#include "blaswrap.h" /* -- translated by f2c (version 19990503). You must link the resulting object file with the libraries: -lf2c -lm (in that order) */ #include "f2c.h" /* Common Block Declarations */ struct { real ops, itcnt; } latime_; #define latime_1 latime_ /* 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 */ 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; /* -- LAPACK routine (instrumented to count operations, version 3.0) -- Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., Courant Institute, Argonne National Lab, and Rice University June 30, 1999 Common block to return operation count and iteration count ITCNT is initialized to 0, OPS is only incremented 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 */ --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"); latime_1.ops += 1; rtoli = ulp * 2.f; nb = ilaenv_(&c__1, "SSTEBZ", " ", n, &c_n1, &c_n1, &c_n1, (ftnlen)6, ( ftnlen)1); 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; latime_1.ops = latime_1.ops + (*n - 1) * 5 + 1; /* 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; latime_1.ops = latime_1.ops + (*n - 1) * 5 + 23; 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.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 */ latime_1.ops = latime_1.ops + 3 + (*n - 2 << 1); /* 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 */ latime_1.ops += 4; 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; latime_1.ops = latime_1.ops + (iend - ibegin << 2) + 13; 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.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 */ latime_1.ops += 8; 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. */ latime_1.ops += iout << 1; 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_ */