/* Subroutine */ int ssbevx_(char *jobz, char *range, char *uplo, integer *n,
integer *kd, real *ab, integer *ldab, real *q, integer *ldq, real *vl,
real *vu, integer *il, integer *iu, real *abstol, integer *m, real *
w, real *z__, integer *ldz, real *work, integer *iwork, integer *
ifail, integer *info)
{
/* System generated locals */
integer ab_dim1, ab_offset, q_dim1, q_offset, z_dim1, z_offset, i__1,
i__2;
real r__1, r__2;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
integer i__, j, jj;
real eps, vll, vuu, tmp1;
integer indd, inde;
real anrm;
integer imax;
real rmin, rmax;
logical test;
integer itmp1, indee;
real sigma;
extern logical lsame_(char *, char *);
integer iinfo;
extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *);
char order[1];
extern /* Subroutine */ int sgemv_(char *, integer *, integer *, real *,
real *, integer *, real *, integer *, real *, real *, integer *);
logical lower;
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *), sswap_(integer *, real *, integer *, real *, integer *
);
logical wantz, alleig, indeig;
integer iscale, indibl;
logical valeig;
extern doublereal slamch_(char *);
real safmin;
extern /* Subroutine */ int xerbla_(char *, integer *);
real abstll, bignum;
extern doublereal slansb_(char *, char *, integer *, integer *, real *,
integer *, real *);
extern /* Subroutine */ int slascl_(char *, integer *, integer *, real *,
real *, integer *, integer *, real *, integer *, integer *);
integer indisp, indiwo;
extern /* Subroutine */ int slacpy_(char *, integer *, integer *, real *,
integer *, real *, integer *);
integer indwrk;
extern /* Subroutine */ int ssbtrd_(char *, char *, integer *, integer *,
real *, integer *, real *, real *, real *, integer *, real *,
integer *), sstein_(integer *, real *, real *,
integer *, real *, integer *, integer *, real *, integer *, real *
, integer *, integer *, integer *), ssterf_(integer *, real *,
real *, integer *);
integer nsplit;
real smlnum;
extern /* Subroutine */ int sstebz_(char *, char *, integer *, real *,
real *, integer *, integer *, real *, real *, real *, integer *,
integer *, real *, integer *, integer *, real *, integer *,
integer *), ssteqr_(char *, integer *, real *,
real *, real *, integer *, real *, integer *);
/* -- LAPACK driver routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SSBEVX computes selected eigenvalues and, optionally, eigenvectors */
/* of a real symmetric band matrix A. Eigenvalues and eigenvectors can */
/* be selected by specifying either a range of values or a range of */
/* indices for the desired eigenvalues. */
/* 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. */
/* UPLO (input) CHARACTER*1 */
/* = 'U': Upper triangle of A is stored; */
/* = 'L': Lower triangle of A is stored. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* KD (input) INTEGER */
/* The number of superdiagonals of the matrix A if UPLO = 'U', */
/* or the number of subdiagonals if UPLO = 'L'. KD >= 0. */
/* AB (input/output) REAL array, dimension (LDAB, N) */
/* On entry, the upper or lower triangle of the symmetric band */
/* matrix A, stored in the first KD+1 rows of the array. The */
/* j-th column of A is stored in the j-th column of the array AB */
/* as follows: */
/* if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; */
/* if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). */
/* On exit, AB is overwritten by values generated during the */
/* reduction to tridiagonal form. If UPLO = 'U', the first */
/* superdiagonal and the diagonal of the tridiagonal matrix T */
/* are returned in rows KD and KD+1 of AB, and if UPLO = 'L', */
/* the diagonal and first subdiagonal of T are returned in the */
/* first two rows of AB. */
/* LDAB (input) INTEGER */
/* The leading dimension of the array AB. LDAB >= KD + 1. */
/* Q (output) REAL array, dimension (LDQ, N) */
/* If JOBZ = 'V', the N-by-N orthogonal matrix used in the */
/* reduction to tridiagonal form. */
/* If JOBZ = 'N', the array Q is not referenced. */
/* LDQ (input) INTEGER */
/* The leading dimension of the array Q. If JOBZ = 'V', then */
/* LDQ >= max(1,N). */
/* 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. */
