/* Subroutine */ int slarrd_(char *range, char *order, integer *n, real *vl,
real *vu, integer *il, integer *iu, real *gers, real *reltol, real *
d__, real *e, real *e2, real *pivmin, integer *nsplit, integer *
isplit, integer *m, real *w, real *werr, real *wl, real *wu, integer *
iblock, integer *indexw, real *work, integer *iwork, integer *info)
{
/* System generated locals */
integer i__1, i__2, i__3;
real r__1, r__2;
/* Builtin functions */
double log(doublereal);
/* Local variables */
integer i__, j, ib, ie, je, nb;
real gl;
integer im, in;
real gu;
integer iw, jee;
real eps;
integer nwl;
real wlu, wul;
integer nwu;
real tmp1, tmp2;
integer iend, jblk, ioff, iout, itmp1, itmp2, jdisc;
extern logical lsame_(char *, char *);
integer iinfo;
real atoli;
integer iwoff, itmax;
real wkill, rtoli, uflow, tnorm;
integer ibegin, irange, idiscl;
extern doublereal slamch_(char *);
integer idumma[1];
real spdiam;
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 *);
logical ncnvrg, toofew;
/* -- LAPACK auxiliary routine (version 3.1) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SLARRD computes the eigenvalues of a symmetric tridiagonal */
/* matrix T to suitable accuracy. This is an auxiliary code to be */
/* called from SSTEMR. */
/* 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 */
/* = '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 */
/* = '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'. */
/* GERS (input) REAL array, dimension (2*N) */
/* The N Gerschgorin intervals (the i-th Gerschgorin interval */
/* is (GERS(2*i-1), GERS(2*i)). */
/* RELTOL (input) REAL */
/* The minimum relative width of an interval. When an interval */
/* is narrower 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. */
/* 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. */
/* E2 (input) REAL array, dimension (N-1) */
/* The (n-1) squared off-diagonal elements of the tridiagonal matrix T. */
/* PIVMIN (input) REAL */
/* The minimum pivot allowed in the Sturm sequence for T. */
/* NSPLIT (input) INTEGER */
/* The number of diagonal blocks in the matrix T. */
/* 1 <= NSPLIT <= N. */
/* ISPLIT (input) 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.) */
/* M (output) INTEGER */
/* The actual number of eigenvalues found. 0 <= M <= N. */
/* (See also the description of INFO=2,3.) */
/* W (output) REAL array, dimension (N) */
/* On exit, the first M elements of W will contain the */
/* eigenvalue approximations. SLARRD computes an interval */
/* I_j = (a_j, b_j] that includes eigenvalue j. The eigenvalue */
/* approximation is given as the interval midpoint */
/* W(j)= ( a_j + b_j)/2. The corresponding error is bounded by */
/* WERR(j) = abs( a_j - b_j)/2 */
/* WERR (output) REAL array, dimension (N) */
/* The error bound on the corresponding eigenvalue approximation */
/* in W. */
/* WL (output) REAL */
/* WU (output) REAL */
/* The interval (WL, WU] contains all the wanted eigenvalues. */
/* If RANGE='V', then WL=VL and WU=VU. */
/* If RANGE='A', then WL and WU are the global Gerschgorin bounds */
/* on the spectrum. */
/* If RANGE='I', then WL and WU are computed by SLAEBZ from the */
/* index range specified. */
/* 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. (SLARRD may use the remaining N-M elements as */
/* workspace.) */
/* INDEXW (output) INTEGER array, dimension (N) */
/* The indices of the eigenvalues within each block (submatrix); */
/* for example, INDEXW(i)= j and IBLOCK(i)=k imply that the */
/* i-th eigenvalue W(i) is the j-th eigenvalue in block k. */
/* 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 */
/* =================== */
/* 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. */
/* Based on contributions by */
/* W. Kahan, University of California, Berkeley, USA */
/* Beresford Parlett, University of California, Berkeley, USA */
/* Jim Demmel, University of California, Berkeley, USA */
