/* sstebz.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"
/* Table of constant values */
static integer c__1 = 1;
static integer c_n1 = -1;
static integer c__3 = 3;
static integer c__2 = 2;
static integer c__0 = 0;
/* Subroutine */ int sstebz_(char *range, char *order, integer *n, real *vl,
real *vu, integer *il, integer *iu, real *abstol, real *d__, real *e,
integer *m, integer *nsplit, real *w, integer *iblock, integer *
isplit, real *work, integer *iwork, integer *info)
{
/* System generated locals */
integer i__1, i__2, i__3;
real r__1, r__2, r__3, r__4, r__5;
/* Builtin functions */
double sqrt(doublereal), log(doublereal);
/* Local variables */
integer j, ib, jb, ie, je, nb;
real gl;
integer im, in;
real gu;
integer iw;
real wl, wu;
integer nwl;
real ulp, wlu, wul;
integer nwu;
real tmp1, tmp2;
integer iend, ioff, iout, itmp1, jdisc;
extern logical lsame_(char *, char *);
integer iinfo;
real atoli;
integer iwoff;
real bnorm;
integer itmax;
real wkill, rtoli, tnorm;
integer ibegin, irange, idiscl;
extern doublereal slamch_(char *);
real safemn;
integer idumma[1];
extern /* Subroutine */ int xerbla_(char *, integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *);
integer idiscu;
extern /* Subroutine */ int slaebz_(integer *, integer *, integer *,
integer *, integer *, integer *, real *, real *, real *, real *,
real *, real *, integer *, real *, real *, integer *, integer *,
real *, integer *, integer *);
integer iorder;
logical ncnvrg;
real pivmin;
logical toofew;
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* 8-18-00: Increase FUDGE factor for T3E (eca) */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SSTEBZ computes the eigenvalues of a symmetric tridiagonal */
/* matrix T. The user may ask for all eigenvalues, all eigenvalues */
/* in the half-open interval (VL, VU], or the IL-th through IU-th */
/* eigenvalues. */
/* To avoid overflow, the matrix must be scaled so that its */
/* largest element is no greater than overflow**(1/2) * */
/* underflow**(1/4) in absolute value, and for greatest */
/* accuracy, it should not be much smaller than that. */
/* See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal */
/* Matrix", Report CS41, Computer Science Dept., Stanford */
/* University, July 21, 1966. */
/* Arguments */
/* ========= */
/* RANGE (input) CHARACTER*1 */
/* = 'A': ("All") all eigenvalues will be found. */
/* = 'V': ("Value") all eigenvalues in the half-open interval */
/* (VL, VU] will be found. */
/* = 'I': ("Index") the IL-th through IU-th eigenvalues (of the */
/* entire matrix) will be found. */
/* ORDER (input) CHARACTER*1 */
/* = 'B': ("By Block") the eigenvalues will be grouped by */
/* split-off block (see IBLOCK, ISPLIT) and */
/* ordered from smallest to largest within */
/* the block. */
/* = 'E': ("Entire matrix") */
/* the eigenvalues for the entire matrix */
/* will be ordered from smallest to */
/* largest. */
/* N (input) INTEGER */
/* The order of the tridiagonal matrix T. N >= 0. */
/* VL (input) REAL */
/* VU (input) REAL */
/* If RANGE='V', the lower and upper bounds of the interval to */
/* be searched for eigenvalues. Eigenvalues less than or equal */
/* to VL, or greater than VU, will not be returned. VL < VU. */
/* Not referenced if RANGE = 'A' or 'I'. */
/* IL (input) INTEGER */
/* IU (input) INTEGER */
/* If RANGE='I', the indices (in ascending order) of the */
