/* Subroutine */ int dstemr_(char *jobz, char *range, integer *n, doublereal *
d__, doublereal *e, doublereal *vl, doublereal *vu, integer *il,
integer *iu, integer *m, doublereal *w, doublereal *z__, integer *ldz,
integer *nzc, integer *isuppz, logical *tryrac, doublereal *work,
integer *lwork, integer *iwork, integer *liwork, integer *info)
{
/* System generated locals */
integer z_dim1, z_offset, i__1, i__2;
doublereal d__1, d__2;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
integer i__, j;
doublereal r1, r2;
integer jj;
doublereal cs;
integer in;
doublereal sn, wl, wu;
integer iil, iiu;
doublereal eps, tmp;
integer indd, iend, jblk, wend;
doublereal rmin, rmax;
integer itmp;
doublereal tnrm;
extern /* Subroutine */ int dlae2_(doublereal *, doublereal *, doublereal
*, doublereal *, doublereal *);
integer inde2, itmp2;
doublereal rtol1, rtol2;
extern /* Subroutine */ int dscal_(integer *, doublereal *, doublereal *,
integer *);
doublereal scale;
integer indgp;
extern logical lsame_(char *, char *);
integer iinfo, iindw, ilast;
extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *,
doublereal *, integer *), dswap_(integer *, doublereal *, integer
*, doublereal *, integer *);
integer lwmin;
logical wantz;
extern /* Subroutine */ int dlaev2_(doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *, doublereal *,
doublereal *);
extern doublereal dlamch_(char *);
logical alleig;
integer ibegin;
logical indeig;
integer iindbl;
logical valeig;
extern /* Subroutine */ int dlarrc_(char *, integer *, doublereal *,
doublereal *, doublereal *, doublereal *, doublereal *, integer *,
integer *, integer *, integer *), dlarre_(char *,
integer *, doublereal *, doublereal *, integer *, integer *,
doublereal *, doublereal *, doublereal *, doublereal *,
doublereal *, doublereal *, integer *, integer *, integer *,
doublereal *, doublereal *, doublereal *, integer *, integer *,
doublereal *, doublereal *, doublereal *, integer *, integer *);
integer wbegin;
doublereal safmin;
extern /* Subroutine */ int dlarrj_(integer *, doublereal *, doublereal *,
integer *, integer *, doublereal *, integer *, doublereal *,
doublereal *, doublereal *, integer *, doublereal *, doublereal *,
integer *), xerbla_(char *, integer *);
doublereal bignum;
integer inderr, iindwk, indgrs, offset;
extern doublereal dlanst_(char *, integer *, doublereal *, doublereal *);
extern /* Subroutine */ int dlarrr_(integer *, doublereal *, doublereal *,
integer *), dlarrv_(integer *, doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *, integer *, integer *,
integer *, integer *, doublereal *, doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *, integer *, integer *,
doublereal *, doublereal *, integer *, integer *, doublereal *,
integer *, integer *), dlasrt_(char *, integer *, doublereal *,
integer *);
doublereal thresh;
integer iinspl, ifirst, indwrk, liwmin, nzcmin;
doublereal pivmin;
integer nsplit;
doublereal smlnum;
logical lquery, zquery;
/* -- LAPACK computational routine (version 3.1) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* DSTEMR computes selected eigenvalues and, optionally, eigenvectors */
/* of a real symmetric tridiagonal matrix T. Any such unreduced matrix has */
/* a well defined set of pairwise different real eigenvalues, the corresponding */
/* real eigenvectors are pairwise orthogonal. */
/* The spectrum may be computed either completely or partially by specifying */
/* either an interval (VL,VU] or a range of indices IL:IU for the desired */
/* eigenvalues. */
/* Depending on the number of desired eigenvalues, these are computed either */
/* by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are */
/* computed by the use of various suitable L D L^T factorizations near clusters */
/* of close eigenvalues (referred to as RRRs, Relatively Robust */
/* Representations). An informal sketch of the algorithm follows. */
/* For each unreduced block (submatrix) of T, */
/* (a) Compute T - sigma I = L D L^T, so that L and D */
/* define all the wanted eigenvalues to high relative accuracy. */
/* This means that small relative changes in the entries of D and L */
/* cause only small relative changes in the eigenvalues and */
/* eigenvectors. The standard (unfactored) representation of the */
/* tridiagonal matrix T does not have this property in general. */
/* (b) Compute the eigenvalues to suitable accuracy. */
/* If the eigenvectors are desired, the algorithm attains full */
/* accuracy of the computed eigenvalues only right before */
/* the corresponding vectors have to be computed, see steps c) and d). */
/* (c) For each cluster of close eigenvalues, select a new */
/* shift close to the cluster, find a new factorization, and refine */
/* the shifted eigenvalues to suitable accuracy. */
/* (d) For each eigenvalue with a large enough relative separation compute */
/* the corresponding eigenvector by forming a rank revealing twisted */
/* factorization. Go back to (c) for any clusters that remain. */
/* For more details, see: */
/* - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations */
/* to compute orthogonal eigenvectors of symmetric tridiagonal matrices," */
/* Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004. */
/* - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and */
/* Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25, */
/* 2004. Also LAPACK Working Note 154. */
/* - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric */
/* tridiagonal eigenvalue/eigenvector problem", */
/* Computer Science Division Technical Report No. UCB/CSD-97-971, */
/* UC Berkeley, May 1997. */
/* Notes: */
/* 1.DSTEMR works only on machines which follow IEEE-754 */
/* floating-point standard in their handling of infinities and NaNs. */
/* This permits the use of efficient inner loops avoiding a check for */
/* zero divisors. */
/* Arguments */
/* ========= */
/* JOBZ (input) CHARACTER*1 */
/* = 'N': Compute eigenvalues only; */
/* = 'V': Compute eigenvalues and eigenvectors. */
/* RANGE (input) CHARACTER*1 */
/* = 'A': all eigenvalues will be found. */
/* = 'V': all eigenvalues in the half-open interval (VL,VU] */
/* will be found. */
/* = 'I': the IL-th through IU-th eigenvalues will be found. */
/* N (input) INTEGER */
/* The order of the matrix. N >= 0. */
/* D (input/output) DOUBLE PRECISION array, dimension (N) */
/* On entry, the N diagonal elements of the tridiagonal matrix */
/* T. On exit, D is overwritten. */
/* E (input/output) DOUBLE PRECISION array, dimension (N) */
/* On entry, the (N-1) subdiagonal elements of the tridiagonal */
/* matrix T in elements 1 to N-1 of E. E(N) need not be set on */
/* input, but is used internally as workspace. */
/* On exit, E is overwritten. */
/* VL (input) DOUBLE PRECISION */
/* VU (input) DOUBLE PRECISION */
/* If RANGE='V', the lower and upper bounds of the interval to */
/* be searched for eigenvalues. 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. */
/* Not referenced if RANGE = 'A' or 'V'. */
/* M (output) INTEGER */
/* The total number of eigenvalues found. 0 <= M <= N. */
/* If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. */
/* W (output) DOUBLE PRECISION array, dimension (N) */
/* The first M elements contain the selected eigenvalues in */
/* ascending order. */
/* Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M) ) */
/* If JOBZ = 'V', and if INFO = 0, then the first M columns of Z */
/* contain the orthonormal eigenvectors of the matrix T */
/* corresponding to the selected eigenvalues, with the i-th */
/* column of Z holding the eigenvector associated with W(i). */
/* If JOBZ = 'N', then Z is not referenced. */
/* Note: the user must ensure that at least max(1,M) columns are */
/* supplied in the array Z; if RANGE = 'V', the exact value of M */
/* is not known in advance and can be computed with a workspace */
/* query by setting NZC = -1, see below. */
/* LDZ (input) INTEGER */
/* The leading dimension of the array Z. LDZ >= 1, and if */
/* JOBZ = 'V', then LDZ >= max(1,N). */
/* NZC (input) INTEGER */
/* The number of eigenvectors to be held in the array Z. */
/* If RANGE = 'A', then NZC >= max(1,N). */
/* If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU]. */
/* If RANGE = 'I', then NZC >= IU-IL+1. */
/* If NZC = -1, then a workspace query is assumed; the */
/* routine calculates the number of columns of the array Z that */
/* are needed to hold the eigenvectors. */
/* This value is returned as the first entry of the Z array, and */
/* no error message related to NZC is issued by XERBLA. */
/* ISUPPZ (output) INTEGER ARRAY, dimension ( 2*max(1,M) ) */
/* The support of the eigenvectors in Z, i.e., the indices */
/* indicating the nonzero elements in Z. The i-th computed eigenvector */
/* is nonzero only in elements ISUPPZ( 2*i-1 ) through */
/* ISUPPZ( 2*i ). This is relevant in the case when the matrix */
/* is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0. */
/* TRYRAC (input/output) LOGICAL */
/* If TRYRAC.EQ..TRUE., indicates that the code should check whether */
/* the tridiagonal matrix defines its eigenvalues to high relative */
/* accuracy. If so, the code uses relative-accuracy preserving */
/* algorithms that might be (a bit) slower depending on the matrix. */
/* If the matrix does not define its eigenvalues to high relative */
/* accuracy, the code can uses possibly faster algorithms. */
/* If TRYRAC.EQ..FALSE., the code is not required to guarantee */
/* relatively accurate eigenvalues and can use the fastest possible */
/* techniques. */
/* On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix */
/* does not define its eigenvalues to high relative accuracy. */
/* WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) */
/* On exit, if INFO = 0, WORK(1) returns the optimal */
/* (and minimal) LWORK. */
/* LWORK (input) INTEGER */
/* The dimension of the array WORK. LWORK >= max(1,18*N) */
/* if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'. */
