#include "f2c.h" #include "blaswrap.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 dstebz_(char *range, char *order, integer *n, doublereal *vl, doublereal *vu, integer *il, integer *iu, doublereal *abstol, doublereal *d__, doublereal *e, integer *m, integer *nsplit, doublereal *w, integer *iblock, integer *isplit, doublereal *work, integer *iwork, integer *info) { /* System generated locals */ integer i__1, i__2, i__3; doublereal d__1, d__2, d__3, d__4, d__5; /* Builtin functions */ double sqrt(doublereal), log(doublereal); /* Local variables */ integer j, ib, jb, ie, je, nb; doublereal gl; integer im, in; doublereal gu; integer iw; doublereal wl, wu; integer nwl; doublereal ulp, wlu, wul; integer nwu; doublereal tmp1, tmp2; integer iend, ioff, iout, itmp1, jdisc; extern logical lsame_(char *, char *); integer iinfo; doublereal atoli; integer iwoff; doublereal bnorm; integer itmax; doublereal wkill, rtoli, tnorm; extern doublereal dlamch_(char *); integer ibegin; extern /* Subroutine */ int dlaebz_(integer *, integer *, integer *, integer *, integer *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, doublereal *, doublereal *, integer *, integer *, doublereal *, integer *, integer *); integer irange, idiscl; doublereal safemn; integer idumma[1]; extern /* Subroutine */ int xerbla_(char *, integer *); extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *); integer idiscu, iorder; logical ncnvrg; doublereal pivmin; logical toofew; /* -- LAPACK routine (version 3.1) -- */ /* 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 */ /* ======= */ /* DSTEBZ 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) DOUBLE PRECISION */ /* VU (input) DOUBLE PRECISION */ /* 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) DOUBLE PRECISION */ /* 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*DLAMCH('S'), not zero. */ /* D (input) DOUBLE PRECISION array, dimension (N) */ /* The n diagonal elements of the tridiagonal matrix T. */ /* E (input) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (N) */ /* On exit, the first M elements of W will contain the */ /* eigenvalues. (DSTEBZ 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. (DSTEBZ 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) DOUBLE PRECISION 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 DOUBLE PRECISION, 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 DOUBLE PRECISION, 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_("DSTEBZ", &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 = dlamch_("S"); ulp = dlamch_("P"); rtoli = ulp * 2.; nb = ilaenv_(&c__1, "DSTEBZ", " ", 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.; pivmin = 1.; /* DIR$ NOVECTOR */ i__1 = *n; for (j = 2; j <= i__1; ++j) { /* Computing 2nd power */ d__1 = e[j - 1]; tmp1 = d__1 * d__1; /* Computing 2nd power */ d__2 = ulp; if ((d__1 = d__[j] * d__[j - 1], abs(d__1)) * (d__2 * d__2) + safemn > tmp1) { isplit[*nsplit] = j - 1; ++(*nsplit); work[j - 1] = 0.; } 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.; i__1 = *n - 1; for (j = 1; j <= i__1; ++j) { tmp2 = sqrt(work[j]); /* Computing MAX */ d__1 = gu, d__2 = d__[j] + tmp1 + tmp2; gu = max(d__1,d__2); /* Computing MIN */ d__1 = gl, d__2 = d__[j] - tmp1 - tmp2; gl = min(d__1,d__2); tmp1 = tmp2; /* L20: */ } /* Computing MAX */ d__1 = gu, d__2 = d__[*n] + tmp1; gu = max(d__1,d__2); /* Computing MIN */ d__1 = gl, d__2 = d__[*n] - tmp1; gl = min(d__1,d__2); /* Computing MAX */ d__1 = abs(gl), d__2 = abs(gu); tnorm = max(d__1,d__2); gl = gl - tnorm * 2.1 * ulp * *n - pivmin * 4.2000000000000002; gu = gu + tnorm * 2.1 * ulp * *n + pivmin * 2.1; /* Compute Iteration parameters */ itmax = (integer) ((log(tnorm + pivmin) - log(pivmin)) / log(2.)) + 2; if (*abstol <= 0.) { 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; dlaebz_(&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 */ d__3 = abs(d__[1]) + abs(e[1]), d__4 = (d__1 = d__[*n], abs(d__1)) + ( d__2 = e[*n - 1], abs(d__2)); tnorm = max(d__3,d__4); i__1 = *n - 1; for (j = 2; j <= i__1; ++j) { /* Computing MAX */ d__4 = tnorm, d__5 = (d__1 = d__[j], abs(d__1)) + (d__2 = e[j - 1] , abs(d__2)) + (d__3 = e[j], abs(d__3)); tnorm = max(d__4,d__5); /* L30: */ } if (*abstol <= 0.) { atoli = ulp * tnorm; } else { atoli = *abstol; } if (irange == 2) { wl = *vl; wu = *vu; } else { wl = 0.; wu = 0.; } } /* 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.; i__2 = iend - 1; for (j = ibegin; j <= i__2; ++j) { tmp2 = (d__1 = e[j], abs(d__1)); /* Computing MAX */ d__1 = gu, d__2 = d__[j] + tmp1 + tmp2; gu = max(d__1,d__2); /* Computing MIN */ d__1 = gl, d__2 = d__[j] - tmp1 - tmp2; gl = min(d__1,d__2); tmp1 = tmp2; /* L40: */ } /* Computing MAX */ d__1 = gu, d__2 = d__[iend] + tmp1; gu = max(d__1,d__2); /* Computing MIN */ d__1 = gl, d__2 = d__[iend] - tmp1; gl = min(d__1,d__2); /* Computing MAX */ d__1 = abs(gl), d__2 = abs(gu); bnorm = max(d__1,d__2); gl = gl - bnorm * 2.1 * ulp * in - pivmin * 2.1; gu = gu + bnorm * 2.1 * ulp * in + pivmin * 2.1; /* Compute ATOLI for the current submatrix */ if (*abstol <= 0.) { /* Computing MAX */ d__1 = abs(gl), d__2 = abs(gu); atoli = ulp * max(d__1,d__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; dlaebz_(&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.) ) + 2; dlaebz_(&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]) * .5; /* 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 DSTEBZ */ } /* dstebz_ */