/* slaebz.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"
/* Subroutine */ int slaebz_(integer *ijob, integer *nitmax, integer *n,
integer *mmax, integer *minp, integer *nbmin, real *abstol, real *
reltol, real *pivmin, real *d__, real *e, real *e2, integer *nval,
real *ab, real *c__, integer *mout, integer *nab, real *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;
real r__1, r__2, r__3, r__4;
/* Local variables */
integer j, kf, ji, kl, jp, jit;
real 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 */
/* ======= */
/* SLAEBZ 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 SLAEBZ 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 SLAEBZ 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) REAL */
/* 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) REAL */
/* 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) REAL */
/* 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) REAL array, dimension (N) */
/* The diagonal elements of the tridiagonal matrix T. */
/* E (input) REAL array, dimension (N) */
/* The offdiagonal elements of the tridiagonal matrix T in */
/* positions 1 through N-1. E(N) is arbitrary. */
/* E2 (input) REAL 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) REAL 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) REAL 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 SLAEBZ 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 SLAEBZ is called. */
/* WORK (workspace) REAL 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, SLAEBZ should have one or */
/* more initial intervals set up in AB, and SLAEBZ 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. SLAEBZ 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). SLAEBZ 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 (dabs(tmp1) < *pivmin) {
tmp1 = -(*pivmin);
}
nab[ji + jp * nab_dim1] = 0;
if (tmp1 <= 0.f) {
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 (dabs(tmp1) < *pivmin) {
tmp1 = -(*pivmin);
}
if (tmp1 <= 0.f) {
++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)]) * .5f;
/* 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 */
r__1 = work[ji], r__2 = -(*pivmin);
work[ji] = dmin(r__1,r__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 */
r__1 = work[ji], r__2 = -(*pivmin);
work[ji] = dmin(r__1,r__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 */
r__1 = tmp2, r__2 = -(*pivmin);
tmp2 = dmin(r__1,r__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 */
r__1 = tmp2, r__2 = -(*pivmin);
tmp2 = dmin(r__1,r__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 = (r__1 = ab[ji + (ab_dim1 << 1)] - ab[ji + ab_dim1], dabs(
r__1));
/* Computing MAX */
r__3 = (r__1 = ab[ji + (ab_dim1 << 1)], dabs(r__1)), r__4 = (r__2
= ab[ji + ab_dim1], dabs(r__2));
tmp2 = dmax(r__3,r__4);
/* Computing MAX */
r__1 = max(*abstol,*pivmin), r__2 = *reltol * tmp2;
if (tmp1 < dmax(r__1,r__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)]) * .5f;
/* 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 SLAEBZ */
} /* slaebz_ */