/* If LWORK = -1, then a workspace query is assumed; the routine */
/* only calculates the optimal size of the WORK array, returns */
/* this value as the first entry of the WORK array, and no error */
/* message related to LWORK is issued by XERBLA. */
/* IWORK (workspace/output) INTEGER array, dimension (LIWORK) */
/* On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. */
/* LIWORK (input) INTEGER */
/* The dimension of the array IWORK. LIWORK >= max(1,10*N) */
/* if the eigenvectors are desired, and LIWORK >= max(1,8*N) */
/* if only the eigenvalues are to be computed. */
/* If LIWORK = -1, then a workspace query is assumed; the */
/* routine only calculates the optimal size of the IWORK array, */
/* returns this value as the first entry of the IWORK array, and */
/* no error message related to LIWORK is issued by XERBLA. */
/* INFO (output) INTEGER */
/* On exit, INFO */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* > 0: if INFO = 1X, internal error in DLARRE, */
/* if INFO = 2X, internal error in DLARRV. */
/* Here, the digit X = ABS( IINFO ) < 10, where IINFO is */
/* the nonzero error code returned by DLARRE or */
/* DLARRV, respectively. */
/* Further Details */
/* =============== */
/* Based on contributions by */
/* 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 .. */
/* .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--d__;
--e;
--w;
z_dim1 = *ldz;
z_offset = 1 + z_dim1;
z__ -= z_offset;
--isuppz;
--work;
--iwork;
/* Function Body */
wantz = lsame_(jobz, "V");
alleig = lsame_(range, "A");
valeig = lsame_(range, "V");
indeig = lsame_(range, "I");
lquery = *lwork == -1 || *liwork == -1;
zquery = *nzc == -1;
*tryrac = *info != 0;
/* DSTEMR needs WORK of size 6*N, IWORK of size 3*N. */
/* In addition, DLARRE needs WORK of size 6*N, IWORK of size 5*N. */
/* Furthermore, DLARRV needs WORK of size 12*N, IWORK of size 7*N. */
if (wantz) {
lwmin = *n * 18;
liwmin = *n * 10;
} else {
/* need less workspace if only the eigenvalues are wanted */
lwmin = *n * 12;
liwmin = *n << 3;
}
wl = 0.;
wu = 0.;
iil = 0;
iiu = 0;
if (valeig) {
/* We do not reference VL, VU in the cases RANGE = 'I','A' */
/* The interval (WL, WU] contains all the wanted eigenvalues. */
/* It is either given by the user or computed in DLARRE. */
wl = *vl;
wu = *vu;
} else if (indeig) {
/* We do not reference IL, IU in the cases RANGE = 'V','A' */
iil = *il;
iiu = *iu;
}
*info = 0;
if (! (wantz || lsame_(jobz, "N"))) {
*info = -1;
} else if (! (alleig || valeig || indeig)) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else if (valeig && *n > 0 && wu <= wl) {
*info = -7;
} else if (indeig && (iil < 1 || iil > *n)) {
*info = -8;
} else if (indeig && (iiu < iil || iiu > *n)) {
*info = -9;
} else if (*ldz < 1 || wantz && *ldz < *n) {
*info = -13;
} else if (*lwork < lwmin && ! lquery) {
*info = -17;
} else if (*liwork < liwmin && ! lquery) {
*info = -19;
}
/* Get machine constants. */
safmin = dlamch_("Safe minimum");
eps = dlamch_("Precision");
smlnum = safmin / eps;
bignum = 1. / smlnum;
rmin = sqrt(smlnum);
/* Computing MIN */
d__1 = sqrt(bignum), d__2 = 1. / sqrt(sqrt(safmin));
rmax = min(d__1,d__2);
if (*info == 0) {
work[1] = (doublereal) lwmin;
iwork[1] = liwmin;
if (wantz && alleig) {
nzcmin = *n;
} else if (wantz && valeig) {
dlarrc_("T", n, vl, vu, &d__[1], &e[1], &safmin, &nzcmin, &itmp, &
itmp2, info);
} else if (wantz && indeig) {
nzcmin = iiu - iil + 1;
} else {
/* WANTZ .EQ. FALSE. */
nzcmin = 0;
}
if (zquery && *info == 0) {
z__[z_dim1 + 1] = (doublereal) nzcmin;
} else if (*nzc < nzcmin && ! zquery) {
*info = -14;
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DSTEMR", &i__1);
return 0;
} else if (lquery || zquery) {
return 0;
}
/* Handle N = 0, 1, and 2 cases immediately */
*m = 0;
if (*n == 0) {
return 0;
}
if (*n == 1) {
if (alleig || indeig) {
*m = 1;
w[1] = d__[1];
} else {
if (wl < d__[1] && wu >= d__[1]) {
*m = 1;
w[1] = d__[1];
}
}
if (wantz && ! zquery) {
z__[z_dim1 + 1] = 1.;
isuppz[1] = 1;
isuppz[2] = 1;
}
return 0;
}
if (*n == 2) {
if (! wantz) {
dlae2_(&d__[1], &e[1], &d__[2], &r1, &r2);
} else if (wantz && ! zquery) {
dlaev2_(&d__[1], &e[1], &d__[2], &r1, &r2, &cs, &sn);
}
if (alleig || valeig && r2 > wl && r2 <= wu || indeig && iil == 1) {
++(*m);
w[*m] = r2;
if (wantz && ! zquery) {
z__[*m * z_dim1 + 1] = -sn;
z__[*m * z_dim1 + 2] = cs;
/* Note: At most one of SN and CS can be zero. */
if (sn != 0.) {
if (cs != 0.) {
isuppz[(*m << 1) - 1] = 1;
isuppz[(*m << 1) - 1] = 2;
} else {
isuppz[(*m << 1) - 1] = 1;
isuppz[(*m << 1) - 1] = 1;
}
} else {
isuppz[(*m << 1) - 1] = 2;
isuppz[*m * 2] = 2;
}
}
}
if (alleig || valeig && r1 > wl && r1 <= wu || indeig && iiu == 2) {
++(*m);
w[*m] = r1;
if (wantz && ! zquery) {
z__[*m * z_dim1 + 1] = cs;
z__[*m * z_dim1 + 2] = sn;
/* Note: At most one of SN and CS can be zero. */
if (sn != 0.) {
if (cs != 0.) {
isuppz[(*m << 1) - 1] = 1;
isuppz[(*m << 1) - 1] = 2;
} else {
isuppz[(*m << 1) - 1] = 1;
isuppz[(*m << 1) - 1] = 1;
}
} else {
isuppz[(*m << 1) - 1] = 2;
isuppz[*m * 2] = 2;
}
}
}
return 0;
}
/* Continue with general N */
indgrs = 1;
inderr = (*n << 1) + 1;
indgp = *n * 3 + 1;
indd = (*n << 2) + 1;
inde2 = *n * 5 + 1;
indwrk = *n * 6 + 1;
iinspl = 1;
iindbl = *n + 1;
iindw = (*n << 1) + 1;
iindwk = *n * 3 + 1;
/* Scale matrix to allowable range, if necessary. */
/* The allowable range is related to the PIVMIN parameter; see the */
/* comments in DLARRD. The preference for scaling small values */
/* up is heuristic; we expect users' matrices not to be close to the */
/* RMAX threshold. */
scale = 1.;
tnrm = dlanst_("M", n, &d__[1], &e[1]);
if (tnrm > 0. && tnrm < rmin) {
scale = rmin / tnrm;
} else if (tnrm > rmax) {
scale = rmax / tnrm;
}
if (scale != 1.) {
dscal_(n, &scale, &d__[1], &c__1);
i__1 = *n - 1;
dscal_(&i__1, &scale, &e[1], &c__1);
tnrm *= scale;
if (valeig) {
/* If eigenvalues in interval have to be found, */
/* scale (WL, WU] accordingly */
wl *= scale;
wu *= scale;
}
}
/* Compute the desired eigenvalues of the tridiagonal after splitting */
/* into smaller subblocks if the corresponding off-diagonal elements */
/* are small */
/* THRESH is the splitting parameter for DLARRE */
/* A negative THRESH forces the old splitting criterion based on the */
/* size of the off-diagonal. A positive THRESH switches to splitting */
/* which preserves relative accuracy. */
if (*tryrac) {
/* Test whether the matrix warrants the more expensive relative approach. */
dlarrr_(n, &d__[1], &e[1], &iinfo);
} else {
/* The user does not care about relative accurately eigenvalues */
iinfo = -1;
}
/* Set the splitting criterion */
if (iinfo == 0) {
thresh = eps;
} else {
thresh = -eps;
/* relative accuracy is desired but T does not guarantee it */
*tryrac = FALSE_;
}
if (*tryrac) {
/* Copy original diagonal, needed to guarantee relative accuracy */
dcopy_(n, &d__[1], &c__1, &work[indd], &c__1);
}
/* Store the squares of the offdiagonal values of T */
i__1 = *n - 1;
for (j = 1; j <= i__1; ++j) {
/* Computing 2nd power */
d__1 = e[j];
work[inde2 + j - 1] = d__1 * d__1;
/* L5: */
}
/* Set the tolerance parameters for bisection */
if (! wantz) {
/* DLARRE computes the eigenvalues to full precision. */
rtol1 = eps * 4.;
rtol2 = eps * 4.;
} else {
/* DLARRE computes the eigenvalues to less than full precision. */
/* DLARRV will refine the eigenvalue approximations, and we can */
/* need less accurate initial bisection in DLARRE. */
/* Note: these settings do only affect the subset case and DLARRE */
rtol1 = sqrt(eps);
/* Computing MAX */
d__1 = sqrt(eps) * .005, d__2 = eps * 4.;
rtol2 = max(d__1,d__2);
}
dlarre_(range, n, &wl, &wu, &iil, &iiu, &d__[1], &e[1], &work[inde2], &
rtol1, &rtol2, &thresh, &nsplit, &iwork[iinspl], m, &w[1], &work[
inderr], &work[indgp], &iwork[iindbl], &iwork[iindw], &work[
indgrs], &pivmin, &work[indwrk], &iwork[iindwk], &iinfo);
if (iinfo != 0) {
*info = abs(iinfo) + 10;
return 0;
}
/* Note that if RANGE .NE. 'V', DLARRE computes bounds on the desired */
/* part of the spectrum. All desired eigenvalues are contained in */
/* (WL,WU] */
if (wantz) {
/* Compute the desired eigenvectors corresponding to the computed */
/* eigenvalues */
dlarrv_(n, &wl, &wu, &d__[1], &e[1], &pivmin, &iwork[iinspl], m, &
c__1, m, &c_b18, &rtol1, &rtol2, &w[1], &work[inderr], &work[
indgp], &iwork[iindbl], &iwork[iindw], &work[indgrs], &z__[
z_offset], ldz, &isuppz[1], &work[indwrk], &iwork[iindwk], &
iinfo);
if (iinfo != 0) {
*info = abs(iinfo) + 20;
return 0;
}
} else {
/* DLARRE computes eigenvalues of the (shifted) root representation */
/* DLARRV returns the eigenvalues of the unshifted matrix. */
/* However, if the eigenvectors are not desired by the user, we need */
/* to apply the corresponding shifts from DLARRE to obtain the */
/* eigenvalues of the original matrix. */
i__1 = *m;
for (j = 1; j <= i__1; ++j) {
itmp = iwork[iindbl + j - 1];
w[j] += e[iwork[iinspl + itmp - 1]];
/* L20: */
}
}
if (*tryrac) {
/* Refine computed eigenvalues so that they are relatively accurate */
/* with respect to the original matrix T. */
ibegin = 1;
wbegin = 1;
i__1 = iwork[iindbl + *m - 1];
for (jblk = 1; jblk <= i__1; ++jblk) {
iend = iwork[iinspl + jblk - 1];
in = iend - ibegin + 1;
wend = wbegin - 1;
/* check if any eigenvalues have to be refined in this block */
L36:
if (wend < *m) {
if (iwork[iindbl + wend] == jblk) {
++wend;
goto L36;
}
}
if (wend < wbegin) {
ibegin = iend + 1;
goto L39;
}
offset = iwork[iindw + wbegin - 1] - 1;
ifirst = iwork[iindw + wbegin - 1];
ilast = iwork[iindw + wend - 1];
rtol2 = eps * 4.;
dlarrj_(&in, &work[indd + ibegin - 1], &work[inde2 + ibegin - 1],
&ifirst, &ilast, &rtol2, &offset, &w[wbegin], &work[
inderr + wbegin - 1], &work[indwrk], &iwork[iindwk], &
pivmin, &tnrm, &iinfo);
ibegin = iend + 1;
wbegin = wend + 1;
L39:
;
}
}
/* If matrix was scaled, then rescale eigenvalues appropriately. */
if (scale != 1.) {
d__1 = 1. / scale;