/* smallest and largest eigenvalues to be returned. */
/* 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. */
/* Not referenced if RANGE = 'A' or 'V'. */
/* ABSTOL (input) REAL */
/* The absolute tolerance for the eigenvalues. An eigenvalue */
/* (or cluster) is considered to be located if it has been */
/* determined to lie in an interval whose width is ABSTOL or */
/* less. If ABSTOL is less than or equal to zero, then ULP*|T| */
/* will be used, where |T| means the 1-norm of T. */
/* Eigenvalues will be computed most accurately when ABSTOL is */
/* set to twice the underflow threshold 2*SLAMCH('S'), not zero. */
/* D (input) REAL array, dimension (N) */
/* The n diagonal elements of the tridiagonal matrix T. */
/* E (input) REAL array, dimension (N-1) */
/* The (n-1) off-diagonal elements of the tridiagonal matrix T. */
/* M (output) INTEGER */
/* The actual number of eigenvalues found. 0 <= M <= N. */
/* (See also the description of INFO=2,3.) */
/* NSPLIT (output) INTEGER */
/* The number of diagonal blocks in the matrix T. */
/* 1 <= NSPLIT <= N. */
/* W (output) REAL array, dimension (N) */
/* On exit, the first M elements of W will contain the */
/* eigenvalues. (SSTEBZ may use the remaining N-M elements as */
/* workspace.) */
/* IBLOCK (output) INTEGER array, dimension (N) */
/* At each row/column j where E(j) is zero or small, the */
/* matrix T is considered to split into a block diagonal */
/* matrix. On exit, if INFO = 0, IBLOCK(i) specifies to which */
/* block (from 1 to the number of blocks) the eigenvalue W(i) */
/* belongs. (SSTEBZ may use the remaining N-M elements as */
/* workspace.) */
/* ISPLIT (output) INTEGER array, dimension (N) */
/* The splitting points, at which T breaks up into submatrices. */
/* The first submatrix consists of rows/columns 1 to ISPLIT(1), */
/* the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), */
/* etc., and the NSPLIT-th consists of rows/columns */
/* ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N. */
/* (Only the first NSPLIT elements will actually be used, but */
/* since the user cannot know a priori what value NSPLIT will */
/* have, N words must be reserved for ISPLIT.) */
/* WORK (workspace) REAL array, dimension (4*N) */
/* IWORK (workspace) INTEGER array, dimension (3*N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* > 0: some or all of the eigenvalues failed to converge or */
/* were not computed: */
/* =1 or 3: Bisection failed to converge for some */
/* eigenvalues; these eigenvalues are flagged by a */
/* negative block number. The effect is that the */
/* eigenvalues may not be as accurate as the */
/* absolute and relative tolerances. This is */
/* generally caused by unexpectedly inaccurate */
/* arithmetic. */
/* =2 or 3: RANGE='I' only: Not all of the eigenvalues */
/* IL:IU were found. */
/* Effect: M < IU+1-IL */
/* Cause: non-monotonic arithmetic, causing the */
/* Sturm sequence to be non-monotonic. */
/* Cure: recalculate, using RANGE='A', and pick */
/* out eigenvalues IL:IU. In some cases, */
/* increasing the PARAMETER "FUDGE" may */
/* make things work. */
/* = 4: RANGE='I', and the Gershgorin interval */
/* initially used was too small. No eigenvalues */
/* were computed. */
/* Probable cause: your machine has sloppy */
/* floating-point arithmetic. */
/* Cure: Increase the PARAMETER "FUDGE", */