/* An approximate eigenvalue is accepted as converged */
/* when it is determined to lie in an interval [a,b] */
/* of width less than or equal to */
/* ABSTOL + EPS * max( |a|,|b| ) , */
/* where EPS is the machine precision. If ABSTOL is less than */
/* or equal to zero, then EPS*|T| will be used in its place, */
/* where |T| is the 1-norm of the tridiagonal matrix obtained */
/* by reducing AB to tridiagonal form. */
/* Eigenvalues will be computed most accurately when ABSTOL is */
/* set to twice the underflow threshold 2*SLAMCH('S'), not zero. */
/* If this routine returns with INFO>0, indicating that some */
/* eigenvectors did not converge, try setting ABSTOL to */
/* 2*SLAMCH('S'). */
/* See "Computing Small Singular Values of Bidiagonal Matrices */
/* with Guaranteed High Relative Accuracy," by Demmel and */
/* Kahan, LAPACK Working Note #3. */
/* 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 A */
/* corresponding to the selected eigenvalues, with the i-th */
/* column of Z holding the eigenvector associated with W(i). */
/* If an eigenvector fails to converge, then that column of Z */
/* contains the latest approximation to the eigenvector, and the */
/* index of the eigenvector is returned in IFAIL. */
/* 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). */
/* WORK (workspace) REAL array, dimension (7*N) */
/* IWORK (workspace) INTEGER array, dimension (5*N) */
/* IFAIL (output) INTEGER array, dimension (N) */
/* If JOBZ = 'V', then if INFO = 0, the first M elements of */
/* IFAIL are zero. If INFO > 0, then IFAIL contains the */
/* indices of the eigenvectors that failed to converge. */
/* If JOBZ = 'N', then IFAIL is not referenced. */
/* INFO (output) INTEGER */
/* = 0: successful exit. */
/* < 0: if INFO = -i, the i-th argument had an illegal value. */
/* > 0: if INFO = i, then i eigenvectors failed to converge. */
/* Their indices are stored in array IFAIL. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
ab_dim1 = *ldab;
ab_offset = 1 + ab_dim1;
ab -= ab_offset;
q_dim1 = *ldq;
q_offset = 1 + q_dim1;
q -= q_offset;
--w;
z_dim1 = *ldz;
z_offset = 1 + z_dim1;
z__ -= z_offset;
--work;
--iwork;
--ifail;
/* Function Body */
wantz = lsame_(jobz, "V");
alleig = lsame_(range, "A");
valeig = lsame_(range, "V");
indeig = lsame_(range, "I");
lower = lsame_(uplo, "L");
*info = 0;
if (! (wantz || lsame_(jobz, "N"))) {
*info = -1;
} else if (! (alleig || valeig || indeig)) {
*info = -2;
} else if (! (lower || lsame_(uplo, "U"))) {
*info = -3;
} else if (*n < 0) {
*info = -4;
} else if (*kd < 0) {
*info = -5;
} else if (*ldab < *kd + 1) {
*info = -7;
} else if (wantz && *ldq < max(1,*n)) {
*info = -9;
} else {
if (valeig) {
if (*n > 0 && *vu <= *vl) {
*info = -11;
}
} else if (indeig) {
if (*il < 1 || *il > max(1,*n)) {
*info = -12;
} else if (*iu < min(*n,*il) || *iu > *n) {
*info = -13;
}
}
}
if (*info == 0) {
if (*ldz < 1 || wantz && *ldz < *n) {
*info = -18;
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SSBEVX", &i__1);
return 0;
}
/* Quick return if possible */
*m = 0;
if (*n == 0) {
return 0;
}
if (*n == 1) {
*m = 1;
if (lower) {
tmp1 = ab[ab_dim1 + 1];
} else {
tmp1 = ab[*kd + 1 + ab_dim1];
}
if (valeig) {
if (! (*vl < tmp1 && *vu >= tmp1)) {
*m = 0;
}
}
if (*m == 1) {
w[1] = tmp1;
if (wantz) {
z__[z_dim1 + 1] = 1.f;
}
}
return 0;
}
/* 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);
/* Scale matrix to allowable range, if necessary. */
iscale = 0;
abstll = *abstol;
if (valeig) {
vll = *vl;
vuu = *vu;
} else {
vll = 0.f;
vuu = 0.f;
}
anrm = slansb_("M", uplo, n, kd, &ab[ab_offset], ldab, &work[1]);
if (anrm > 0.f && anrm < rmin) {
iscale = 1;
sigma = rmin / anrm;
} else if (anrm > rmax) {
iscale = 1;
sigma = rmax / anrm;
}
if (iscale == 1) {
if (lower) {
slascl_("B", kd, kd, &c_b14, &sigma, n, n, &ab[ab_offset], ldab,
info);
} else {
slascl_("Q", kd, kd, &c_b14, &sigma, n, n, &ab[ab_offset], ldab,
info);
}
if (*abstol > 0.f) {
abstll = *abstol * sigma;
}
if (valeig) {
vll = *vl * sigma;
vuu = *vu * sigma;
}
}
/* Call SSBTRD to reduce symmetric band matrix to tridiagonal form. */
indd = 1;
inde = indd + *n;
indwrk = inde + *n;
ssbtrd_(jobz, uplo, n, kd, &ab[ab_offset], ldab, &work[indd], &work[inde],