dscal_(m, &d__1, &w[1], &c__1);
}
/* If eigenvalues are not in increasing order, then sort them, */
/* possibly along with eigenvectors. */
if (nsplit > 1) {
if (! wantz) {
dlasrt_("I", m, &w[1], &iinfo);
if (iinfo != 0) {
*info = 3;
return 0;
}
} else {
i__1 = *m - 1;
for (j = 1; j <= i__1; ++j) {
i__ = 0;
tmp = w[j];
i__2 = *m;
for (jj = j + 1; jj <= i__2; ++jj) {
if (w[jj] < tmp) {
i__ = jj;
tmp = w[jj];
}
/* L50: */
}
if (i__ != 0) {
w[i__] = w[j];
w[j] = tmp;
if (wantz) {
dswap_(n, &z__[i__ * z_dim1 + 1], &c__1, &z__[j *
z_dim1 + 1], &c__1);
itmp = isuppz[(i__ << 1) - 1];
isuppz[(i__ << 1) - 1] = isuppz[(j << 1) - 1];
isuppz[(j << 1) - 1] = itmp;
itmp = isuppz[i__ * 2];
isuppz[i__ * 2] = isuppz[j * 2];
isuppz[j * 2] = itmp;
}
}
/* L60: */
}
}
}
work[1] = (doublereal) lwmin;
iwork[1] = liwmin;
return 0;
/* End of DSTEMR */
} /* dstemr_ */
/* Subroutine */ int dlasq2_(integer *n, doublereal *z__, integer *info)
{
/* System generated locals */
integer i__1, i__2, i__3;
doublereal d__1, d__2;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
static doublereal d__, e;
static integer k;
static doublereal s, t;
static integer i0, i4, n0, pp;
static doublereal eps, tol;
static integer ipn4;
static doublereal tol2;
static logical ieee;
static integer nbig;
static doublereal dmin__, emin, emax;
static integer ndiv, iter;
static doublereal qmin, temp, qmax, zmax;
static integer splt, nfail;
static doublereal desig, trace, sigma;
static integer iinfo;
extern /* Subroutine */ int dlasq3_(integer *, integer *, doublereal *,
integer *, doublereal *, doublereal *, doublereal *, doublereal *,
integer *, integer *, integer *, logical *);
extern doublereal dlamch_(char *, ftnlen);
static integer iwhila, iwhilb;
static doublereal oldemn, safmin;
extern /* Subroutine */ int xerbla_(char *, integer *, ftnlen);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *, ftnlen, ftnlen);
extern /* Subroutine */ int dlasrt_(char *, integer *, doublereal *,
integer *, ftnlen);
/* -- LAPACK routine (version 3.0) -- */
/* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., */
/* Courant Institute, Argonne National Lab, and Rice University */
/* October 31, 1999 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* DLASQ2 computes all the eigenvalues of the symmetric positive */
/* definite tridiagonal matrix associated with the qd array Z to high */
/* relative accuracy are computed to high relative accuracy, in the */
/* absence of denormalization, underflow and overflow. */
/* To see the relation of Z to the tridiagonal matrix, let L be a */
/* unit lower bidiagonal matrix with subdiagonals Z(2,4,6,,..) and */
/* let U be an upper bidiagonal matrix with 1's above and diagonal */
/* Z(1,3,5,,..). The tridiagonal is L*U or, if you prefer, the */
/* symmetric tridiagonal to which it is similar. */
/* Note : DLASQ2 defines a logical variable, IEEE, which is true */
/* on machines which follow ieee-754 floating-point standard in their */
/* handling of infinities and NaNs, and false otherwise. This variable */
/* is passed to DLASQ3. */
/* Arguments */
/* ========= */
/* N (input) INTEGER */
/* The number of rows and columns in the matrix. N >= 0. */
/* Z (workspace) DOUBLE PRECISION array, dimension ( 4*N ) */
/* On entry Z holds the qd array. On exit, entries 1 to N hold */
/* the eigenvalues in decreasing order, Z( 2*N+1 ) holds the */
/* trace, and Z( 2*N+2 ) holds the sum of the eigenvalues. If */
/* N > 2, then Z( 2*N+3 ) holds the iteration count, Z( 2*N+4 ) */
/* holds NDIVS/NIN^2, and Z( 2*N+5 ) holds the percentage of */
/* shifts that failed. */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if the i-th argument is a scalar and had an illegal */
/* value, then INFO = -i, if the i-th argument is an */
/* array and the j-entry had an illegal value, then */
/* INFO = -(i*100+j) */
/* > 0: the algorithm failed */
/* = 1, a split was marked by a positive value in E */
/* = 2, current block of Z not diagonalized after 30*N */
/* iterations (in inner while loop) */
/* = 3, termination criterion of outer while loop not met */
/* (program created more than N unreduced blocks) */
/* Further Details */
/* =============== */
/* Local Variables: I0:N0 defines a current unreduced segment of Z. */
/* The shifts are accumulated in SIGMA. Iteration count is in ITER. */
/* Ping-pong is controlled by PP (alternates between 0 and 1). */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input arguments. */
/* (in case DLASQ2 is not called by DLASQ1) */
/* Parameter adjustments */
--z__;
/* Function Body */
*info = 0;
eps = dlamch_("Precision", (ftnlen)9);
safmin = dlamch_("Safe minimum", (ftnlen)12);
tol = eps * 100.;
/* Computing 2nd power */
d__1 = tol;
tol2 = d__1 * d__1;
if (*n < 0) {
*info = -1;
xerbla_("DLASQ2", &c__1, (ftnlen)6);
return 0;
} else if (*n == 0) {
return 0;
} else if (*n == 1) {
/* 1-by-1 case. */
if (z__[1] < 0.) {
*info = -201;
xerbla_("DLASQ2", &c__2, (ftnlen)6);
}
return 0;
} else if (*n == 2) {
/* 2-by-2 case. */
if (z__[2] < 0. || z__[3] < 0.) {
*info = -2;
xerbla_("DLASQ2", &c__2, (ftnlen)6);
return 0;
} else if (z__[3] > z__[1]) {
d__ = z__[3];
z__[3] = z__[1];
z__[1] = d__;
}
z__[5] = z__[1] + z__[2] + z__[3];
if (z__[2] > z__[3] * tol2) {
t = (z__[1] - z__[3] + z__[2]) * .5;
s = z__[3] * (z__[2] / t);
if (s <= t) {
s = z__[3] * (z__[2] / (t * (sqrt(s / t + 1.) + 1.)));
} else {
s = z__[3] * (z__[2] / (t + sqrt(t) * sqrt(t + s)));
}
t = z__[1] + (s + z__[2]);
z__[3] *= z__[1] / t;
z__[1] = t;
}
z__[2] = z__[3];
z__[6] = z__[2] + z__[1];
return 0;
}
/* Check for negative data and compute sums of q's and e's. */
z__[*n * 2] = 0.;
emin = z__[2];
qmax = 0.;
zmax = 0.;
d__ = 0.;
e = 0.;
i__1 = *n - 1 << 1;
for (k = 1; k <= i__1; k += 2) {
if (z__[k] < 0.) {
*info = -(k + 200);
xerbla_("DLASQ2", &c__2, (ftnlen)6);
return 0;
} else if (z__[k + 1] < 0.) {
*info = -(k + 201);
xerbla_("DLASQ2", &c__2, (ftnlen)6);
return 0;
}
d__ += z__[k];
e += z__[k + 1];
/* Computing MAX */
d__1 = qmax, d__2 = z__[k];
qmax = max(d__1,d__2);
/* Computing MIN */
d__1 = emin, d__2 = z__[k + 1];
emin = min(d__1,d__2);
/* Computing MAX */
d__1 = max(qmax,zmax), d__2 = z__[k + 1];
zmax = max(d__1,d__2);
/* L10: */
}
if (z__[(*n << 1) - 1] < 0.) {
*info = -((*n << 1) + 199);
xerbla_("DLASQ2", &c__2, (ftnlen)6);
return 0;
}
d__ += z__[(*n << 1) - 1];
/* Computing MAX */
d__1 = qmax, d__2 = z__[(*n << 1) - 1];
qmax = max(d__1,d__2);
zmax = max(qmax,zmax);
/* Check for diagonality. */
if (e == 0.) {
i__1 = *n;
for (k = 2; k <= i__1; ++k) {
z__[k] = z__[(k << 1) - 1];
/* L20: */
}
dlasrt_("D", n, &z__[1], &iinfo, (ftnlen)1);
z__[(*n << 1) - 1] = d__;
return 0;
}
trace = d__ + e;
/* Check for zero data. */
if (trace == 0.) {
z__[(*n << 1) - 1] = 0.;
return 0;
}
/* Check whether the machine is IEEE conformable. */
ieee = ilaenv_(&c__10, "DLASQ2", "N", &c__1, &c__2, &c__3, &c__4, (ftnlen)
6, (ftnlen)1) == 1 && ilaenv_(&c__11, "DLASQ2", "N", &c__1, &c__2,
&c__3, &c__4, (ftnlen)6, (ftnlen)1) == 1;
/* Rearrange data for locality: Z=(q1,qq1,e1,ee1,q2,qq2,e2,ee2,...). */
for (k = *n << 1; k >= 2; k += -2) {
z__[k * 2] = 0.;
z__[(k << 1) - 1] = z__[k];
z__[(k << 1) - 2] = 0.;
z__[(k << 1) - 3] = z__[k - 1];
/* L30: */
}
i0 = 1;
n0 = *n;
/* Reverse the qd-array, if warranted. */
if (z__[(i0 << 2) - 3] * 1.5 < z__[(n0 << 2) - 3]) {
ipn4 = i0 + n0 << 2;
i__1 = i0 + n0 - 1 << 1;
for (i4 = i0 << 2; i4 <= i__1; i4 += 4) {
temp = z__[i4 - 3];
z__[i4 - 3] = z__[ipn4 - i4 - 3];
z__[ipn4 - i4 - 3] = temp;
temp = z__[i4 - 1];
z__[i4 - 1] = z__[ipn4 - i4 - 5];
z__[ipn4 - i4 - 5] = temp;
/* L40: */
}
}
/* Initial split checking via dqd and Li's test. */
pp = 0;
for (k = 1; k <= 2; ++k) {
d__ = z__[(n0 << 2) + pp - 3];
i__1 = (i0 << 2) + pp;
for (i4 = (n0 - 1 << 2) + pp; i4 >= i__1; i4 += -4) {
if (z__[i4 - 1] <= tol2 * d__) {
z__[i4 - 1] = -0.;
d__ = z__[i4 - 3];
} else {
d__ = z__[i4 - 3] * (d__ / (d__ + z__[i4 - 1]));
}
/* L50: */
}
/* dqd maps Z to ZZ plus Li's test. */
emin = z__[(i0 << 2) + pp + 1];
d__ = z__[(i0 << 2) + pp - 3];
i__1 = (n0 - 1 << 2) + pp;
for (i4 = (i0 << 2) + pp; i4 <= i__1; i4 += 4) {
z__[i4 - (pp << 1) - 2] = d__ + z__[i4 - 1];
if (z__[i4 - 1] <= tol2 * d__) {
z__[i4 - 1] = -0.;
z__[i4 - (pp << 1) - 2] = d__;
z__[i4 - (pp << 1)] = 0.;
d__ = z__[i4 + 1];
} else if (safmin * z__[i4 + 1] < z__[i4 - (pp << 1) - 2] &&
safmin * z__[i4 - (pp << 1) - 2] < z__[i4 + 1]) {
temp = z__[i4 + 1] / z__[i4 - (pp << 1) - 2];
z__[i4 - (pp << 1)] = z__[i4 - 1] * temp;
d__ *= temp;
} else {
z__[i4 - (pp << 1)] = z__[i4 + 1] * (z__[i4 - 1] / z__[i4 - (
pp << 1) - 2]);
d__ = z__[i4 + 1] * (d__ / z__[i4 - (pp << 1) - 2]);
}
/* Computing MIN */
d__1 = emin, d__2 = z__[i4 - (pp << 1)];
emin = min(d__1,d__2);
/* L60: */
}
z__[(n0 << 2) - pp - 2] = d__;
/* Now find qmax. */
qmax = z__[(i0 << 2) - pp - 2];
i__1 = (n0 << 2) - pp - 2;
for (i4 = (i0 << 2) - pp + 2; i4 <= i__1; i4 += 4) {
/* Computing MAX */
d__1 = qmax, d__2 = z__[i4];
qmax = max(d__1,d__2);
/* L70: */
}
/* Prepare for the next iteration on K. */
pp = 1 - pp;
/* L80: */
}
iter = 2;
nfail = 0;
ndiv = n0 - i0 << 1;
i__1 = *n + 1;
for (iwhila = 1; iwhila <= i__1; ++iwhila) {
if (n0 < 1) {
goto L150;
}
/* While array unfinished do */
/* E(N0) holds the value of SIGMA when submatrix in I0:N0 */
/* splits from the rest of the array, but is negated. */
desig = 0.;
if (n0 == *n) {
sigma = 0.;
} else {
sigma = -z__[(n0 << 2) - 1];
}
if (sigma < 0.) {
*info = 1;
return 0;
}
/* Find last unreduced submatrix's top index I0, find QMAX and */
/* EMIN. Find Gershgorin-type bound if Q's much greater than E's. */
emax = 0.;
if (n0 > i0) {
emin = (d__1 = z__[(n0 << 2) - 5], abs(d__1));
} else {
emin = 0.;
}
qmin = z__[(n0 << 2) - 3];
qmax = qmin;
for (i4 = n0 << 2; i4 >= 8; i4 += -4) {
if (z__[i4 - 5] <= 0.) {
goto L100;
}
if (qmin >= emax * 4.) {
/* Computing MIN */
d__1 = qmin, d__2 = z__[i4 - 3];
qmin = min(d__1,d__2);
/* Computing MAX */
d__1 = emax, d__2 = z__[i4 - 5];