/* Inderjit Dhillon, University of Texas, Austin, USA */
/* Osni Marques, LBNL/NERSC, USA */
/* Christof Voemel, University of California, Berkeley, USA */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Parameter adjustments */
--iwork;
--work;
--indexw;
--iblock;
--werr;
--w;
--isplit;
--e2;
--e;
--d__;
--gers;
/* 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;
}
/* Check for Errors */
if (irange <= 0) {
*info = -1;
} else if (! (lsame_(order, "B") || lsame_(order,
"E"))) {
*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) {
return 0;
}
/* Initialize error flags */
*info = 0;
ncnvrg = FALSE_;
toofew = FALSE_;
/* Quick return if possible */
*m = 0;
if (*n == 0) {
return 0;
}
/* Simplification: */
if (irange == 3 && *il == 1 && *iu == *n) {
irange = 1;
}
/* Get machine constants */
eps = slamch_("P");
uflow = slamch_("U");
/* Special Case when N=1 */
/* Treat case of 1x1 matrix for quick return */
if (*n == 1) {
if (irange == 1 || irange == 2 && d__[1] > *vl && d__[1] <= *vu ||
irange == 3 && *il == 1 && *iu == 1) {
*m = 1;
w[1] = d__[1];
/* The computation error of the eigenvalue is zero */
werr[1] = 0.f;
iblock[1] = 1;
indexw[1] = 1;
}
return 0;
}
/* NB is the minimum vector length for vector bisection, or 0 */
/* if only scalar is to be done. */
nb = ilaenv_(&c__1, "SSTEBZ", " ", n, &c_n1, &c_n1, &c_n1);
if (nb <= 1) {
nb = 0;
}
/* Find global spectral radius */
gl = d__[1];
gu = d__[1];
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Computing MIN */
r__1 = gl, r__2 = gers[(i__ << 1) - 1];
gl = dmin(r__1,r__2);
/* Computing MAX */
r__1 = gu, r__2 = gers[i__ * 2];
gu = dmax(r__1,r__2);
/* L5: */
}
/* Compute global Gerschgorin bounds and spectral diameter */
/* Computing MAX */
r__1 = dabs(gl), r__2 = dabs(gu);
tnorm = dmax(r__1,r__2);
gl = gl - tnorm * 2.f * eps * *n - *pivmin * 4.f;
gu = gu + tnorm * 2.f * eps * *n + *pivmin * 4.f;
spdiam = gu - gl;
/* Input arguments for SLAEBZ: */
/* The relative tolerance. An interval (a,b] lies within */
/* "relative tolerance" if b-a < RELTOL*max(|a|,|b|), */
rtoli = *reltol;
/* Set the absolute tolerance for interval convergence to zero to force */
/* interval convergence based on relative size of the interval. */
/* This is dangerous because intervals might not converge when RELTOL is */
/* small. But at least a very small number should be selected so that for */
/* strongly graded matrices, the code can get relatively accurate */
/* eigenvalues. */
atoli = uflow * 4.f + *pivmin * 4.f;
if (irange == 3) {
/* RANGE='I': Compute an interval containing eigenvalues */
/* IL through IU. The initial interval [GL,GU] from the global */
/* Gerschgorin bounds GL and GU is refined by SLAEBZ. */
itmax = (integer) ((log(tnorm + *pivmin) - log(*pivmin)) / log(2.f))
+ 2;
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], &e2[1], &iwork[5], &work[*n + 1], &work[*n + 5]
, &iout, &iwork[1], &w[1], &iblock[1], &iinfo);
if (iinfo != 0) {
*info = iinfo;
return 0;
}
/* On exit, output intervals may not be ordered by ascending negcount */
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];
}
/* On exit, the interval [WL, WLU] contains a value with negcount NWL, */
/* and [WUL, WU] contains a value with negcount NWU. */
if (nwl < 0 || nwl >= *n || nwu < 1 || nwu > *n) {
*info = 4;
return 0;
}
} else if (irange == 2) {
*wl = *vl;
*wu = *vu;
} else if (irange == 1) {
*wl = gl;
*wu = gu;
}
/* 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 (jblk = 1; jblk <= i__1; ++jblk) {
ioff = iend;
ibegin = ioff + 1;
iend = isplit[jblk];
in = iend - ioff;
if (in == 1) {
/* 1x1 block */
if (*wl >= d__[ibegin] - *pivmin) {
++nwl;
}
if (*wu >= d__[ibegin] - *pivmin) {
++nwu;
}
if (irange == 1 || *wl < d__[ibegin] - *pivmin && *wu >= d__[
ibegin] - *pivmin) {
++(*m);
w[*m] = d__[ibegin];
werr[*m] = 0.f;
/* The gap for a single block doesn't matter for the later */
/* algorithm and is assigned an arbitrary large value */