&q[q_offset], ldq, &work[indwrk], &iinfo);
/* If all eigenvalues are desired and ABSTOL is less than or equal */
/* to zero, then call SSTERF or SSTEQR. If this fails for some */
/* eigenvalue, then try SSTEBZ. */
test = FALSE_;
if (indeig) {
if (*il == 1 && *iu == *n) {
test = TRUE_;
}
}
if ((alleig || test) && *abstol <= 0.f) {
scopy_(n, &work[indd], &c__1, &w[1], &c__1);
indee = indwrk + (*n << 1);
if (! wantz) {
i__1 = *n - 1;
scopy_(&i__1, &work[inde], &c__1, &work[indee], &c__1);
ssterf_(n, &w[1], &work[indee], info);
} else {
slacpy_("A", n, n, &q[q_offset], ldq, &z__[z_offset], ldz);
i__1 = *n - 1;
scopy_(&i__1, &work[inde], &c__1, &work[indee], &c__1);
ssteqr_(jobz, n, &w[1], &work[indee], &z__[z_offset], ldz, &work[
indwrk], info);
if (*info == 0) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
ifail[i__] = 0;
/* L10: */
}
}
}
if (*info == 0) {
*m = *n;
goto L30;
}
*info = 0;
}
/* Otherwise, call SSTEBZ and, if eigenvectors are desired, SSTEIN. */
if (wantz) {
*(unsigned char *)order = 'B';
} else {
*(unsigned char *)order = 'E';
}
indibl = 1;
indisp = indibl + *n;
indiwo = indisp + *n;
sstebz_(range, order, n, &vll, &vuu, il, iu, &abstll, &work[indd], &work[
inde], m, &nsplit, &w[1], &iwork[indibl], &iwork[indisp], &work[
indwrk], &iwork[indiwo], info);
if (wantz) {
sstein_(n, &work[indd], &work[inde], m, &w[1], &iwork[indibl], &iwork[
indisp], &z__[z_offset], ldz, &work[indwrk], &iwork[indiwo], &
ifail[1], info);
/* Apply orthogonal matrix used in reduction to tridiagonal */
/* form to eigenvectors returned by SSTEIN. */
i__1 = *m;
for (j = 1; j <= i__1; ++j) {
scopy_(n, &z__[j * z_dim1 + 1], &c__1, &work[1], &c__1);
sgemv_("N", n, n, &c_b14, &q[q_offset], ldq, &work[1], &c__1, &
c_b34, &z__[j * z_dim1 + 1], &c__1);
/* L20: */
}
}
/* If matrix was scaled, then rescale eigenvalues appropriately. */
L30:
if (iscale == 1) {
if (*info == 0) {
imax = *m;
} else {
imax = *info - 1;
}
r__1 = 1.f / sigma;
sscal_(&imax, &r__1, &w[1], &c__1);
}
/* If eigenvalues are not in order, then sort them, along with */
/* eigenvectors. */
if (wantz) {
i__1 = *m - 1;
for (j = 1; j <= i__1; ++j) {
i__ = 0;
tmp1 = w[j];
i__2 = *m;
for (jj = j + 1; jj <= i__2; ++jj) {
if (w[jj] < tmp1) {
i__ = jj;
tmp1 = w[jj];
}
/* L40: */
}
if (i__ != 0) {
itmp1 = iwork[indibl + i__ - 1];
w[i__] = w[j];
iwork[indibl + i__ - 1] = iwork[indibl + j - 1];
w[j] = tmp1;
iwork[indibl + j - 1] = itmp1;
sswap_(n, &z__[i__ * z_dim1 + 1], &c__1, &z__[j * z_dim1 + 1],
&c__1);
if (*info != 0) {
itmp1 = ifail[i__];
ifail[i__] = ifail[j];
ifail[j] = itmp1;
}
}
/* L50: */
}
}
return 0;
/* End of SSBEVX */
} /* ssbevx_ */
/* Subroutine */ int cheevr_(char *jobz, char *range, char *uplo, integer *n,
complex *a, integer *lda, real *vl, real *vu, integer *il, integer *
iu, real *abstol, integer *m, real *w, complex *z__, integer *ldz,
integer *isuppz, complex *work, integer *lwork, real *rwork, integer *
lrwork, integer *iwork, integer *liwork, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, z_dim1, z_offset, i__1, i__2;
real r__1, r__2;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
integer i__, j, nb, jj;
real eps, vll, vuu, tmp1, anrm;
integer imax;
real rmin, rmax;
logical test;
integer itmp1, indrd, indre;
real sigma;
extern logical lsame_(char *, char *);
integer iinfo;
extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *);
char order[1];
integer indwk;
extern /* Subroutine */ int cswap_(integer *, complex *, integer *,
complex *, integer *);
integer lwmin;
logical lower;
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *);
logical wantz, alleig, indeig;
integer iscale, ieeeok, indibl, indrdd, indifl, indree;
logical valeig;
extern doublereal slamch_(char *);
extern /* Subroutine */ int chetrd_(char *, integer *, complex *, integer
*, real *, real *, complex *, complex *, integer *, integer *), csscal_(integer *, real *, complex *, integer *);
real safmin;
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *);
extern /* Subroutine */ int xerbla_(char *, integer *);
real abstll, bignum;
integer indtau, indisp;
extern /* Subroutine */ int cstein_(integer *, real *, real *, integer *,
real *, integer *, integer *, complex *, integer *, real *,