emax = max(d__1,d__2);
}
/* Computing MAX */
d__1 = qmax, d__2 = z__[i4 - 7] + z__[i4 - 5];
qmax = max(d__1,d__2);
/* Computing MIN */
d__1 = emin, d__2 = z__[i4 - 5];
emin = min(d__1,d__2);
/* L90: */
}
i4 = 4;
L100:
i0 = i4 / 4;
/* Store EMIN for passing to DLASQ3. */
z__[(n0 << 2) - 1] = emin;
/* Put -(initial shift) into DMIN. */
/* Computing MAX */
d__1 = 0., d__2 = qmin - sqrt(qmin) * 2. * sqrt(emax);
dmin__ = -max(d__1,d__2);
/* Now I0:N0 is unreduced. PP = 0 for ping, PP = 1 for pong. */
pp = 0;
nbig = (n0 - i0 + 1) * 30;
i__2 = nbig;
for (iwhilb = 1; iwhilb <= i__2; ++iwhilb) {
if (i0 > n0) {
goto L130;
}
/* While submatrix unfinished take a good dqds step. */
dlasq3_(&i0, &n0, &z__[1], &pp, &dmin__, &sigma, &desig, &qmax, &
nfail, &iter, &ndiv, &ieee);
pp = 1 - pp;
/* When EMIN is very small check for splits. */
if (pp == 0 && n0 - i0 >= 3) {
if (z__[n0 * 4] <= tol2 * qmax || z__[(n0 << 2) - 1] <= tol2 *
sigma) {
splt = i0 - 1;
qmax = z__[(i0 << 2) - 3];
emin = z__[(i0 << 2) - 1];
oldemn = z__[i0 * 4];
i__3 = n0 - 3 << 2;
for (i4 = i0 << 2; i4 <= i__3; i4 += 4) {
if (z__[i4] <= tol2 * z__[i4 - 3] || z__[i4 - 1] <=
tol2 * sigma) {
z__[i4 - 1] = -sigma;
splt = i4 / 4;
qmax = 0.;
emin = z__[i4 + 3];
oldemn = z__[i4 + 4];
} else {
/* Computing MAX */
d__1 = qmax, d__2 = z__[i4 + 1];
qmax = max(d__1,d__2);
/* Computing MIN */
d__1 = emin, d__2 = z__[i4 - 1];
emin = min(d__1,d__2);
/* Computing MIN */
d__1 = oldemn, d__2 = z__[i4];
oldemn = min(d__1,d__2);
}
/* L110: */
}
z__[(n0 << 2) - 1] = emin;
z__[n0 * 4] = oldemn;
i0 = splt + 1;
}
}
/* L120: */
}
*info = 2;
return 0;
/* end IWHILB */
L130:
/* L140: */
;
}
*info = 3;
return 0;
/* end IWHILA */
L150:
/* Move q's to the front. */
i__1 = *n;
for (k = 2; k <= i__1; ++k) {
z__[k] = z__[(k << 2) - 3];
/* L160: */
}
/* Sort and compute sum of eigenvalues. */
dlasrt_("D", n, &z__[1], &iinfo, (ftnlen)1);
e = 0.;
for (k = *n; k >= 1; --k) {
e += z__[k];
/* L170: */
}
/* Store trace, sum(eigenvalues) and information on performance. */
z__[(*n << 1) + 1] = trace;
z__[(*n << 1) + 2] = e;
z__[(*n << 1) + 3] = (doublereal) iter;
/* Computing 2nd power */
i__1 = *n;
z__[(*n << 1) + 4] = (doublereal) ndiv / (doublereal) (i__1 * i__1);
z__[(*n << 1) + 5] = nfail * 100. / (doublereal) iter;
return 0;
/* End of DLASQ2 */
} /* dlasq2_ */
/* Subroutine */ int dlalsd_(char *uplo, integer *smlsiz, integer *n, integer
*nrhs, doublereal *d__, doublereal *e, doublereal *b, integer *ldb,
doublereal *rcond, integer *rank, doublereal *work, integer *iwork,
integer *info)
{
/* System generated locals */
integer b_dim1, b_offset, i__1, i__2;
doublereal d__1;
/* Builtin functions */
double log(doublereal), d_sign(doublereal *, doublereal *);
/* Local variables */
static integer difl, difr, perm, nsub;
extern /* Subroutine */ int drot_(integer *, doublereal *, integer *,
doublereal *, integer *, doublereal *, doublereal *);
static integer nlvl, sqre, bxst, c__, i__, j, k;
static doublereal r__;
static integer s, u;
extern /* Subroutine */ int dgemm_(char *, char *, integer *, integer *,
integer *, doublereal *, doublereal *, integer *, doublereal *,
integer *, doublereal *, doublereal *, integer *);
static integer z__;
extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *,
doublereal *, integer *);
static integer poles, sizei, nsize, nwork, icmpq1, icmpq2;
static doublereal cs;
extern doublereal dlamch_(char *);
extern /* Subroutine */ int dlasda_(integer *, integer *, integer *,
integer *, doublereal *, doublereal *, doublereal *, integer *,
doublereal *, integer *, doublereal *, doublereal *, doublereal *,
doublereal *, integer *, integer *, integer *, integer *,
doublereal *, doublereal *, doublereal *, doublereal *, integer *,
integer *);
static integer bx;
extern /* Subroutine */ int dlalsa_(integer *, integer *, integer *,
integer *, doublereal *, integer *, doublereal *, integer *,
doublereal *, integer *, doublereal *, integer *, doublereal *,
doublereal *, doublereal *, doublereal *, integer *, integer *,
integer *, integer *, doublereal *, doublereal *, doublereal *,
doublereal *, integer *, integer *);
static doublereal sn;
extern /* Subroutine */ int dlascl_(char *, integer *, integer *,
doublereal *, doublereal *, integer *, integer *, doublereal *,
integer *, integer *);
extern integer idamax_(integer *, doublereal *, integer *);
static integer st;
extern /* Subroutine */ int dlasdq_(char *, integer *, integer *, integer
*, integer *, integer *, doublereal *, doublereal *, doublereal *,
integer *, doublereal *, integer *, doublereal *, integer *,
doublereal *, integer *);
static integer vt;
extern /* Subroutine */ int dlacpy_(char *, integer *, integer *,
doublereal *, integer *, doublereal *, integer *),
dlartg_(doublereal *, doublereal *, doublereal *, doublereal *,
doublereal *), dlaset_(char *, integer *, integer *, doublereal *,
doublereal *, doublereal *, integer *), xerbla_(char *,
integer *);
static integer givcol;
extern doublereal dlanst_(char *, integer *, doublereal *, doublereal *);
extern /* Subroutine */ int dlasrt_(char *, integer *, doublereal *,
integer *);
static doublereal orgnrm;
static integer givnum, givptr, nm1, smlszp, st1;
static doublereal eps;
static integer iwk;
static doublereal tol;
#define b_ref(a_1,a_2) b[(a_2)*b_dim1 + a_1]
/* -- LAPACK routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
October 31, 1999
Purpose
=======
DLALSD uses the singular value decomposition of A to solve the least
squares problem of finding X to minimize the Euclidean norm of each
column of A*X-B, where A is N-by-N upper bidiagonal, and X and B
are N-by-NRHS. The solution X overwrites B.
The singular values of A smaller than RCOND times the largest
singular value are treated as zero in solving the least squares
problem; in this case a minimum norm solution is returned.
The actual singular values are returned in D in ascending order.
This code makes very mild assumptions about floating point
arithmetic. It will work on machines with a guard digit in
add/subtract, or on those binary machines without guard digits
which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2.
It could conceivably fail on hexadecimal or decimal machines
without guard digits, but we know of none.
Arguments
=========
UPLO (input) CHARACTER*1
= 'U': D and E define an upper bidiagonal matrix.
= 'L': D and E define a lower bidiagonal matrix.
SMLSIZ (input) INTEGER
The maximum size of the subproblems at the bottom of the
computation tree.
N (input) INTEGER
The dimension of the bidiagonal matrix. N >= 0.
NRHS (input) INTEGER
The number of columns of B. NRHS must be at least 1.
D (input/output) DOUBLE PRECISION array, dimension (N)
On entry D contains the main diagonal of the bidiagonal
matrix. On exit, if INFO = 0, D contains its singular values.
E (input) DOUBLE PRECISION array, dimension (N-1)
Contains the super-diagonal entries of the bidiagonal matrix.
On exit, E has been destroyed.
B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
On input, B contains the right hand sides of the least
squares problem. On output, B contains the solution X.
LDB (input) INTEGER
The leading dimension of B in the calling subprogram.
LDB must be at least max(1,N).
RCOND (input) DOUBLE PRECISION
The singular values of A less than or equal to RCOND times
the largest singular value are treated as zero in solving
the least squares problem. If RCOND is negative,
machine precision is used instead.
For example, if diag(S)*X=B were the least squares problem,
where diag(S) is a diagonal matrix of singular values, the
solution would be X(i) = B(i) / S(i) if S(i) is greater than
RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to
RCOND*max(S).
RANK (output) INTEGER
The number of singular values of A greater than RCOND times
the largest singular value.
WORK (workspace) DOUBLE PRECISION array, dimension at least
(9*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2),
where NLVL = max(0, INT(log_2 (N/(SMLSIZ+1))) + 1).
IWORK (workspace) INTEGER array, dimension at least
(3*N*NLVL + 11*N)
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: The algorithm failed to compute an singular value while
working on the submatrix lying in rows and columns
INFO/(N+1) through MOD(INFO,N+1).
Further Details
===============
Based on contributions by
Ming Gu and Ren-Cang Li, Computer Science Division, University of
California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA
=====================================================================
Test the input parameters.
Parameter adjustments */
--d__;
--e;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
--work;
--iwork;
/* Function Body */
*info = 0;
if (*n < 0) {
*info = -3;
} else if (*nrhs < 1) {
*info = -4;
} else if (*ldb < 1 || *ldb < *n) {
*info = -8;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DLALSD", &i__1);
return 0;
}
eps = dlamch_("Epsilon");
/* Set up the tolerance. */
if (*rcond <= 0. || *rcond >= 1.) {
*rcond = eps;
}
*rank = 0;
/* Quick return if possible. */
if (*n == 0) {
return 0;
} else if (*n == 1) {
if (d__[1] == 0.) {
dlaset_("A", &c__1, nrhs, &c_b6, &c_b6, &b[b_offset], ldb);
} else {
*rank = 1;
dlascl_("G", &c__0, &c__0, &d__[1], &c_b11, &c__1, nrhs, &b[
b_offset], ldb, info);
d__[1] = abs(d__[1]);
}
return 0;
}
/* Rotate the matrix if it is lower bidiagonal. */
if (*(unsigned char *)uplo == 'L') {
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
dlartg_(&d__[i__], &e[i__], &cs, &sn, &r__);
d__[i__] = r__;
e[i__] = sn * d__[i__ + 1];
d__[i__ + 1] = cs * d__[i__ + 1];
if (*nrhs == 1) {
drot_(&c__1, &b_ref(i__, 1), &c__1, &b_ref(i__ + 1, 1), &c__1,
&cs, &sn);
} else {
work[(i__ << 1) - 1] = cs;
work[i__ * 2] = sn;
}
/* L10: */
}
if (*nrhs > 1) {
i__1 = *nrhs;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = *n - 1;
for (j = 1; j <= i__2; ++j) {
cs = work[(j << 1) - 1];
sn = work[j * 2];
drot_(&c__1, &b_ref(j, i__), &c__1, &b_ref(j + 1, i__), &
