = itmp2;
		    if (*ijob == 3) {
			itmp1 = nval[ji];
			nval[ji] = nval[kfnew];
			nval[kfnew] = itmp1;
		    }
		}
		++kfnew;
	    }
/* L110: */
	}
	kf = kfnew;

/*        Choose Midpoints */

	i__2 = kl;
	for (ji = kf; ji <= i__2; ++ji) {
	    c__[ji] = (ab[ji + ab_dim1] + ab[ji + (ab_dim1 << 1)]) * .5;
/* L120: */
	}

/*        If no more intervals to refine, quit. */

	if (kf > kl) {
	    goto L140;
	}
/* L130: */
    }

/*     Converged */

L140:
/* Computing MAX */
    i__1 = kl + 1 - kf;
    *info = max(i__1,0);
    *mout = kl;

    return 0;

/*     End of DLAEBZ */

} /* dlaebz_ */