/* Subroutine */ int sstegr_(char *jobz, char *range, integer *n, real *d__,
real *e, real *vl, real *vu, integer *il, integer *iu, real *abstol,
integer *m, real *w, real *z__, integer *ldz, integer *isuppz, real *
work, integer *lwork, integer *iwork, integer *liwork, integer *info,
ftnlen jobz_len, ftnlen range_len)
{
/* System generated locals */
integer z_dim1, z_offset, i__1, i__2;
real r__1, r__2;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
static integer i__, j, jj;
static real eps, tol, tmp;
static integer iend;
static real rmin, rmax;
static integer itmp;
static real tnrm, scale;
extern logical lsame_(char *, char *, ftnlen, ftnlen);
static integer iinfo;
extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *);
static integer lwmin;
extern /* Subroutine */ int sswap_(integer *, real *, integer *, real *,
integer *);
static logical wantz, alleig;
static integer ibegin;
static logical indeig;
static integer iindbl;
static logical valeig;
extern doublereal slamch_(char *, ftnlen);
static real safmin;
extern /* Subroutine */ int xerbla_(char *, integer *, ftnlen);
static real bignum;
static integer iindwk, indgrs, indwof;
extern /* Subroutine */ int slarre_(integer *, real *, real *, real *,
integer *, integer *, integer *, real *, real *, real *, real *,
integer *), slaset_(char *, integer *, integer *, real *, real *,
real *, integer *, ftnlen);
static real thresh;
static integer iinspl, indwrk, liwmin;
extern doublereal slanst_(char *, integer *, real *, real *, ftnlen);
static integer nsplit;
extern /* Subroutine */ int slarrv_(integer *, real *, real *, integer *,
integer *, real *, integer *, real *, real *, real *, integer *,
integer *, real *, integer *, integer *);
static real smlnum;
static logical lquery;
/* -- LAPACK computational routine (version 3.0) -- */
/* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., */
/* Courant Institute, Argonne National Lab, and Rice University */
/* June 30, 1999 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SSTEGR computes selected eigenvalues and, optionally, eigenvectors */
/* of a real symmetric tridiagonal matrix T. Eigenvalues and */
/* eigenvectors can be selected by specifying either a range of values */
/* or a range of indices for the desired eigenvalues. The eigenvalues */
/* are computed by the dqds algorithm, while orthogonal eigenvectors are */
/* computed from various ``good'' L D L^T representations (also known as */
/* Relatively Robust Representations). Gram-Schmidt orthogonalization is */
/* avoided as far as possible. More specifically, the various steps of */
/* the algorithm are as follows. For the i-th unreduced block of T, */
/* (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T */
/* is a relatively robust representation, */
/* (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high */
/* relative accuracy by the dqds algorithm, */
/* (c) If there is a cluster of close eigenvalues, "choose" sigma_i */
/* close to the cluster, and go to step (a), */
/* (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T, */
/* compute the corresponding eigenvector by forming a */
/* rank-revealing twisted factorization. */
/* The desired accuracy of the output can be specified by the input */
/* parameter ABSTOL. */
/* For more details, see "A new O(n^2) algorithm for the symmetric */
/* tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon, */
/* Computer Science Division Technical Report No. UCB/CSD-97-971, */
/* UC Berkeley, May 1997. */
/* Note 1 : Currently SSTEGR is only set up to find ALL the n */
/* eigenvalues and eigenvectors of T in O(n^2) time */
/* Note 2 : Currently the routine SSTEIN is called when an appropriate */
/* sigma_i cannot be chosen in step (c) above. SSTEIN invokes modified */
/* Gram-Schmidt when eigenvalues are close. */
/* Note 3 : SSTEGR works only on machines which follow ieee-754 */
/* floating-point standard in their handling of infinities and NaNs. */
/* Normal execution of SSTEGR may create NaNs and infinities and hence */
/* may abort due to a floating point exception in environments which */
/* do not conform to the ieee standard. */
/* 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. */
/* ********* Only RANGE = 'A' is currently supported ********************* */
/* N (input) INTEGER */
/* The order of the matrix. N >= 0. */
/* D (input/output) REAL array, dimension (N) */
/* On entry, the n diagonal elements of the tridiagonal matrix */