c__1, &cs, &sn);
/* L20: */
}
/* L30: */
}
}
}
/* Scale. */
nm1 = *n - 1;
orgnrm = dlanst_("M", n, &d__[1], &e[1]);
if (orgnrm == 0.) {
dlaset_("A", n, nrhs, &c_b6, &c_b6, &b[b_offset], ldb);
return 0;
}
dlascl_("G", &c__0, &c__0, &orgnrm, &c_b11, n, &c__1, &d__[1], n, info);
dlascl_("G", &c__0, &c__0, &orgnrm, &c_b11, &nm1, &c__1, &e[1], &nm1,
info);
/* If N is smaller than the minimum divide size SMLSIZ, then solve
the problem with another solver. */
if (*n <= *smlsiz) {
nwork = *n * *n + 1;
dlaset_("A", n, n, &c_b6, &c_b11, &work[1], n);
dlasdq_("U", &c__0, n, n, &c__0, nrhs, &d__[1], &e[1], &work[1], n, &
work[1], n, &b[b_offset], ldb, &work[nwork], info);
if (*info != 0) {
return 0;
}
tol = *rcond * (d__1 = d__[idamax_(n, &d__[1], &c__1)], abs(d__1));
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
if (d__[i__] <= tol) {
dlaset_("A", &c__1, nrhs, &c_b6, &c_b6, &b_ref(i__, 1), ldb);
} else {
dlascl_("G", &c__0, &c__0, &d__[i__], &c_b11, &c__1, nrhs, &
b_ref(i__, 1), ldb, info);
++(*rank);
}
/* L40: */
}
dgemm_("T", "N", n, nrhs, n, &c_b11, &work[1], n, &b[b_offset], ldb, &
c_b6, &work[nwork], n);
dlacpy_("A", n, nrhs, &work[nwork], n, &b[b_offset], ldb);
/* Unscale. */
dlascl_("G", &c__0, &c__0, &c_b11, &orgnrm, n, &c__1, &d__[1], n,
info);
dlasrt_("D", n, &d__[1], info);
dlascl_("G", &c__0, &c__0, &orgnrm, &c_b11, n, nrhs, &b[b_offset],
ldb, info);
return 0;
}
/* Book-keeping and setting up some constants. */
nlvl = (integer) (log((doublereal) (*n) / (doublereal) (*smlsiz + 1)) /
log(2.)) + 1;
smlszp = *smlsiz + 1;
u = 1;
vt = *smlsiz * *n + 1;
difl = vt + smlszp * *n;
difr = difl + nlvl * *n;
z__ = difr + (nlvl * *n << 1);
c__ = z__ + nlvl * *n;
s = c__ + *n;
poles = s + *n;
givnum = poles + (nlvl << 1) * *n;
bx = givnum + (nlvl << 1) * *n;
nwork = bx + *n * *nrhs;
sizei = *n + 1;
k = sizei + *n;
givptr = k + *n;
perm = givptr + *n;
givcol = perm + nlvl * *n;
iwk = givcol + (nlvl * *n << 1);
st = 1;
sqre = 0;
icmpq1 = 1;
icmpq2 = 0;
nsub = 0;
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
if ((d__1 = d__[i__], abs(d__1)) < eps) {
d__[i__] = d_sign(&eps, &d__[i__]);
}
/* L50: */
}
i__1 = nm1;
for (i__ = 1; i__ <= i__1; ++i__) {
if ((d__1 = e[i__], abs(d__1)) < eps || i__ == nm1) {
++nsub;
iwork[nsub] = st;
/* Subproblem found. First determine its size and then
apply divide and conquer on it. */
if (i__ < nm1) {
/* A subproblem with E(I) small for I < NM1. */
nsize = i__ - st + 1;
iwork[sizei + nsub - 1] = nsize;
} else if ((d__1 = e[i__], abs(d__1)) >= eps) {
/* A subproblem with E(NM1) not too small but I = NM1. */
nsize = *n - st + 1;
iwork[sizei + nsub - 1] = nsize;
} else {
/* A subproblem with E(NM1) small. This implies an
1-by-1 subproblem at D(N), which is not solved
explicitly. */
nsize = i__ - st + 1;
iwork[sizei + nsub - 1] = nsize;
++nsub;
iwork[nsub] = *n;
iwork[sizei + nsub - 1] = 1;
dcopy_(nrhs, &b_ref(*n, 1), ldb, &work[bx + nm1], n);
}
st1 = st - 1;
if (nsize == 1) {
/* This is a 1-by-1 subproblem and is not solved
explicitly. */
dcopy_(nrhs, &b_ref(st, 1), ldb, &work[bx + st1], n);
} else if (nsize <= *smlsiz) {
/* This is a small subproblem and is solved by DLASDQ. */
dlaset_("A", &nsize, &nsize, &c_b6, &c_b11, &work[vt + st1],
n);
dlasdq_("U", &c__0, &nsize, &nsize, &c__0, nrhs, &d__[st], &e[
st], &work[vt + st1], n, &work[nwork], n, &b_ref(st,
1), ldb, &work[nwork], info);
if (*info != 0) {
return 0;
}
dlacpy_("A", &nsize, nrhs, &b_ref(st, 1), ldb, &work[bx + st1]
, n);
} else {
/* A large problem. Solve it using divide and conquer. */
dlasda_(&icmpq1, smlsiz, &nsize, &sqre, &d__[st], &e[st], &
work[u + st1], n, &work[vt + st1], &iwork[k + st1], &
work[difl + st1], &work[difr + st1], &work[z__ + st1],
&work[poles + st1], &iwork[givptr + st1], &iwork[
givcol + st1], n, &iwork[perm + st1], &work[givnum +
st1], &work[c__ + st1], &work[s + st1], &work[nwork],
&iwork[iwk], info);
if (*info != 0) {
return 0;
}
bxst = bx + st1;
dlalsa_(&icmpq2, smlsiz, &nsize, nrhs, &b_ref(st, 1), ldb, &
work[bxst], n, &work[u + st1], n, &work[vt + st1], &
iwork[k + st1], &work[difl + st1], &work[difr + st1],
&work[z__ + st1], &work[poles + st1], &iwork[givptr +
st1], &iwork[givcol + st1], n, &iwork[perm + st1], &
work[givnum + st1], &work[c__ + st1], &work[s + st1],
&work[nwork], &iwork[iwk], info);
if (*info != 0) {
return 0;
}
}
st = i__ + 1;
}
/* L60: */
}
/* Apply the singular values and treat the tiny ones as zero. */
tol = *rcond * (d__1 = d__[idamax_(n, &d__[1], &c__1)], abs(d__1));
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Some of the elements in D can be negative because 1-by-1
subproblems were not solved explicitly. */
if ((d__1 = d__[i__], abs(d__1)) <= tol) {
dlaset_("A", &c__1, nrhs, &c_b6, &c_b6, &work[bx + i__ - 1], n);
} else {
++(*rank);
dlascl_("G", &c__0, &c__0, &d__[i__], &c_b11, &c__1, nrhs, &work[
bx + i__ - 1], n, info);
}
d__[i__] = (d__1 = d__[i__], abs(d__1));
/* L70: */
}
/* Now apply back the right singular vectors. */
icmpq2 = 1;
i__1 = nsub;
for (i__ = 1; i__ <= i__1; ++i__) {
st = iwork[i__];
st1 = st - 1;
nsize = iwork[sizei + i__ - 1];
bxst = bx + st1;
if (nsize == 1) {
dcopy_(nrhs, &work[bxst], n, &b_ref(st, 1), ldb);
} else if (nsize <= *smlsiz) {
dgemm_("T", "N", &nsize, nrhs, &nsize, &c_b11, &work[vt + st1], n,
&work[bxst], n, &c_b6, &b_ref(st, 1), ldb);
} else {
dlalsa_(&icmpq2, smlsiz, &nsize, nrhs, &work[bxst], n, &b_ref(st,
1), ldb, &work[u + st1], n, &work[vt + st1], &iwork[k +
st1], &work[difl + st1], &work[difr + st1], &work[z__ +
st1], &work[poles + st1], &iwork[givptr + st1], &iwork[
givcol + st1], n, &iwork[perm + st1], &work[givnum + st1],
&work[c__ + st1], &work[s + st1], &work[nwork], &iwork[
iwk], info);
if (*info != 0) {
return 0;
}
}
/* L80: */
}
/* Unscale and sort the singular values. */
dlascl_("G", &c__0, &c__0, &c_b11, &orgnrm, n, &c__1, &d__[1], n, info);
dlasrt_("D", n, &d__[1], info);
dlascl_("G", &c__0, &c__0, &orgnrm, &c_b11, n, nrhs, &b[b_offset], ldb,
info);
return 0;
/* End of DLALSD */
} /* dlalsd_ */
/* Subroutine */ int dsteqr_(char *compz, integer *n, doublereal *d__,
doublereal *e, doublereal *z__, integer *ldz, doublereal *work,
integer *info)
{
/* System generated locals */
integer z_dim1, z_offset, i__1, i__2;
doublereal d__1, d__2;
/* Builtin functions */
double sqrt(doublereal), d_sign(doublereal *, doublereal *);
/* Local variables */
doublereal b, c__, f, g;
integer i__, j, k, l, m;
doublereal p, r__, s;
integer l1, ii, mm, lm1, mm1, nm1;
doublereal rt1, rt2, eps;
integer lsv;
doublereal tst, eps2;
integer lend, jtot;
extern /* Subroutine */ int dlae2_(doublereal *, doublereal *, doublereal
*, doublereal *, doublereal *);
extern logical lsame_(char *, char *);
extern /* Subroutine */ int dlasr_(char *, char *, char *, integer *,
integer *, doublereal *, doublereal *, doublereal *, integer *);
doublereal anorm;
extern /* Subroutine */ int dswap_(integer *, doublereal *, integer *,
doublereal *, integer *), dlaev2_(doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *, doublereal *,
doublereal *);
integer lendm1, lendp1;
extern doublereal dlapy2_(doublereal *, doublereal *), dlamch_(char *);
integer iscale;
extern /* Subroutine */ int dlascl_(char *, integer *, integer *,
doublereal *, doublereal *, integer *, integer *, doublereal *,
integer *, integer *), dlaset_(char *, integer *, integer
*, doublereal *, doublereal *, doublereal *, integer *);
doublereal safmin;
extern /* Subroutine */ int dlartg_(doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *);
doublereal safmax;
extern /* Subroutine */ int xerbla_(char *, integer *);
extern doublereal dlanst_(char *, integer *, doublereal *, doublereal *);
extern /* Subroutine */ int dlasrt_(char *, integer *, doublereal *,
integer *);
integer lendsv;
doublereal ssfmin;
integer nmaxit, icompz;
doublereal ssfmax;
/* -- LAPACK routine (version 3.1) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* DSTEQR computes all eigenvalues and, optionally, eigenvectors of a */
/* symmetric tridiagonal matrix using the implicit QL or QR method. */
/* The eigenvectors of a full or band symmetric matrix can also be found */
/* if DSYTRD or DSPTRD or DSBTRD has been used to reduce this matrix to */
/* tridiagonal form. */
/* Arguments */
/* ========= */
/* COMPZ (input) CHARACTER*1 */
/* = 'N': Compute eigenvalues only. */
/* = 'V': Compute eigenvalues and eigenvectors of the original */
/* symmetric matrix. On entry, Z must contain the */
/* orthogonal matrix used to reduce the original matrix */
/* to tridiagonal form. */
/* = 'I': Compute eigenvalues and eigenvectors of the */
/* tridiagonal matrix. Z is initialized to the identity */
/* matrix. */
/* N (input) INTEGER */
/* The order of the matrix. N >= 0. */
/* D (input/output) DOUBLE PRECISION array, dimension (N) */
/* On entry, the diagonal elements of the tridiagonal matrix. */
/* On exit, if INFO = 0, the eigenvalues in ascending order. */
/* E (input/output) DOUBLE PRECISION array, dimension (N-1) */
/* On entry, the (n-1) subdiagonal elements of the tridiagonal */
/* matrix. */
/* On exit, E has been destroyed. */
/* Z (input/output) DOUBLE PRECISION array, dimension (LDZ, N) */
/* On entry, if COMPZ = 'V', then Z contains the orthogonal */
/* matrix used in the reduction to tridiagonal form. */
/* On exit, if INFO = 0, then if COMPZ = 'V', Z contains the */
/* orthonormal eigenvectors of the original symmetric matrix, */
/* and if COMPZ = 'I', Z contains the orthonormal eigenvectors */
/* of the symmetric tridiagonal matrix. */
/* If COMPZ = 'N', then Z is not referenced. */
/* LDZ (input) INTEGER */
/* The leading dimension of the array Z. LDZ >= 1, and if */
/* eigenvectors are desired, then LDZ >= max(1,N). */
/* WORK (workspace) DOUBLE PRECISION array, dimension (max(1,2*N-2)) */
/* If COMPZ = 'N', then WORK is not referenced. */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* > 0: the algorithm has failed to find all the eigenvalues in */