/* recompile, and try again. */
/* Internal Parameters */
/* =================== */
/* RELFAC REAL, default = 2.0e0 */
/* The relative tolerance. An interval (a,b] lies within */
/* "relative tolerance" if b-a < RELFAC*ulp*max(|a|,|b|), */
/* where "ulp" is the machine precision (distance from 1 to */
/* the next larger floating point number.) */
/* FUDGE REAL, default = 2 */
/* A "fudge factor" to widen the Gershgorin intervals. Ideally, */
/* a value of 1 should work, but on machines with sloppy */
/* arithmetic, this needs to be larger. The default for */
/* publicly released versions should be large enough to handle */
/* the worst machine around. Note that this has no effect */
/* on accuracy of the solution. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Parameter adjustments */
--iwork;
--work;
--isplit;
--iblock;
--w;
--e;
--d__;
/* Function Body */
*info = 0;
/* Decode RANGE */
if (lsame_(range, "A")) {
irange = 1;
} else if (lsame_(range, "V")) {
irange = 2;
} else if (lsame_(range, "I")) {
irange = 3;
} else {
irange = 0;
}
/* Decode ORDER */
if (lsame_(order, "B")) {
iorder = 2;
} else if (lsame_(order, "E")) {
iorder = 1;
} else {
iorder = 0;
}
/* Check for Errors */
if (irange <= 0) {
*info = -1;
} else if (iorder <= 0) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else if (irange == 2) {
if (*vl >= *vu) {
*info = -5;
}
} else if (irange == 3 && (*il < 1 || *il > max(1,*n))) {
*info = -6;
} else if (irange == 3 && (*iu < min(*n,*il) || *iu > *n)) {
*info = -7;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SSTEBZ", &i__1);
return 0;
}
/* Initialize error flags */
*info = 0;
ncnvrg = FALSE_;
toofew = FALSE_;
/* Quick return if possible */
*m = 0;
if (*n == 0) {
return 0;
}
/* Simplifications: */
if (irange == 3 && *il == 1 && *iu == *n) {
irange = 1;
}
/* Get machine constants */
/* NB is the minimum vector length for vector bisection, or 0 */
/* if only scalar is to be done. */
safemn = slamch_("S");
ulp = slamch_("P");
rtoli = ulp * 2.f;
nb = ilaenv_(&c__1, "SSTEBZ", " ", n, &c_n1, &c_n1, &c_n1);
if (nb <= 1) {
nb = 0;
}
/* Special Case when N=1 */
if (*n == 1) {
*nsplit = 1;
isplit[1] = 1;
if (irange == 2 && (*vl >= d__[1] || *vu < d__[1])) {
*m = 0;
} else {
w[1] = d__[1];
iblock[1] = 1;
*m = 1;
}
return 0;
}
/* Compute Splitting Points */
*nsplit = 1;
work[*n] = 0.f;
pivmin = 1.f;
/* DIR$ NOVECTOR */
i__1 = *n;
for (j = 2; j <= i__1; ++j) {
/* Computing 2nd power */
r__1 = e[j - 1];
tmp1 = r__1 * r__1;
/* Computing 2nd power */
r__2 = ulp;
if ((r__1 = d__[j] * d__[j - 1], dabs(r__1)) * (r__2 * r__2) + safemn
> tmp1) {
isplit[*nsplit] = j - 1;
++(*nsplit);
work[j - 1] = 0.f;
} else {
work[j - 1] = tmp1;
pivmin = dmax(pivmin,tmp1);
}
/* L10: */
}
isplit[*nsplit] = *n;
pivmin *= safemn;
/* Compute Interval and ATOLI */
if (irange == 3) {
/* RANGE='I': Compute the interval containing eigenvalues */
/* IL through IU. */
/* Compute Gershgorin interval for entire (split) matrix */
/* and use it as the initial interval */
gu = d__[1];
gl = d__[1];
tmp1 = 0.f;
i__1 = *n - 1;
for (j = 1; j <= i__1; ++j) {
tmp2 = sqrt(work[j]);
/* Computing MAX */
r__1 = gu, r__2 = d__[j] + tmp1 + tmp2;
gu = dmax(r__1,r__2);
/* Computing MIN */
r__1 = gl, r__2 = d__[j] - tmp1 - tmp2;
gl = dmin(r__1,r__2);
tmp1 = tmp2;
/* L20: */
}
/* Computing MAX */
r__1 = gu, r__2 = d__[*n] + tmp1;
gu = dmax(r__1,r__2);