/* T. On exit, D is overwritten. */
/* E (input/output) REAL 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 exit, E is overwritten. */
/* VL (input) REAL */
/* VU (input) REAL */
/* 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; IL = 1 and IU = 0 if N = 0. */
/* Not referenced if RANGE = 'A' or 'V'. */
/* ABSTOL (input) REAL */
/* The absolute error tolerance for the */
/* eigenvalues/eigenvectors. IF JOBZ = 'V', the eigenvalues and */
/* eigenvectors output have residual norms bounded by ABSTOL, */
/* and the dot products between different eigenvectors are */
/* bounded by ABSTOL. If ABSTOL is less than N*EPS*|T|, then */
/* N*EPS*|T| will be used in its place, where EPS is the */
/* machine precision and |T| is the 1-norm of the tridiagonal */
/* matrix. The eigenvalues are computed to an accuracy of */
/* EPS*|T| irrespective of ABSTOL. If high relative accuracy */
/* is important, set ABSTOL to DLAMCH( 'Safe minimum' ). */
/* See Barlow and Demmel "Computing Accurate Eigensystems of */
/* Scaled Diagonally Dominant Matrices", LAPACK Working Note #7 */
/* for a discussion of which matrices define their eigenvalues */
/* to high relative accuracy. */
/* 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) REAL array, dimension (N) */
/* The first M elements contain the selected eigenvalues in */
/* ascending order. */
/* Z (output) REAL array, dimension (LDZ, max(1,M) ) */
/* If JOBZ = 'V', then if INFO = 0, 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 an upper bound must be used. */
/* LDZ (input) INTEGER */
/* The leading dimension of the array Z. LDZ >= 1, and if */
/* JOBZ = 'V', LDZ >= max(1,N). */
/* 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 eigenvector */
/* is nonzero only in elements ISUPPZ( 2*i-1 ) through */
/* ISUPPZ( 2*i ). */
/* WORK (workspace/output) REAL 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 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 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 */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* > 0: if INFO = 1, internal error in SLARRE, */
/* if INFO = 2, internal error in SLARRV. */
/* Further Details */
/* =============== */
/* Based on contributions by */
/* Inderjit Dhillon, IBM Almaden, USA */
/* Osni Marques, LBNL/NERSC, 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", (ftnlen)1, (ftnlen)1);
alleig = lsame_(range, "A", (ftnlen)1, (ftnlen)1);
valeig = lsame_(range, "V", (ftnlen)1, (ftnlen)1);
indeig = lsame_(range, "I", (ftnlen)1, (ftnlen)1);
lquery = *lwork == -1 || *liwork == -1;
lwmin = *n * 18;
liwmin = *n * 10;
*info = 0;
if (! (wantz || lsame_(jobz, "N", (ftnlen)1, (ftnlen)1))) {
*info = -1;
} else if (! (alleig || valeig || indeig)) {
*info = -2;
/* The following two lines need to be removed once the */
/* RANGE = 'V' and RANGE = 'I' options are provided. */
} else if (valeig || indeig) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else if (valeig && *n > 0 && *vu <= *vl) {
*info = -7;
} else if (indeig && *il < 1) {
*info = -8;
/* The following change should be made in DSTEVX also, otherwise */
/* IL can be specified as N+1 and IU as N. */
/* ELSE IF( INDEIG .AND. ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) ) THEN */
} else if (indeig && (*iu < *il || *iu > *n)) {
*info = -9;
} else if (*ldz < 1 || wantz && *ldz < *n) {
*info = -14;
} else if (*lwork < lwmin && ! lquery) {
*info = -17;
} else if (*liwork < liwmin && ! lquery) {
*info = -19;
}
if (*info == 0) {
work[1] = (real) lwmin;
iwork[1] = liwmin;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SSTEGR", &i__1, (ftnlen)6);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
*m = 0;
if (*n == 0) {
return 0;
}
if (*n == 1) {
if (alleig || indeig) {
*m = 1;
w[1] = d__[1];
} else {
if (*vl < d__[1] && *vu >= d__[1]) {
*m = 1;
w[1] = d__[1];
}
}
if (wantz) {
z__[z_dim1 + 1] = 1.f;
}
return 0;
}
/* Get machine constants. */
safmin = slamch_("Safe minimum", (ftnlen)12);
eps = slamch_("Precision", (ftnlen)9);
smlnum = safmin / eps;
bignum = 1.f / smlnum;
rmin = sqrt(smlnum);
/* Computing MIN */
r__1 = sqrt(bignum), r__2 = 1.f / sqrt(sqrt(safmin));
rmax = dmin(r__1,r__2);
/* Scale matrix to allowable range, if necessary. */
scale = 1.f;
tnrm = slanst_("M", n, &d__[1], &e[1], (ftnlen)1);
if (tnrm > 0.f && tnrm < rmin) {
scale = rmin / tnrm;
} else if (tnrm > rmax) {
scale = rmax / tnrm;
}
if (scale != 1.f) {