iblock[*m] = jblk;
indexw[*m] = 1;
}
/* Disabled 2x2 case because of a failure on the following matrix */
/* RANGE = 'I', IL = IU = 4 */
/* Original Tridiagonal, d = [ */
/* -0.150102010615740E+00 */
/* -0.849897989384260E+00 */
/* -0.128208148052635E-15 */
/* 0.128257718286320E-15 */
/* ]; */
/* e = [ */
/* -0.357171383266986E+00 */
/* -0.180411241501588E-15 */
/* -0.175152352710251E-15 */
/* ]; */
/* ELSE IF( IN.EQ.2 ) THEN */
/* * 2x2 block */
/* DISC = SQRT( (HALF*(D(IBEGIN)-D(IEND)))**2 + E(IBEGIN)**2 ) */
/* TMP1 = HALF*(D(IBEGIN)+D(IEND)) */
/* L1 = TMP1 - DISC */
/* IF( WL.GE. L1-PIVMIN ) */
/* $ NWL = NWL + 1 */
/* IF( WU.GE. L1-PIVMIN ) */
/* $ NWU = NWU + 1 */
/* IF( IRANGE.EQ.ALLRNG .OR. ( WL.LT.L1-PIVMIN .AND. WU.GE. */
/* $ L1-PIVMIN ) ) THEN */
/* M = M + 1 */
/* W( M ) = L1 */
/* * The uncertainty of eigenvalues of a 2x2 matrix is very small */
/* WERR( M ) = EPS * ABS( W( M ) ) * TWO */
/* IBLOCK( M ) = JBLK */
/* INDEXW( M ) = 1 */
/* ENDIF */
/* L2 = TMP1 + DISC */
/* IF( WL.GE. L2-PIVMIN ) */
/* $ NWL = NWL + 1 */
/* IF( WU.GE. L2-PIVMIN ) */
/* $ NWU = NWU + 1 */
/* IF( IRANGE.EQ.ALLRNG .OR. ( WL.LT.L2-PIVMIN .AND. WU.GE. */
/* $ L2-PIVMIN ) ) THEN */
/* M = M + 1 */
/* W( M ) = L2 */
/* * The uncertainty of eigenvalues of a 2x2 matrix is very small */
/* WERR( M ) = EPS * ABS( W( M ) ) * TWO */
/* IBLOCK( M ) = JBLK */
/* INDEXW( M ) = 2 */
/* ENDIF */
} else {
/* General Case - block of size IN >= 2 */
/* Compute local Gerschgorin interval and use it as the initial */
/* interval for SLAEBZ */
gu = d__[ibegin];
gl = d__[ibegin];
tmp1 = 0.f;
i__2 = iend;
for (j = ibegin; j <= i__2; ++j) {
/* Computing MIN */
r__1 = gl, r__2 = gers[(j << 1) - 1];
gl = dmin(r__1,r__2);
/* Computing MAX */
r__1 = gu, r__2 = gers[j * 2];
gu = dmax(r__1,r__2);
/* L40: */
}
spdiam = gu - gl;
gl = gl - spdiam * 2.f * eps * in - *pivmin * 2.f;
gu = gu + spdiam * 2.f * eps * in + *pivmin * 2.f;
if (irange > 1) {
if (gu < *wl) {
/* the local block contains none of the wanted eigenvalues */
nwl += in;
nwu += in;
goto L70;
}
/* refine search interval if possible, only range (WL,WU] matters */
gl = dmax(gl,*wl);
gu = dmin(gu,*wu);
if (gl >= gu) {
goto L70;
}
}
/* Find negcount of initial interval boundaries GL and GU */
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], &e2[ibegin], idumma, &
work[*n + 1], &work[*n + (in << 1) + 1], &im, &iwork[1], &
w[*m + 1], &iblock[*m + 1], &iinfo);
if (iinfo != 0) {
*info = iinfo;
return 0;
}
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], &e2[ibegin], idumma, &
work[*n + 1], &work[*n + (in << 1) + 1], &iout, &iwork[1],
&w[*m + 1], &iblock[*m + 1], &iinfo);
if (iinfo != 0) {
*info = iinfo;
return 0;
}
/* Copy eigenvalues into W and IBLOCK */
/* Use -JBLK for block number for unconverged eigenvalues. */
/* Loop over the number of output intervals from SLAEBZ */
i__2 = iout;
for (j = 1; j <= i__2; ++j) {
/* eigenvalue approximation is middle point of interval */
tmp1 = (work[j + *n] + work[j + in + *n]) * .5f;
/* semi length of error interval */
tmp2 = (r__1 = work[j + *n] - work[j + in + *n], dabs(r__1)) *
.5f;
if (j > iout - iinfo) {
/* Flag non-convergence. */
ncnvrg = TRUE_;
ib = -jblk;
} else {
ib = jblk;
}
i__3 = iwork[j + in] + iwoff;
for (je = iwork[j] + 1 + iwoff; je <= i__3; ++je) {
w[je] = tmp1;
werr[je] = tmp2;
indexw[je] = je - iwoff;
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) {
idiscl = *il - 1 - nwl;
idiscu = nwu - *iu;
if (idiscl > 0) {
im = 0;
i__1 = *m;
for (je = 1; je <= i__1; ++je) {
/* Remove some of the smallest eigenvalues from the left so that */
/* at the end IDISCL =0. Move all eigenvalues up to the left. */
if (w[je] <= wlu && idiscl > 0) {
--idiscl;
} else {
++im;
w[im] = w[je];
werr[im] = werr[je];
indexw[im] = indexw[je];
iblock[im] = iblock[je];
}
/* L80: */
}
*m = im;
}
if (idiscu > 0) {
/* Remove some of the largest eigenvalues from the right so that */