/* Computing MIN */
r__1 = gl, r__2 = d__[*n] - tmp1;
gl = dmin(r__1,r__2);
/* Computing MAX */
r__1 = dabs(gl), r__2 = dabs(gu);
tnorm = dmax(r__1,r__2);
gl = gl - tnorm * 2.1f * ulp * *n - pivmin * 4.2000000000000002f;
gu = gu + tnorm * 2.1f * ulp * *n + pivmin * 2.1f;
/* Compute Iteration parameters */
itmax = (integer) ((log(tnorm + pivmin) - log(pivmin)) / log(2.f)) +
2;
if (*abstol <= 0.f) {
atoli = ulp * tnorm;
} else {
atoli = *abstol;
}
work[*n + 1] = gl;
work[*n + 2] = gl;
work[*n + 3] = gu;
work[*n + 4] = gu;
work[*n + 5] = gl;
work[*n + 6] = gu;
iwork[1] = -1;
iwork[2] = -1;
iwork[3] = *n + 1;
iwork[4] = *n + 1;
iwork[5] = *il - 1;
iwork[6] = *iu;
slaebz_(&c__3, &itmax, n, &c__2, &c__2, &nb, &atoli, &rtoli, &pivmin,
&d__[1], &e[1], &work[1], &iwork[5], &work[*n + 1], &work[*n
+ 5], &iout, &iwork[1], &w[1], &iblock[1], &iinfo);
if (iwork[6] == *iu) {
wl = work[*n + 1];
wlu = work[*n + 3];
nwl = iwork[1];
wu = work[*n + 4];
wul = work[*n + 2];
nwu = iwork[4];
} else {
wl = work[*n + 2];
wlu = work[*n + 4];
nwl = iwork[2];
wu = work[*n + 3];
wul = work[*n + 1];
nwu = iwork[3];
}
if (nwl < 0 || nwl >= *n || nwu < 1 || nwu > *n) {
*info = 4;
return 0;
}
} else {
/* RANGE='A' or 'V' -- Set ATOLI */
/* Computing MAX */
r__3 = dabs(d__[1]) + dabs(e[1]), r__4 = (r__1 = d__[*n], dabs(r__1))
+ (r__2 = e[*n - 1], dabs(r__2));
tnorm = dmax(r__3,r__4);
i__1 = *n - 1;
for (j = 2; j <= i__1; ++j) {
/* Computing MAX */
r__4 = tnorm, r__5 = (r__1 = d__[j], dabs(r__1)) + (r__2 = e[j -
1], dabs(r__2)) + (r__3 = e[j], dabs(r__3));
tnorm = dmax(r__4,r__5);
/* L30: */
}
if (*abstol <= 0.f) {
atoli = ulp * tnorm;
} else {
atoli = *abstol;
}
if (irange == 2) {
wl = *vl;
wu = *vu;
} else {
wl = 0.f;
wu = 0.f;
}
}
/* Find Eigenvalues -- Loop Over Blocks and recompute NWL and NWU. */
/* NWL accumulates the number of eigenvalues .le. WL, */
/* NWU accumulates the number of eigenvalues .le. WU */
*m = 0;
iend = 0;
*info = 0;
nwl = 0;
nwu = 0;
i__1 = *nsplit;
for (jb = 1; jb <= i__1; ++jb) {
ioff = iend;
ibegin = ioff + 1;
iend = isplit[jb];
in = iend - ioff;
if (in == 1) {
/* Special Case -- IN=1 */
if (irange == 1 || wl >= d__[ibegin] - pivmin) {
++nwl;
}
if (irange == 1 || wu >= d__[ibegin] - pivmin) {
++nwu;
}
if (irange == 1 || wl < d__[ibegin] - pivmin && wu >= d__[ibegin]
- pivmin) {
++(*m);
w[*m] = d__[ibegin];
iblock[*m] = jb;
}
} else {
/* General Case -- IN > 1 */
/* Compute Gershgorin Interval */
/* and use it as the initial interval */
gu = d__[ibegin];
gl = d__[ibegin];
tmp1 = 0.f;
i__2 = iend - 1;
for (j = ibegin; j <= i__2; ++j) {
tmp2 = (r__1 = e[j], dabs(r__1));
/* Computing MAX */
r__1 = gu, r__2 = d__[j] + tmp1 + tmp2;
gu = dmax(r__1,r__2);
/* Computing MIN */
r__1 = gl, r__2 = d__[j] - tmp1 - tmp2;
gl = dmin(r__1,r__2);
tmp1 = tmp2;
/* L40: */
}
/* Computing MAX */
r__1 = gu, r__2 = d__[iend] + tmp1;
gu = dmax(r__1,r__2);
/* Computing MIN */
r__1 = gl, r__2 = d__[iend] - tmp1;
gl = dmin(r__1,r__2);
/* Computing MAX */
r__1 = dabs(gl), r__2 = dabs(gu);
bnorm = dmax(r__1,r__2);
gl = gl - bnorm * 2.1f * ulp * in - pivmin * 2.1f;
gu = gu + bnorm * 2.1f * ulp * in + pivmin * 2.1f;
/* Compute ATOLI for the current submatrix */
if (*abstol <= 0.f) {
/* Computing MAX */
r__1 = dabs(gl), r__2 = dabs(gu);
atoli = ulp * dmax(r__1,r__2);
} else {
atoli = *abstol;
}
if (irange > 1) {
if (gu < wl) {
nwl += in;
nwu += in;
goto L70;
}