sscal_(n, &scale, &d__[1], &c__1);
i__1 = *n - 1;
sscal_(&i__1, &scale, &e[1], &c__1);
tnrm *= scale;
}
indgrs = 1;
indwof = (*n << 1) + 1;
indwrk = *n * 3 + 1;
iinspl = 1;
iindbl = *n + 1;
iindwk = (*n << 1) + 1;
slaset_("Full", n, n, &c_b14, &c_b14, &z__[z_offset], ldz, (ftnlen)4);
/* Compute the desired eigenvalues of the tridiagonal after splitting */
/* into smaller subblocks if the corresponding of-diagonal elements */
/* are small */
thresh = eps * tnrm;
slarre_(n, &d__[1], &e[1], &thresh, &nsplit, &iwork[iinspl], m, &w[1], &
work[indwof], &work[indgrs], &work[indwrk], &iinfo);
if (iinfo != 0) {
*info = 1;
return 0;
}
if (wantz) {
/* Compute the desired eigenvectors corresponding to the computed */
/* eigenvalues */
/* Computing MAX */
r__1 = *abstol, r__2 = (real) (*n) * thresh;
tol = dmax(r__1,r__2);
ibegin = 1;
i__1 = nsplit;
for (i__ = 1; i__ <= i__1; ++i__) {
iend = iwork[iinspl + i__ - 1];
i__2 = iend;
for (j = ibegin; j <= i__2; ++j) {
iwork[iindbl + j - 1] = i__;
/* L10: */
}
ibegin = iend + 1;
/* L20: */
}
slarrv_(n, &d__[1], &e[1], &iwork[iinspl], m, &w[1], &iwork[iindbl], &
work[indgrs], &tol, &z__[z_offset], ldz, &isuppz[1], &work[
indwrk], &iwork[iindwk], &iinfo);
if (iinfo != 0) {
*info = 2;
return 0;
}
}
ibegin = 1;
i__1 = nsplit;
for (i__ = 1; i__ <= i__1; ++i__) {
iend = iwork[iinspl + i__ - 1];
i__2 = iend;
for (j = ibegin; j <= i__2; ++j) {
w[j] += work[indwof + i__ - 1];
/* L30: */
}
ibegin = iend + 1;
/* L40: */
}
/* If matrix was scaled, then rescale eigenvalues appropriately. */
if (scale != 1.f) {
r__1 = 1.f / scale;
sscal_(m, &r__1, &w[1], &c__1);
}
/* If eigenvalues are not in order, then sort them, along with */
/* eigenvectors. */
if (nsplit > 1) {
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) {
sswap_(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] = (real) lwmin;
iwork[1] = liwmin;
return 0;
/* End of SSTEGR */
} /* sstegr_ */
/* Subroutine */ int sstemr_(char *jobz, char *range, integer *n, real *d__,
real *e, real *vl, real *vu, integer *il, integer *iu, integer *m,
real *w, real *z__, integer *ldz, integer *nzc, integer *isuppz,
logical *tryrac, real *work, integer *lwork, integer *iwork, integer *
liwork, integer *info)
{
/* System generated locals */
integer z_dim1, z_offset, i__1, i__2;
real r__1, r__2;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
integer i__, j;
real r1, r2;
integer jj;
real cs;
integer in;
real sn, wl, wu;
integer iil, iiu;
real eps, tmp;
integer indd, iend, jblk, wend;
real rmin, rmax;
integer itmp;
real tnrm;
integer inde2;
extern /* Subroutine */ int slae2_(real *, real *, real *, real *, real *)
;
integer itmp2;
real rtol1, rtol2, scale;
integer indgp;
extern logical lsame_(char *, char *);
integer iinfo;
extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *);
integer iindw, ilast, lwmin;
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *), sswap_(integer *, real *, integer *, real *, integer *
);
logical wantz;
extern /* Subroutine */ int slaev2_(real *, real *, real *, real *, real *
, real *, real *);
logical alleig;
integer ibegin;
logical indeig;
integer iindbl;
logical valeig;
extern doublereal slamch_(char *);
integer wbegin;
real safmin;
extern /* Subroutine */ int xerbla_(char *, integer *);
real bignum;
integer inderr, iindwk, indgrs, offset;
extern /* Subroutine */ int slarrc_(char *, integer *, real *, real *,
real *, real *, real *, integer *, integer *, integer *, integer *
), slarre_(char *, integer *, real *, real *, integer *,
integer *, real *, real *, real *, real *, real *, real *,
integer *, integer *, integer *, real *, real *, real *, integer *
, integer *, real *, real *, real *, integer *, integer *)
;
real thresh;
integer iinspl, indwrk, ifirst, liwmin, nzcmin;
real pivmin;
extern doublereal slanst_(char *, integer *, real *, real *);
extern /* Subroutine */ int slarrj_(integer *, real *, real *, integer *,
integer *, real *, integer *, real *, real *, real *, integer *,
real *, real *, integer *), slarrr_(integer *, real *, real *,
integer *);
integer nsplit;
extern /* Subroutine */ int slarrv_(integer *, real *, real *, real *,
real *, real *, integer *, integer *, integer *, integer *, real *
, real *, real *, real *, real *, real *, integer *, integer *,
real *, real *, integer *, integer *, real *, integer *, integer *
);
real smlnum;
extern /* Subroutine */ int slasrt_(char *, integer *, real *, integer *);