/* at the end IDISCU =0. Move all eigenvalues up to the left. */
im = *m + 1;
for (je = *m; je >= 1; --je) {
if (w[je] >= wul && idiscu > 0) {
--idiscu;
} else {
--im;
w[im] = w[je];
werr[im] = werr[je];
indexw[im] = indexw[je];
iblock[im] = iblock[je];
}
/* L81: */
}
jee = 0;
i__1 = *m;
for (je = im; je <= i__1; ++je) {
++jee;
w[jee] = w[je];
werr[jee] = werr[je];
indexw[jee] = indexw[je];
iblock[jee] = iblock[je];
/* L82: */
}
*m = *m - im + 1;
}
if (idiscl > 0 || idiscu > 0) {
/* Code to deal with effects of bad arithmetic. (If N(w) is */
/* monotone non-decreasing, this should never happen.) */
/* 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 marking the corresponding IBLOCK = 0 */
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: */
}
}
/* Now erase all eigenvalues with IBLOCK set to zero */
im = 0;
i__1 = *m;
for (je = 1; je <= i__1; ++je) {
if (iblock[je] != 0) {
++im;
w[im] = w[je];
werr[im] = werr[je];
indexw[im] = indexw[je];
iblock[im] = iblock[je];
}
/* L130: */
}
*m = im;
}
if (idiscl < 0 || idiscu < 0) {
toofew = TRUE_;
}
}
if (irange == 1 && *m != *n || irange == 3 && *m != *iu - *il + 1) {
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 (lsame_(order, "E") && *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) {
tmp2 = werr[ie];
itmp1 = iblock[ie];
itmp2 = indexw[ie];
w[ie] = w[je];
werr[ie] = werr[je];
iblock[ie] = iblock[je];
indexw[ie] = indexw[je];
w[je] = tmp1;
werr[je] = tmp2;
iblock[je] = itmp1;
indexw[je] = itmp2;
}
/* L150: */
}
}
*info = 0;
if (ncnvrg) {
++(*info);
}
if (toofew) {
*info += 2;
}
return 0;
/* End of SLARRD */
} /* slarrd_ */
int sstebz_(char *range, char *order, int *n, float *vl,
float *vu, int *il, int *iu, float *abstol, float *d__, float *e,
int *m, int *nsplit, float *w, int *iblock, int *
isplit, float *work, int *iwork, int *info)
{
/* System generated locals */
int i__1, i__2, i__3;
float r__1, r__2, r__3, r__4, r__5;
/* Builtin functions */
double sqrt(double), log(double);
/* Local variables */
int j, ib, jb, ie, je, nb;
float gl;
int im, in;
float gu;
int iw;
float wl, wu;
int nwl;
float ulp, wlu, wul;
int nwu;
float tmp1, tmp2;
int iend, ioff, iout, itmp1, jdisc;
extern int lsame_(char *, char *);
int iinfo;
float atoli;
int iwoff;
float bnorm;
int itmax;
float wkill, rtoli, tnorm;
int ibegin, irange, idiscl;
extern double slamch_(char *);
float safemn;
int idumma[1];
extern int xerbla_(char *, int *);
extern int ilaenv_(int *, char *, char *, int *, int *,
int *, int *);
int idiscu;
extern int slaebz_(int *, int *, int *,
int *, int *, int *, float *, float *, float *, float *,
float *, float *, int *, float *, float *, int *, int *,
float *, int *, int *);
int iorder;
int ncnvrg;
float pivmin;
int 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], ABS(r__1)) * (r__2 * r__2) + safemn
> tmp1) {
isplit[*nsplit] = j - 1;
++(*nsplit);
work[j - 1] = 0.f;
} else {
work[j - 1] = tmp1;
pivmin = MAX(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 = MAX(r__1,r__2);
/* Computing MIN */
r__1 = gl, r__2 = d__[j] - tmp1 - tmp2;
gl = MIN(r__1,r__2);
tmp1 = tmp2;
/* L20: */
}
/* Computing MAX */
r__1 = gu, r__2 = d__[*n] + tmp1;
gu = MAX(r__1,r__2);
/* Computing MIN */
r__1 = gl, r__2 = d__[*n] - tmp1;
gl = MIN(r__1,r__2);
/* Computing MAX */
r__1 = ABS(gl), r__2 = ABS(gu);
tnorm = MAX(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 = (int) ((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 = ABS(d__[1]) + ABS(e[1]), r__4 = (r__1 = d__[*n], ABS(r__1))
+ (r__2 = e[*n - 1], ABS(r__2));
tnorm = MAX(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], ABS(r__1)) + (r__2 = e[j -
1], ABS(r__2)) + (r__3 = e[j], ABS(r__3));
tnorm = MAX(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], ABS(r__1));
/* Computing MAX */
r__1 = gu, r__2 = d__[j] + tmp1 + tmp2;
gu = MAX(r__1,r__2);
/* Computing MIN */
r__1 = gl, r__2 = d__[j] - tmp1 - tmp2;
gl = MIN(r__1,r__2);
tmp1 = tmp2;