/* a total of 30*N iterations; if INFO = i, then i */
/* elements of E have not converged to zero; on exit, D */
/* and E contain the elements of a symmetric tridiagonal */
/* matrix which is orthogonally similar to the original */
/* matrix. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--d__;
--e;
z_dim1 = *ldz;
z_offset = 1 + z_dim1;
z__ -= z_offset;
--work;
/* Function Body */
*info = 0;
if (lsame_(compz, "N")) {
icompz = 0;
} else if (lsame_(compz, "V")) {
icompz = 1;
} else if (lsame_(compz, "I")) {
icompz = 2;
} else {
icompz = -1;
}
if (icompz < 0) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*ldz < 1 || icompz > 0 && *ldz < max(1,*n)) {
*info = -6;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DSTEQR", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
if (*n == 1) {
if (icompz == 2) {
z__[z_dim1 + 1] = 1.;
}
return 0;
}
/* Determine the unit roundoff and over/underflow thresholds. */
eps = dlamch_("E");
/* Computing 2nd power */
d__1 = eps;
eps2 = d__1 * d__1;
safmin = dlamch_("S");
safmax = 1. / safmin;
ssfmax = sqrt(safmax) / 3.;
ssfmin = sqrt(safmin) / eps2;
/* Compute the eigenvalues and eigenvectors of the tridiagonal */
/* matrix. */
if (icompz == 2) {
dlaset_("Full", n, n, &c_b9, &c_b10, &z__[z_offset], ldz);
}
nmaxit = *n * 30;
jtot = 0;
/* Determine where the matrix splits and choose QL or QR iteration */
/* for each block, according to whether top or bottom diagonal */
/* element is smaller. */
l1 = 1;
nm1 = *n - 1;
L10:
if (l1 > *n) {
goto L160;
}
if (l1 > 1) {
e[l1 - 1] = 0.;
}
if (l1 <= nm1) {
i__1 = nm1;
for (m = l1; m <= i__1; ++m) {
tst = (d__1 = e[m], abs(d__1));
if (tst == 0.) {
goto L30;
}
if (tst <= sqrt((d__1 = d__[m], abs(d__1))) * sqrt((d__2 = d__[m
+ 1], abs(d__2))) * eps) {
e[m] = 0.;
goto L30;
}
/* L20: */
}
}
m = *n;
L30:
l = l1;
lsv = l;
lend = m;
lendsv = lend;
l1 = m + 1;
if (lend == l) {
goto L10;
}
/* Scale submatrix in rows and columns L to LEND */
i__1 = lend - l + 1;
anorm = dlanst_("I", &i__1, &d__[l], &e[l]);
iscale = 0;
if (anorm == 0.) {
goto L10;
}
if (anorm > ssfmax) {
iscale = 1;
i__1 = lend - l + 1;
dlascl_("G", &c__0, &c__0, &anorm, &ssfmax, &i__1, &c__1, &d__[l], n,
info);
i__1 = lend - l;
dlascl_("G", &c__0, &c__0, &anorm, &ssfmax, &i__1, &c__1, &e[l], n,
info);
} else if (anorm < ssfmin) {
iscale = 2;
i__1 = lend - l + 1;
dlascl_("G", &c__0, &c__0, &anorm, &ssfmin, &i__1, &c__1, &d__[l], n,
info);
i__1 = lend - l;
dlascl_("G", &c__0, &c__0, &anorm, &ssfmin, &i__1, &c__1, &e[l], n,
info);
}
/* Choose between QL and QR iteration */
if ((d__1 = d__[lend], abs(d__1)) < (d__2 = d__[l], abs(d__2))) {
lend = lsv;
l = lendsv;
}
if (lend > l) {
/* QL Iteration */
/* Look for small subdiagonal element. */
L40:
if (l != lend) {
lendm1 = lend - 1;
i__1 = lendm1;
for (m = l; m <= i__1; ++m) {
/* Computing 2nd power */
d__2 = (d__1 = e[m], abs(d__1));
tst = d__2 * d__2;
if (tst <= eps2 * (d__1 = d__[m], abs(d__1)) * (d__2 = d__[m
+ 1], abs(d__2)) + safmin) {
goto L60;
}
/* L50: */
}
}
m = lend;
L60:
if (m < lend) {
e[m] = 0.;
}
p = d__[l];
if (m == l) {
goto L80;
}
/* If remaining matrix is 2-by-2, use DLAE2 or SLAEV2 */
/* to compute its eigensystem. */
if (m == l + 1) {
if (icompz > 0) {
dlaev2_(&d__[l], &e[l], &d__[l + 1], &rt1, &rt2, &c__, &s);
work[l] = c__;
work[*n - 1 + l] = s;
dlasr_("R", "V", "B", n, &c__2, &work[l], &work[*n - 1 + l], &
z__[l * z_dim1 + 1], ldz);
} else {
dlae2_(&d__[l], &e[l], &d__[l + 1], &rt1, &rt2);
}
d__[l] = rt1;
d__[l + 1] = rt2;
e[l] = 0.;
l += 2;
if (l <= lend) {
goto L40;
}
goto L140;
}
if (jtot == nmaxit) {
goto L140;
}
++jtot;
/* Form shift. */
g = (d__[l + 1] - p) / (e[l] * 2.);
r__ = dlapy2_(&g, &c_b10);
g = d__[m] - p + e[l] / (g + d_sign(&r__, &g));
s = 1.;
c__ = 1.;
p = 0.;
/* Inner loop */
mm1 = m - 1;
i__1 = l;
for (i__ = mm1; i__ >= i__1; --i__) {
f = s * e[i__];
b = c__ * e[i__];
dlartg_(&g, &f, &c__, &s, &r__);
if (i__ != m - 1) {
e[i__ + 1] = r__;
}
g = d__[i__ + 1] - p;
r__ = (d__[i__] - g) * s + c__ * 2. * b;
p = s * r__;
d__[i__ + 1] = g + p;
g = c__ * r__ - b;
/* If eigenvectors are desired, then save rotations. */
if (icompz > 0) {
work[i__] = c__;
work[*n - 1 + i__] = -s;
}
/* L70: */
}
/* If eigenvectors are desired, then apply saved rotations. */
if (icompz > 0) {
mm = m - l + 1;
dlasr_("R", "V", "B", n, &mm, &work[l], &work[*n - 1 + l], &z__[l
* z_dim1 + 1], ldz);
}
d__[l] -= p;
e[l] = g;
goto L40;
/* Eigenvalue found. */
L80:
d__[l] = p;
++l;
if (l <= lend) {
goto L40;
}
goto L140;
} else {
/* QR Iteration */
/* Look for small superdiagonal element. */
L90:
if (l != lend) {
lendp1 = lend + 1;
i__1 = lendp1;
for (m = l; m >= i__1; --m) {
/* Computing 2nd power */
d__2 = (d__1 = e[m - 1], abs(d__1));
tst = d__2 * d__2;
if (tst <= eps2 * (d__1 = d__[m], abs(d__1)) * (d__2 = d__[m
- 1], abs(d__2)) + safmin) {
goto L110;
}
/* L100: */
}
}
m = lend;
L110:
if (m > lend) {
e[m - 1] = 0.;
}
p = d__[l];
if (m == l) {
goto L130;
}
/* If remaining matrix is 2-by-2, use DLAE2 or SLAEV2 */
/* to compute its eigensystem. */
if (m == l - 1) {
if (icompz > 0) {
dlaev2_(&d__[l - 1], &e[l - 1], &d__[l], &rt1, &rt2, &c__, &s)
;
work[m] = c__;
work[*n - 1 + m] = s;
dlasr_("R", "V", "F", n, &c__2, &work[m], &work[*n - 1 + m], &
z__[(l - 1) * z_dim1 + 1], ldz);
} else {
dlae2_(&d__[l - 1], &e[l - 1], &d__[l], &rt1, &rt2);
}
d__[l - 1] = rt1;
d__[l] = rt2;
e[l - 1] = 0.;
l += -2;
if (l >= lend) {
goto L90;
}
goto L140;
}
if (jtot == nmaxit) {
goto L140;
}
++jtot;
/* Form shift. */
g = (d__[l - 1] - p) / (e[l - 1] * 2.);
r__ = dlapy2_(&g, &c_b10);
g = d__[m] - p + e[l - 1] / (g + d_sign(&r__, &g));
s = 1.;
c__ = 1.;
p = 0.;
/* Inner loop */
lm1 = l - 1;
i__1 = lm1;
for (i__ = m; i__ <= i__1; ++i__) {
f = s * e[i__];
b = c__ * e[i__];
dlartg_(&g, &f, &c__, &s, &r__);
if (i__ != m) {
e[i__ - 1] = r__;
}
g = d__[i__] - p;
r__ = (d__[i__ + 1] - g) * s + c__ * 2. * b;
p = s * r__;
d__[i__] = g + p;
g = c__ * r__ - b;
/* If eigenvectors are desired, then save rotations. */
if (icompz > 0) {
work[i__] = c__;
work[*n - 1 + i__] = s;
}
/* L120: */
}
/* If eigenvectors are desired, then apply saved rotations. */
if (icompz > 0) {
mm = l - m + 1;
dlasr_("R", "V", "F", n, &mm, &work[m], &work[*n - 1 + m], &z__[m
* z_dim1 + 1], ldz);
}
d__[l] -= p;
e[lm1] = g;
goto L90;
/* Eigenvalue found. */
L130:
d__[l] = p;
--l;
if (l >= lend) {
goto L90;
}
goto L140;
}
/* Undo scaling if necessary */
L140:
if (iscale == 1) {
i__1 = lendsv - lsv + 1;
dlascl_("G", &c__0, &c__0, &ssfmax, &anorm, &i__1, &c__1, &d__[lsv],
n, info);
i__1 = lendsv - lsv;
dlascl_("G", &c__0, &c__0, &ssfmax, &anorm, &i__1, &c__1, &e[lsv], n,
info);
} else if (iscale == 2) {
i__1 = lendsv - lsv + 1;
dlascl_("G", &c__0, &c__0, &ssfmin, &anorm, &i__1, &c__1, &d__[lsv],
n, info);
i__1 = lendsv - lsv;
dlascl_("G", &c__0, &c__0, &ssfmin, &anorm, &i__1, &c__1, &e[lsv], n,
info);
}
/* Check for no convergence to an eigenvalue after a total */
/* of N*MAXIT iterations. */
if (jtot < nmaxit) {
goto L10;
}
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
if (e[i__] != 0.) {
++(*info);
}
/* L150: */
}
goto L190;
/* Order eigenvalues and eigenvectors. */
L160:
if (icompz == 0) {
/* Use Quick Sort */
dlasrt_("I", n, &d__[1], info);
} else {
/* Use Selection Sort to minimize swaps of eigenvectors */
i__1 = *n;
for (ii = 2; ii <= i__1; ++ii) {
i__ = ii - 1;
k = i__;
p = d__[i__];
i__2 = *n;
for (j = ii; j <= i__2; ++j) {
if (d__[j] < p) {
k = j;
p = d__[j];
}
/* L170: */
}
if (k != i__) {
d__[k] = d__[i__];
d__[i__] = p;
dswap_(n, &z__[i__ * z_dim1 + 1], &c__1, &z__[k * z_dim1 + 1],
&c__1);
}
/* L180: */
}
}
L190:
return 0;
/* End of DSTEQR */
} /* dsteqr_ */
/* Subroutine */ int zsteqr_(char *compz, integer *n, doublereal *d,
doublereal *e, doublecomplex *z, integer *ldz, doublereal *work,
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
=======
ZSTEQR computes all eigenvalues and, optionally, eigenvectors of a
symmetric tridiagonal matrix using the implicit QL or QR method.
The eigenvectors of a full or band complex Hermitian matrix can also
be found if ZHETRD or ZHPTRD or ZHBTRD has been used to reduce this
matrix to tridiagonal form.
Arguments
=========
COMPZ (input) CHARACTER*1
= 'N': Compute eigenvalues only.
= 'V': Compute eigenvalues and eigenvectors of the original
Hermitian matrix. On entry, Z must contain the
unitary matrix used to reduce the original matrix
to tridiagonal form.
= 'I': Compute eigenvalues and eigenvectors of the
tridiagonal matrix. Z is initialized to the identity
matrix.
N (input) INTEGER
The order of the matrix. N >= 0.
D (input/output) DOUBLE PRECISION array, dimension (N)
On entry, the diagonal elements of the tridiagonal matrix.
On exit, if INFO = 0, the eigenvalues in ascending order.
E (input/output) DOUBLE PRECISION array, dimension (N-1)
On entry, the (n-1) subdiagonal elements of the tridiagonal
matrix.
On exit, E has been destroyed.
Z (input/output) COMPLEX*16 array, dimension (LDZ, N)
On entry, if COMPZ = 'V', then Z contains the unitary
matrix used in the reduction to tridiagonal form.
On exit, if INFO = 0, then if COMPZ = 'V', Z contains the
orthonormal eigenvectors of the original Hermitian matrix,
and if COMPZ = 'I', Z contains the orthonormal eigenvectors
of the symmetric tridiagonal matrix.
If COMPZ = 'N', then Z is not referenced.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1, and if
eigenvectors are desired, then LDZ >= max(1,N).
WORK (workspace) DOUBLE PRECISION array, dimension (max(1,2*N-2))
If COMPZ = 'N', then WORK is not referenced.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: the algorithm has failed to find all the eigenvalues in
a total of 30*N iterations; if INFO = i, then i
elements of E have not converged to zero; on exit, D
and E contain the elements of a symmetric tridiagonal
matrix which is unitarily similar to the original
matrix.
=====================================================================
Test the input parameters.