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 */
/* ======= */
/* SSTEMR 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.SSTEMR 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) REAL array, dimension (N) */
/* On entry, the N diagonal elements of the tridiagonal matrix */
/* T. On exit, D is overwritten. */
/* E (input/output) REAL 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) REAL */
/* VU (input) REAL */
/* 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) REAL array, dimension (N) */
/* The first M elements contain the selected eigenvalues in */
/* ascending order. */
/* Z (output) REAL 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) REAL 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 SLARRE, */
/* if INFO = 2X, internal error in SLARRV. */
/* Here, the digit X = ABS( IINFO ) < 10, where IINFO is */
/* the nonzero error code returned by SLARRE or */
/* SLARRV, 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;
/* SSTEMR needs WORK of size 6*N, IWORK of size 3*N. */
/* In addition, SLARRE needs WORK of size 6*N, IWORK of size 5*N. */
/* Furthermore, SLARRV 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.f;
wu = 0.f;
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 SLARRE. */
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 = slamch_("Safe minimum");
eps = slamch_("Precision");
smlnum = safmin / eps;
bignum = 1.f / smlnum;
rmin = sqrt(smlnum);
/* Computing MIN */
r__1 = sqrt(bignum), r__2 = 1.f / sqrt(sqrt(safmin));
rmax = dmin(r__1,r__2);
if (*info == 0) {
work[1] = (real) lwmin;
iwork[1] = liwmin;
if (wantz && alleig) {
nzcmin = *n;
} else if (wantz && valeig) {
slarrc_("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] = (real) nzcmin;
} else if (*nzc < nzcmin && ! zquery) {
*info = -14;
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SSTEMR", &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.f;
isuppz[1] = 1;
isuppz[2] = 1;
}
return 0;
}
if (*n == 2) {
if (! wantz) {
slae2_(&d__[1], &e[1], &d__[2], &r1, &r2);
} else if (wantz && ! zquery) {
slaev2_(&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.f) {
if (cs != 0.f) {
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.f) {
if (cs != 0.f) {
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 SLARRD. The preference for scaling small values */
/* up is heuristic; we expect users' matrices not to be close to the */
/* RMAX threshold. */
scale = 1.f;
tnrm = slanst_("M", n, &d__[1], &e[1]);
if (tnrm > 0.f && tnrm < rmin) {
scale = rmin / tnrm;
} else if (tnrm > rmax) {
scale = rmax / tnrm;
}
if (scale != 1.f) {
sscal_(n, &scale, &d__[1], &c__1);
i__1 = *n - 1;
sscal_(&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 SLARRE */
/* 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. */
slarrr_(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 */
scopy_(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 */
r__1 = e[j];
work[inde2 + j - 1] = r__1 * r__1;
/* L5: */
}
/* Set the tolerance parameters for bisection */
if (! wantz) {
/* SLARRE computes the eigenvalues to full precision. */
rtol1 = eps * 4.f;
rtol2 = eps * 4.f;
} else {
/* SLARRE computes the eigenvalues to less than full precision. */
/* SLARRV will refine the eigenvalue approximations, and we can */
/* need less accurate initial bisection in SLARRE. */
/* Note: these settings do only affect the subset case and SLARRE */
/* Computing MAX */
r__1 = sqrt(eps) * .05f, r__2 = eps * 4.f;
rtol1 = dmax(r__1,r__2);
/* Computing MAX */
r__1 = sqrt(eps) * .005f, r__2 = eps * 4.f;
rtol2 = dmax(r__1,r__2);
}
slarre_(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', SLARRE 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 */
slarrv_(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 {
/* SLARRE computes eigenvalues of the (shifted) root representation */
/* SLARRV 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 SLARRE 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.f;
slarrj_(&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.f) {
r__1 = 1.f / scale;
sscal_(m, &r__1, &w[1], &c__1);
}
/* If eigenvalues are not in increasing order, then sort them, */
/* possibly along with eigenvectors. */
if (nsplit > 1) {
if (! wantz) {
slasrt_("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) {
sswap_(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] = (real) lwmin;
iwork[1] = liwmin;
return 0;
/* End of SSTEMR */
} /* sstemr_ */