/* L40: */
}
/* Computing MAX */
r__1 = gu, r__2 = d__[iend] + tmp1;
gu = MAX(r__1,r__2);
/* Computing MIN */
r__1 = gl, r__2 = d__[iend] - tmp1;
gl = MIN(r__1,r__2);
/* Computing MAX */
r__1 = ABS(gl), r__2 = ABS(gu);
bnorm = MAX(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 = ABS(gl), r__2 = ABS(gu);
atoli = ulp * MAX(r__1,r__2);
} else {
atoli = *abstol;
}
if (irange > 1) {
if (gu < wl) {
nwl += in;
nwu += in;
goto L70;
}
gl = MAX(gl,wl);
gu = MIN(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 = (int) ((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_ */
/* 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)
{
/* -- LAPACK routine (version 2.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
September 30, 1994
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
= '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
= '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.
=====================================================================
Parameter adjustments
Function Body */
/* 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;
/* 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 */
static integer iend, ioff, iout, itmp1, j, jdisc;
extern logical lsame_(char *, char *);
static integer iinfo;
static real atoli;
static integer iwoff;
static real bnorm;
static integer itmax;
static real wkill, rtoli, tnorm;
static integer ib, jb, ie, je, nb;
static real gl;
static integer im, in, ibegin;
static real gu;
static integer iw;
static real wl;
static integer irange, idiscl;
extern doublereal slamch_(char *);
static real safemn, wu;
static integer idumma[1];
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *, ftnlen, ftnlen);
extern /* Subroutine */ int xerbla_(char *, integer *);
static 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 *);
static integer iorder;
static logical ncnvrg;
static real pivmin;
static logical toofew;
static integer nwl;
static real ulp, wlu, wul;
static integer nwu;
static real tmp1, tmp2;
#define IDUMMA(I) idumma[(I)]
#define IWORK(I) iwork[(I)-1]
#define WORK(I) work[(I)-1]
#define ISPLIT(I) isplit[(I)-1]
#define IBLOCK(I) iblock[(I)-1]
#define W(I) w[(I)-1]
#define E(I) e[(I)-1]
#define D(I) d[(I)-1]
*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 && *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, 6L, 1L);
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;
i__1 = *n;
for (j = 2; j <= *n; ++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 <= *n-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.f * ulp * *n - pivmin * 4.f;
gu = gu + tnorm * 2.f * ulp * *n + pivmin * 2.f;
/* 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 <= *n-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;
}
}
/* 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 <= *nsplit; ++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 <= iend-1; ++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.f * ulp * in - pivmin * 2.f;
gu = gu + bnorm * 2.f * ulp * in + pivmin * 2.f;
/* 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 <= iout; ++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 <= IWORK(j+in)+iwoff; ++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 <= *m; ++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,W
LU],
or high eigenvalues to be discarded are not in (WUL,W
U]
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 neve
r
happen.) */
if (idiscl > 0) {
wkill = wu;
i__1 = idiscl;
for (jdisc = 1; jdisc <= idiscl; ++jdisc) {
iw = 0;
i__2 = *m;
for (je = 1; je <= *m; ++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 <= idiscu; ++jdisc) {
iw = 0;
i__2 = *m;
for (je = 1; je <= *m; ++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 <= *m; ++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 <= *m-1; ++je) {
ie = 0;
tmp1 = W(je);
i__2 = *m;
for (j = je + 1; j <= *m; ++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_ */