Parameter adjustments
Function Body */
/* Table of constant values */
static doublecomplex c_b1 = {0.,0.};
static doublecomplex c_b2 = {1.,0.};
static integer c__0 = 0;
static integer c__1 = 1;
static integer c__2 = 2;
static doublereal c_b41 = 1.;
/* System generated locals */
integer z_dim1, z_offset, i__1, i__2;
doublereal d__1, d__2;
/* Builtin functions */
double sqrt(doublereal), d_sign(doublereal *, doublereal *);
/* Local variables */
static integer lend, jtot;
extern /* Subroutine */ int dlae2_(doublereal *, doublereal *, doublereal
*, doublereal *, doublereal *);
static doublereal b, c, f, g;
static integer i, j, k, l, m;
static doublereal p, r, s;
extern logical lsame_(char *, char *);
static doublereal anorm;
extern /* Subroutine */ int zlasr_(char *, char *, char *, integer *,
integer *, doublereal *, doublereal *, doublecomplex *, integer *);
static integer l1;
extern /* Subroutine */ int zswap_(integer *, doublecomplex *, integer *,
doublecomplex *, integer *), dlaev2_(doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *, doublereal *,
doublereal *);
static integer lendm1, lendp1;
extern doublereal dlapy2_(doublereal *, doublereal *);
static integer ii;
extern doublereal dlamch_(char *);
static integer mm, iscale;
extern /* Subroutine */ int dlascl_(char *, integer *, integer *,
doublereal *, doublereal *, integer *, integer *, doublereal *,
integer *, integer *);
static doublereal safmin;
extern /* Subroutine */ int dlartg_(doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *);
static doublereal safmax;
extern /* Subroutine */ int xerbla_(char *, integer *);
extern doublereal dlanst_(char *, integer *, doublereal *, doublereal *);
extern /* Subroutine */ int dlasrt_(char *, integer *, doublereal *,
integer *);
static integer lendsv;
static doublereal ssfmin;
static integer nmaxit, icompz;
static doublereal ssfmax;
extern /* Subroutine */ int zlaset_(char *, integer *, integer *,
doublecomplex *, doublecomplex *, doublecomplex *, integer *);
static integer lm1, mm1, nm1;
static doublereal rt1, rt2, eps;
static integer lsv;
static doublereal tst, eps2;
#define D(I) d[(I)-1]
#define E(I) e[(I)-1]
#define WORK(I) work[(I)-1]
#define Z(I,J) z[(I)-1 + ((J)-1)* ( *ldz)]
*info = 0;
if (lsame_(compz, "N")) {
icompz = 0;
} else if (lsame_(compz, "V")) {
icompz = 1;
} else if (lsame_(compz, "I")) {
icompz = 2;
} else {
icompz = -1;
}
if (icompz < 0) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*ldz < 1 || icompz > 0 && *ldz < max(1,*n)) {
*info = -6;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZSTEQR", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
if (*n == 1) {
if (icompz == 2) {
i__1 = z_dim1 + 1;
Z(1,1).r = 1., Z(1,1).i = 0.;
}
return 0;
}
/* Determine the unit roundoff and over/underflow thresholds. */
eps = dlamch_("E");
/* Computing 2nd power */
d__1 = eps;
eps2 = d__1 * d__1;
safmin = dlamch_("S");
safmax = 1. / safmin;
ssfmax = sqrt(safmax) / 3.;
ssfmin = sqrt(safmin) / eps2;
/* Compute the eigenvalues and eigenvectors of the tridiagonal
matrix. */
if (icompz == 2) {
zlaset_("Full", n, n, &c_b1, &c_b2, &Z(1,1), ldz);
}
nmaxit = *n * 30;
jtot = 0;
/* Determine where the matrix splits and choose QL or QR iteration
for each block, according to whether top or bottom diagonal
element is smaller. */
l1 = 1;
nm1 = *n - 1;
L10:
if (l1 > *n) {
goto L160;
}
if (l1 > 1) {
E(l1 - 1) = 0.;
}
if (l1 <= nm1) {
i__1 = nm1;
for (m = l1; m <= nm1; ++m) {
tst = (d__1 = E(m), abs(d__1));
if (tst == 0.) {
goto L30;
}
if (tst <= sqrt((d__1 = D(m), abs(d__1))) * sqrt((d__2 = D(m + 1),
abs(d__2))) * eps) {
E(m) = 0.;
goto L30;
}
/* L20: */
}
}
m = *n;
L30:
l = l1;
lsv = l;
lend = m;
lendsv = lend;
l1 = m + 1;
if (lend == l) {
goto L10;
}
/* Scale submatrix in rows and columns L to LEND */
i__1 = lend - l + 1;
anorm = dlanst_("I", &i__1, &D(l), &E(l));
iscale = 0;
if (anorm == 0.) {
goto L10;
}
if (anorm > ssfmax) {
iscale = 1;
i__1 = lend - l + 1;
dlascl_("G", &c__0, &c__0, &anorm, &ssfmax, &i__1, &c__1, &D(l), n,
info);
i__1 = lend - l;
dlascl_("G", &c__0, &c__0, &anorm, &ssfmax, &i__1, &c__1, &E(l), n,
info);
} else if (anorm < ssfmin) {
iscale = 2;
i__1 = lend - l + 1;
dlascl_("G", &c__0, &c__0, &anorm, &ssfmin, &i__1, &c__1, &D(l), n,
info);
i__1 = lend - l;
dlascl_("G", &c__0, &c__0, &anorm, &ssfmin, &i__1, &c__1, &E(l), n,
info);
}
/* Choose between QL and QR iteration */
if ((d__1 = D(lend), abs(d__1)) < (d__2 = D(l), abs(d__2))) {
lend = lsv;
l = lendsv;
}
if (lend > l) {
/* QL Iteration
Look for small subdiagonal element. */
L40:
if (l != lend) {
lendm1 = lend - 1;
i__1 = lendm1;
for (m = l; m <= lendm1; ++m) {
/* Computing 2nd power */
d__2 = (d__1 = E(m), abs(d__1));
tst = d__2 * d__2;
if (tst <= eps2 * (d__1 = D(m), abs(d__1)) * (d__2 = D(m + 1),
abs(d__2)) + safmin) {
goto L60;
}
/* L50: */
}
}
m = lend;
L60:
if (m < lend) {
E(m) = 0.;
}
p = D(l);
if (m == l) {
goto L80;
}
/* If remaining matrix is 2-by-2, use DLAE2 or SLAEV2
to compute its eigensystem. */
if (m == l + 1) {
if (icompz > 0) {
dlaev2_(&D(l), &E(l), &D(l + 1), &rt1, &rt2, &c, &s);
WORK(l) = c;
WORK(*n - 1 + l) = s;
zlasr_("R", "V", "B", n, &c__2, &WORK(l), &WORK(*n - 1 + l), &
Z(1,l), ldz);
} else {
dlae2_(&D(l), &E(l), &D(l + 1), &rt1, &rt2);
}
D(l) = rt1;
D(l + 1) = rt2;
E(l) = 0.;
l += 2;
if (l <= lend) {
goto L40;
}
goto L140;
}
if (jtot == nmaxit) {
goto L140;
}
++jtot;
/* Form shift. */
g = (D(l + 1) - p) / (E(l) * 2.);
r = dlapy2_(&g, &c_b41);
g = D(m) - p + E(l) / (g + d_sign(&r, &g));
s = 1.;
c = 1.;
p = 0.;
/* Inner loop */
mm1 = m - 1;
i__1 = l;
for (i = mm1; i >= l; --i) {
f = s * E(i);
b = c * E(i);
dlartg_(&g, &f, &c, &s, &r);
if (i != m - 1) {
E(i + 1) = r;
}
g = D(i + 1) - p;
r = (D(i) - g) * s + c * 2. * b;
p = s * r;
D(i + 1) = g + p;
g = c * r - b;
/* If eigenvectors are desired, then save rotations. */
if (icompz > 0) {
WORK(i) = c;
WORK(*n - 1 + i) = -s;
}
/* L70: */
}
/* If eigenvectors are desired, then apply saved rotations. */
if (icompz > 0) {
mm = m - l + 1;
zlasr_("R", "V", "B", n, &mm, &WORK(l), &WORK(*n - 1 + l), &Z(1,l), ldz);
}
D(l) -= p;
E(l) = g;
goto L40;
/* Eigenvalue found. */
L80:
D(l) = p;
++l;
if (l <= lend) {
goto L40;
}
goto L140;
} else {
/* QR Iteration
Look for small superdiagonal element. */
L90:
if (l != lend) {
lendp1 = lend + 1;
i__1 = lendp1;
for (m = l; m >= lendp1; --m) {
/* Computing 2nd power */
d__2 = (d__1 = E(m - 1), abs(d__1));
tst = d__2 * d__2;
if (tst <= eps2 * (d__1 = D(m), abs(d__1)) * (d__2 = D(m - 1),
abs(d__2)) + safmin) {
goto L110;
}
/* L100: */
}
}
m = lend;
L110:
if (m > lend) {
E(m - 1) = 0.;
}
p = D(l);
if (m == l) {
goto L130;
}
/* If remaining matrix is 2-by-2, use DLAE2 or SLAEV2
to compute its eigensystem. */
if (m == l - 1) {
if (icompz > 0) {
dlaev2_(&D(l - 1), &E(l - 1), &D(l), &rt1, &rt2, &c, &s);
WORK(m) = c;
WORK(*n - 1 + m) = s;
zlasr_("R", "V", "F", n, &c__2, &WORK(m), &WORK(*n - 1 + m), &
Z(1,l-1), ldz);
} else {
dlae2_(&D(l - 1), &E(l - 1), &D(l), &rt1, &rt2);
}
D(l - 1) = rt1;
D(l) = rt2;
E(l - 1) = 0.;
l += -2;
if (l >= lend) {
goto L90;
}
goto L140;
}
if (jtot == nmaxit) {
goto L140;
}
++jtot;
/* Form shift. */
g = (D(l - 1) - p) / (E(l - 1) * 2.);
r = dlapy2_(&g, &c_b41);
g = D(m) - p + E(l - 1) / (g + d_sign(&r, &g));
s = 1.;
c = 1.;
p = 0.;
/* Inner loop */
lm1 = l - 1;
i__1 = lm1;
for (i = m; i <= lm1; ++i) {
f = s * E(i);
b = c * E(i);
dlartg_(&g, &f, &c, &s, &r);
if (i != m) {
E(i - 1) = r;
}
g = D(i) - p;
r = (D(i + 1) - g) * s + c * 2. * b;
p = s * r;
D(i) = g + p;
g = c * r - b;
/* If eigenvectors are desired, then save rotations. */
if (icompz > 0) {
WORK(i) = c;
WORK(*n - 1 + i) = s;
}
/* L120: */
}
/* If eigenvectors are desired, then apply saved rotations. */
if (icompz > 0) {
mm = l - m + 1;
zlasr_("R", "V", "F", n, &mm, &WORK(m), &WORK(*n - 1 + m), &Z(1,m), ldz);
}
D(l) -= p;
E(lm1) = g;
goto L90;
/* Eigenvalue found. */
L130:
D(l) = p;
--l;
if (l >= lend) {
goto L90;
}
goto L140;
}
/* Undo scaling if necessary */
L140:
if (iscale == 1) {
i__1 = lendsv - lsv + 1;
dlascl_("G", &c__0, &c__0, &ssfmax, &anorm, &i__1, &c__1, &D(lsv), n,
info);
i__1 = lendsv - lsv;
dlascl_("G", &c__0, &c__0, &ssfmax, &anorm, &i__1, &c__1, &E(lsv), n,
info);
} else if (iscale == 2) {
i__1 = lendsv - lsv + 1;
dlascl_("G", &c__0, &c__0, &ssfmin, &anorm, &i__1, &c__1, &D(lsv), n,
info);
i__1 = lendsv - lsv;
dlascl_("G", &c__0, &c__0, &ssfmin, &anorm, &i__1, &c__1, &E(lsv), n,
info);
}
/* Check for no convergence to an eigenvalue after a total
of N*MAXIT iterations. */
if (jtot == nmaxit) {
i__1 = *n - 1;
for (i = 1; i <= *n-1; ++i) {
if (E(i) != 0.) {
++(*info);
}
/* L150: */
}
return 0;
}
goto L10;
/* Order eigenvalues and eigenvectors. */
L160:
if (icompz == 0) {
/* Use Quick Sort */
dlasrt_("I", n, &D(1), info);
} else {
/* Use Selection Sort to minimize swaps of eigenvectors */
i__1 = *n;
for (ii = 2; ii <= *n; ++ii) {
i = ii - 1;
k = i;
p = D(i);
i__2 = *n;
for (j = ii; j <= *n; ++j) {
if (D(j) < p) {
k = j;
p = D(j);
}
/* L170: */
}
if (k != i) {
D(k) = D(i);
D(i) = p;
zswap_(n, &Z(1,i), &c__1, &Z(1,k), &
c__1);
}
/* L180: */
}
}
return 0;
/* End of ZSTEQR */
} /* zsteqr_ */
/* Subroutine */ int dstqrb_(integer *n, doublereal *d__, doublereal *e,
doublereal *z__, doublereal *work, integer *info)
{
/* System generated locals */
integer i__1, i__2;
doublereal d__1, d__2;
/* Local variables */
static doublereal b, c__, f, g;
static integer i__, j, k, l, m;
static doublereal p, r__, s;
static integer l1, ii, mm, lm1, mm1, nm1;
static doublereal rt1, rt2, eps;
static integer lsv;
static doublereal tst, eps2;
static integer lend, jtot;
extern /* Subroutine */ int dlae2_(doublereal *, doublereal *, doublereal
*, doublereal *, doublereal *), dlasr_(char *, char *, char *,
integer *, integer *, doublereal *, doublereal *, doublereal *,
integer *, ftnlen, ftnlen, ftnlen);
static doublereal anorm;
extern /* Subroutine */ int dlaev2_(doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *, doublereal *,
doublereal *);
static integer lendm1, lendp1;