gl = dmax(gl,wl);
gu = dmin(gu,wu);
if (gl >= gu) {
goto L70;
}
}
/* Set Up Initial Interval */
work[*n + 1] = gl;
work[*n + in + 1] = gu;
slaebz_(&c__1, &c__0, &in, &in, &c__1, &nb, &atoli, &rtoli, &
pivmin, &d__[ibegin], &e[ibegin], &work[ibegin], idumma, &
work[*n + 1], &work[*n + (in << 1) + 1], &im, &iwork[1], &
w[*m + 1], &iblock[*m + 1], &iinfo);
nwl += iwork[1];
nwu += iwork[in + 1];
iwoff = *m - iwork[1];
/* Compute Eigenvalues */
itmax = (integer) ((log(gu - gl + pivmin) - log(pivmin)) / log(
2.f)) + 2;
slaebz_(&c__2, &itmax, &in, &in, &c__1, &nb, &atoli, &rtoli, &
pivmin, &d__[ibegin], &e[ibegin], &work[ibegin], idumma, &
work[*n + 1], &work[*n + (in << 1) + 1], &iout, &iwork[1],
&w[*m + 1], &iblock[*m + 1], &iinfo);
/* Copy Eigenvalues Into W and IBLOCK */
/* Use -JB for block number for unconverged eigenvalues. */
i__2 = iout;
for (j = 1; j <= i__2; ++j) {
tmp1 = (work[j + *n] + work[j + in + *n]) * .5f;
/* Flag non-convergence. */
if (j > iout - iinfo) {
ncnvrg = TRUE_;
ib = -jb;
} else {
ib = jb;
}
i__3 = iwork[j + in] + iwoff;
for (je = iwork[j] + 1 + iwoff; je <= i__3; ++je) {
w[je] = tmp1;
iblock[je] = ib;
/* L50: */
}
/* L60: */
}
*m += im;
}
L70:
;
}
/* If RANGE='I', then (WL,WU) contains eigenvalues NWL+1,...,NWU */
/* If NWL+1 < IL or NWU > IU, discard extra eigenvalues. */
if (irange == 3) {
im = 0;
idiscl = *il - 1 - nwl;
idiscu = nwu - *iu;
if (idiscl > 0 || idiscu > 0) {
i__1 = *m;
for (je = 1; je <= i__1; ++je) {
if (w[je] <= wlu && idiscl > 0) {
--idiscl;
} else if (w[je] >= wul && idiscu > 0) {
--idiscu;
} else {
++im;
w[im] = w[je];
iblock[im] = iblock[je];
}
/* L80: */
}
*m = im;
}
if (idiscl > 0 || idiscu > 0) {
/* Code to deal with effects of bad arithmetic: */
/* Some low eigenvalues to be discarded are not in (WL,WLU], */
/* or high eigenvalues to be discarded are not in (WUL,WU] */
/* so just kill off the smallest IDISCL/largest IDISCU */
/* eigenvalues, by simply finding the smallest/largest */
/* eigenvalue(s). */
/* (If N(w) is monotone non-decreasing, this should never */
/* happen.) */
if (idiscl > 0) {
wkill = wu;
i__1 = idiscl;
for (jdisc = 1; jdisc <= i__1; ++jdisc) {
iw = 0;
i__2 = *m;
for (je = 1; je <= i__2; ++je) {
if (iblock[je] != 0 && (w[je] < wkill || iw == 0)) {
iw = je;
wkill = w[je];
}
/* L90: */
}
iblock[iw] = 0;
/* L100: */
}
}
if (idiscu > 0) {
wkill = wl;
i__1 = idiscu;
for (jdisc = 1; jdisc <= i__1; ++jdisc) {
iw = 0;
i__2 = *m;
for (je = 1; je <= i__2; ++je) {
if (iblock[je] != 0 && (w[je] > wkill || iw == 0)) {
iw = je;
wkill = w[je];
}
/* L110: */
}
iblock[iw] = 0;
/* L120: */
}
}
im = 0;
i__1 = *m;
for (je = 1; je <= i__1; ++je) {
if (iblock[je] != 0) {
++im;
w[im] = w[je];
iblock[im] = iblock[je];
}
/* L130: */
}
*m = im;
}
if (idiscl < 0 || idiscu < 0) {
toofew = TRUE_;
}
}
/* If ORDER='B', do nothing -- the eigenvalues are already sorted */
/* by block. */
/* If ORDER='E', sort the eigenvalues from smallest to largest */
if (iorder == 1 && *nsplit > 1) {
i__1 = *m - 1;
for (je = 1; je <= i__1; ++je) {
ie = 0;
tmp1 = w[je];
i__2 = *m;
for (j = je + 1; j <= i__2; ++j) {
if (w[j] < tmp1) {
ie = j;
tmp1 = w[j];
}
/* L140: */
}
if (ie != 0) {
itmp1 = iblock[ie];
w[ie] = w[je];
iblock[ie] = iblock[je];
w[je] = tmp1;
iblock[je] = itmp1;
}
/* L150: */
}
}
*info = 0;
if (ncnvrg) {
++(*info);
}
if (toofew) {
*info += 2;
}
return 0;
/* End of SSTEBZ */
} /* sstebz_ */