extern doublereal dlapy2_(doublereal *, doublereal *), dlamch_(char *,
ftnlen);
static integer iscale;
extern /* Subroutine */ int dlascl_(char *, integer *, integer *,
doublereal *, doublereal *, integer *, integer *, doublereal *,
integer *, integer *, ftnlen);
static doublereal safmin;
extern /* Subroutine */ int dlartg_(doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *);
static doublereal safmax;
extern doublereal dlanst_(char *, integer *, doublereal *, doublereal *,
ftnlen);
extern /* Subroutine */ int dlasrt_(char *, integer *, doublereal *,
integer *, ftnlen);
static integer lendsv, nmaxit, icompz;
static doublereal ssfmax, ssfmin;
/* %------------------% */
/* | Scalar Arguments | */
/* %------------------% */
/* %-----------------% */
/* | Array Arguments | */
/* %-----------------% */
/* .. parameters .. */
/* .. */
/* .. local scalars .. */
/* .. */
/* .. external functions .. */
/* .. */
/* .. external subroutines .. */
/* .. */
/* .. intrinsic functions .. */
/* .. */
/* .. executable statements .. */
/* test the input parameters. */
/* Parameter adjustments */
--work;
--z__;
--e;
--d__;
/* Function Body */
*info = 0;
/* $$$ IF( LSAME( COMPZ, 'N' ) ) THEN */
/* $$$ ICOMPZ = 0 */
/* $$$ ELSE IF( LSAME( COMPZ, 'V' ) ) THEN */
/* $$$ ICOMPZ = 1 */
/* $$$ ELSE IF( LSAME( COMPZ, 'I' ) ) THEN */
/* $$$ ICOMPZ = 2 */
/* $$$ ELSE */
/* $$$ ICOMPZ = -1 */
/* $$$ END IF */
/* $$$ IF( ICOMPZ.LT.0 ) THEN */
/* $$$ INFO = -1 */
/* $$$ ELSE IF( N.LT.0 ) THEN */
/* $$$ INFO = -2 */
/* $$$ ELSE IF( ( LDZ.LT.1 ) .OR. ( ICOMPZ.GT.0 .AND. LDZ.LT.MAX( 1, */
/* $$$ $ N ) ) ) THEN */
/* $$$ INFO = -6 */
/* $$$ END IF */
/* $$$ IF( INFO.NE.0 ) THEN */
/* $$$ CALL XERBLA( 'SSTEQR', -INFO ) */
/* $$$ RETURN */
/* $$$ END IF */
/* *** New starting with version 2.5 *** */
icompz = 2;
/* ************************************* */
/* quick return if possible */
if (*n == 0) {
return 0;
}
if (*n == 1) {
if (icompz == 2) {
z__[1] = 1.;
}
return 0;
}
/* determine the unit roundoff and over/underflow thresholds. */
eps = dlamch_("e", (ftnlen)1);
/* Computing 2nd power */
d__1 = eps;
eps2 = d__1 * d__1;
safmin = dlamch_("s", (ftnlen)1);
safmax = 1. / safmin;
ssfmax = sqrt(safmax) / 3.;
ssfmin = sqrt(safmin) / eps2;
/* compute the eigenvalues and eigenvectors of the tridiagonal */
/* matrix. */
/* $$ if( icompz.eq.2 ) */
/* $$$ $ call dlaset( 'full', n, n, zero, one, z, ldz ) */
/* *** New starting with version 2.5 *** */
if (icompz == 2) {
i__1 = *n - 1;
for (j = 1; j <= i__1; ++j) {
z__[j] = 0.;
/* L5: */
}
z__[*n] = 1.;
}
/* ************************************* */
nmaxit = *n * 30;
jtot = 0;
/* determine where the matrix splits and choose ql or qr iteration */
/* for each block, according to whether top or bottom diagonal */
/* element is smaller. */
l1 = 1;
nm1 = *n - 1;
L10:
if (l1 > *n) {
goto L160;
}
if (l1 > 1) {
e[l1 - 1] = 0.;
}
if (l1 <= nm1) {
i__1 = nm1;
for (m = l1; m <= i__1; ++m) {
tst = (d__1 = e[m], abs(d__1));
if (tst == 0.) {
goto L30;
}
if (tst <= sqrt((d__1 = d__[m], abs(d__1))) * sqrt((d__2 = d__[m
+ 1], abs(d__2))) * eps) {
e[m] = 0.;
goto L30;
}
/* L20: */
}
}
m = *n;
L30:
l = l1;
lsv = l;
lend = m;
lendsv = lend;
l1 = m + 1;
if (lend == l) {
goto L10;
}
/* scale submatrix in rows and columns l to lend */
i__1 = lend - l + 1;
anorm = dlanst_("i", &i__1, &d__[l], &e[l], (ftnlen)1);
iscale = 0;
if (anorm == 0.) {
goto L10;
}
if (anorm > ssfmax) {
iscale = 1;
i__1 = lend - l + 1;
dlascl_("g", &c__0, &c__0, &anorm, &ssfmax, &i__1, &c__1, &d__[l], n,
info, (ftnlen)1);
i__1 = lend - l;
dlascl_("g", &c__0, &c__0, &anorm, &ssfmax, &i__1, &c__1, &e[l], n,
info, (ftnlen)1);
} else if (anorm < ssfmin) {
iscale = 2;
i__1 = lend - l + 1;
dlascl_("g", &c__0, &c__0, &anorm, &ssfmin, &i__1, &c__1, &d__[l], n,
info, (ftnlen)1);
i__1 = lend - l;
dlascl_("g", &c__0, &c__0, &anorm, &ssfmin, &i__1, &c__1, &e[l], n,
info, (ftnlen)1);
}
/* choose between ql and qr iteration */
if ((d__1 = d__[lend], abs(d__1)) < (d__2 = d__[l], abs(d__2))) {
lend = lsv;
l = lendsv;
}
if (lend > l) {
/* ql iteration */
/* look for small subdiagonal element. */
L40:
if (l != lend) {
lendm1 = lend - 1;
i__1 = lendm1;
for (m = l; m <= i__1; ++m) {
/* Computing 2nd power */
d__2 = (d__1 = e[m], abs(d__1));
tst = d__2 * d__2;
if (tst <= eps2 * (d__1 = d__[m], abs(d__1)) * (d__2 = d__[m
+ 1], abs(d__2)) + safmin) {
goto L60;
}
/* L50: */
}
}
m = lend;
L60:
if (m < lend) {
e[m] = 0.;
}
p = d__[l];
if (m == l) {
goto L80;
}
/* if remaining matrix is 2-by-2, use dlae2 or dlaev2 */
/* to compute its eigensystem. */
if (m == l + 1) {
if (icompz > 0) {
dlaev2_(&d__[l], &e[l], &d__[l + 1], &rt1, &rt2, &c__, &s);
work[l] = c__;
work[*n - 1 + l] = s;
/* $$$ call dlasr( 'r', 'v', 'b', n, 2, work( l ), */
/* $$$ $ work( n-1+l ), z( 1, l ), ldz ) */
/* *** New starting with version 2.5 *** */
tst = z__[l + 1];
z__[l + 1] = c__ * tst - s * z__[l];
z__[l] = s * tst + c__ * z__[l];
/* ************************************* */
} else {
dlae2_(&d__[l], &e[l], &d__[l + 1], &rt1, &rt2);
}
d__[l] = rt1;
d__[l + 1] = rt2;
e[l] = 0.;
l += 2;
if (l <= lend) {
goto L40;
}
goto L140;
}
if (jtot == nmaxit) {
goto L140;
}
++jtot;
/* form shift. */
g = (d__[l + 1] - p) / (e[l] * 2.);
r__ = dlapy2_(&g, &c_b31);
g = d__[m] - p + e[l] / (g + d_sign(&r__, &g));
s = 1.;
c__ = 1.;
p = 0.;
/* inner loop */
mm1 = m - 1;
i__1 = l;
for (i__ = mm1; i__ >= i__1; --i__) {
f = s * e[i__];
b = c__ * e[i__];
dlartg_(&g, &f, &c__, &s, &r__);
if (i__ != m - 1) {
e[i__ + 1] = r__;
}
g = d__[i__ + 1] - p;
r__ = (d__[i__] - g) * s + c__ * 2. * b;
p = s * r__;
d__[i__ + 1] = g + p;
g = c__ * r__ - b;
/* if eigenvectors are desired, then save rotations. */
if (icompz > 0) {
work[i__] = c__;
work[*n - 1 + i__] = -s;
}
/* L70: */
}
/* if eigenvectors are desired, then apply saved rotations. */
if (icompz > 0) {
mm = m - l + 1;
/* $$$ call dlasr( 'r', 'v', 'b', n, mm, work( l ), work( n-1+l ), */
/* $$$ $ z( 1, l ), ldz ) */
/* *** New starting with version 2.5 *** */
dlasr_("r", "v", "b", &c__1, &mm, &work[l], &work[*n - 1 + l], &
z__[l], &c__1, (ftnlen)1, (ftnlen)1, (ftnlen)1);
/* ************************************* */
}
d__[l] -= p;
e[l] = g;
goto L40;
/* eigenvalue found. */
L80:
d__[l] = p;
++l;
if (l <= lend) {
goto L40;
}
goto L140;
} else {
/* qr iteration */
/* look for small superdiagonal element. */
L90:
if (l != lend) {
lendp1 = lend + 1;
i__1 = lendp1;
for (m = l; m >= i__1; --m) {
/* Computing 2nd power */
d__2 = (d__1 = e[m - 1], abs(d__1));
tst = d__2 * d__2;
if (tst <= eps2 * (d__1 = d__[m], abs(d__1)) * (d__2 = d__[m
- 1], abs(d__2)) + safmin) {
goto L110;
}
/* L100: */
}
}
m = lend;
L110:
if (m > lend) {
e[m - 1] = 0.;
}
p = d__[l];
if (m == l) {
goto L130;
}
/* if remaining matrix is 2-by-2, use dlae2 or dlaev2 */
/* to compute its eigensystem. */
if (m == l - 1) {
if (icompz > 0) {
dlaev2_(&d__[l - 1], &e[l - 1], &d__[l], &rt1, &rt2, &c__, &s)
;
/* $$$ work( m ) = c */
/* $$$ work( n-1+m ) = s */
/* $$$ call dlasr( 'r', 'v', 'f', n, 2, work( m ), */
/* $$$ $ work( n-1+m ), z( 1, l-1 ), ldz ) */
/* *** New starting with version 2.5 *** */
tst = z__[l];
z__[l] = c__ * tst - s * z__[l - 1];
z__[l - 1] = s * tst + c__ * z__[l - 1];
/* ************************************* */
} else {
dlae2_(&d__[l - 1], &e[l - 1], &d__[l], &rt1, &rt2);
}
d__[l - 1] = rt1;
d__[l] = rt2;
e[l - 1] = 0.;
l += -2;
if (l >= lend) {
goto L90;
}
goto L140;
}
if (jtot == nmaxit) {
goto L140;
}
++jtot;
/* form shift. */
g = (d__[l - 1] - p) / (e[l - 1] * 2.);
r__ = dlapy2_(&g, &c_b31);
g = d__[m] - p + e[l - 1] / (g + d_sign(&r__, &g));
s = 1.;
c__ = 1.;
p = 0.;
/* inner loop */
lm1 = l - 1;
i__1 = lm1;
for (i__ = m; i__ <= i__1; ++i__) {
f = s * e[i__];
b = c__ * e[i__];
dlartg_(&g, &f, &c__, &s, &r__);
if (i__ != m) {
e[i__ - 1] = r__;
}
g = d__[i__] - p;
r__ = (d__[i__ + 1] - g) * s + c__ * 2. * b;
p = s * r__;
d__[i__] = g + p;
g = c__ * r__ - b;
/* if eigenvectors are desired, then save rotations. */
if (icompz > 0) {
work[i__] = c__;
work[*n - 1 + i__] = s;
}
/* L120: */
}
/* if eigenvectors are desired, then apply saved rotations. */
if (icompz > 0) {
mm = l - m + 1;
/* $$$ call dlasr( 'r', 'v', 'f', n, mm, work( m ), work( n-1+m ), */
/* $$$ $ z( 1, m ), ldz ) */
/* *** New starting with version 2.5 *** */
dlasr_("r", "v", "f", &c__1, &mm, &work[m], &work[*n - 1 + m], &
z__[m], &c__1, (ftnlen)1, (ftnlen)1, (ftnlen)1);
/* ************************************* */
}
d__[l] -= p;
e[lm1] = g;
goto L90;
/* eigenvalue found. */
L130:
d__[l] = p;
--l;
if (l >= lend) {
goto L90;
}
goto L140;
}
/* undo scaling if necessary */
L140:
if (iscale == 1) {
i__1 = lendsv - lsv + 1;
dlascl_("g", &c__0, &c__0, &ssfmax, &anorm, &i__1, &c__1, &d__[lsv],
n, info, (ftnlen)1);
i__1 = lendsv - lsv;
dlascl_("g", &c__0, &c__0, &ssfmax, &anorm, &i__1, &c__1, &e[lsv], n,
info, (ftnlen)1);
} else if (iscale == 2) {
i__1 = lendsv - lsv + 1;
dlascl_("g", &c__0, &c__0, &ssfmin, &anorm, &i__1, &c__1, &d__[lsv],
n, info, (ftnlen)1);
i__1 = lendsv - lsv;
dlascl_("g", &c__0, &c__0, &ssfmin, &anorm, &i__1, &c__1, &e[lsv], n,
info, (ftnlen)1);
}
/* check for no convergence to an eigenvalue after a total */
/* of n*maxit iterations. */
if (jtot < nmaxit) {
goto L10;
}
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
if (e[i__] != 0.) {
++(*info);
}
/* L150: */
}
goto L190;
/* order eigenvalues and eigenvectors. */
L160:
if (icompz == 0) {
/* use quick sort */
dlasrt_("i", n, &d__[1], info, (ftnlen)1);
} else {
/* use selection sort to minimize swaps of eigenvectors */
i__1 = *n;
for (ii = 2; ii <= i__1; ++ii) {
i__ = ii - 1;
k = i__;
p = d__[i__];
i__2 = *n;
for (j = ii; j <= i__2; ++j) {
if (d__[j] < p) {
k = j;
p = d__[j];
}
/* L170: */
}
if (k != i__) {
d__[k] = d__[i__];
d__[i__] = p;
/* $$$ call dswap( n, z( 1, i ), 1, z( 1, k ), 1 ) */
/* *** New starting with version 2.5 *** */
p = z__[k];
z__[k] = z__[i__];
z__[i__] = p;
/* ************************************* */
}
/* L180: */
}
}
L190:
return 0;
/* %---------------% */
/* | End of dstqrb | */
/* %---------------% */
} /* dstqrb_ */