integer *, integer *, integer *);
integer indiwo, indwkn;
extern doublereal clansy_(char *, char *, integer *, complex *, integer *,
real *);
extern /* Subroutine */ int cstemr_(char *, char *, integer *, real *,
real *, real *, real *, integer *, integer *, integer *, real *,
complex *, integer *, integer *, integer *, logical *, real *,
integer *, integer *, integer *, integer *);
integer indrwk, liwmin;
logical tryrac;
extern /* Subroutine */ int ssterf_(integer *, real *, real *, integer *);
integer lrwmin, llwrkn, llwork, nsplit;
real smlnum;
extern /* Subroutine */ int cunmtr_(char *, char *, char *, integer *,
integer *, complex *, integer *, complex *, complex *, integer *,
complex *, integer *, integer *), sstebz_(
char *, char *, integer *, real *, real *, integer *, integer *,
real *, real *, real *, integer *, integer *, real *, integer *,
integer *, real *, integer *, integer *);
logical lquery;
integer lwkopt, llrwork;
/* -- LAPACK driver routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CHEEVR computes selected eigenvalues and, optionally, eigenvectors */
/* of a complex Hermitian matrix A. Eigenvalues and eigenvectors can */
/* be selected by specifying either a range of values or a range of */
/* indices for the desired eigenvalues. */
/* CHEEVR first reduces the matrix A to tridiagonal form T with a call */
/* to CHETRD. Then, whenever possible, CHEEVR calls CSTEMR to compute */
/* the eigenspectrum using Relatively Robust Representations. CSTEMR */
/* computes eigenvalues 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 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. */
/* The desired accuracy of the output can be specified by the input */
/* parameter ABSTOL. */
/* For more details, see DSTEMR's documentation and: */
/* - 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. */
/* Note 1 : CHEEVR calls CSTEMR when the full spectrum is requested */
/* on machines which conform to the ieee-754 floating point standard. */
/* CHEEVR calls SSTEBZ and CSTEIN on non-ieee machines and */
/* when partial spectrum requests are made. */
/* Normal execution of CSTEMR may create NaNs and infinities and */
/* hence may abort due to a floating point exception in environments */
/* which do not handle NaNs and infinities in the ieee standard default */
/* manner. */
/* 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. */
/* ********* For RANGE = 'V' or 'I' and IU - IL < N - 1, SSTEBZ and */
/* ********* CSTEIN are called */
/* UPLO (input) CHARACTER*1 */
/* = 'U': Upper triangle of A is stored; */
/* = 'L': Lower triangle of A is stored. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* A (input/output) COMPLEX array, dimension (LDA, N) */
/* On entry, the Hermitian matrix A. If UPLO = 'U', the */
/* leading N-by-N upper triangular part of A contains the */
/* upper triangular part of the matrix A. If UPLO = 'L', */
/* the leading N-by-N lower triangular part of A contains */
/* the lower triangular part of the matrix A. */
/* On exit, the lower triangle (if UPLO='L') or the upper */
/* triangle (if UPLO='U') of A, including the diagonal, is */
/* destroyed. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,N). */
/* 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. */
/* An approximate eigenvalue is accepted as converged */
/* when it is determined to lie in an interval [a,b] */
/* of width less than or equal to */
/* ABSTOL + EPS * max( |a|,|b| ) , */
/* where EPS is the machine precision. If ABSTOL is less than */
/* or equal to zero, then EPS*|T| will be used in its place, */
/* where |T| is the 1-norm of the tridiagonal matrix obtained */
/* by reducing A to tridiagonal form. */
/* See "Computing Small Singular Values of Bidiagonal Matrices */
/* with Guaranteed High Relative Accuracy," by Demmel and */
/* Kahan, LAPACK Working Note #3. */
/* If high relative accuracy is important, set ABSTOL to */
/* SLAMCH( 'Safe minimum' ). Doing so will guarantee that */
/* eigenvalues are computed to high relative accuracy when */
/* possible in future releases. The current code does not */
/* make any guarantees about high relative accuracy, but */
/* furutre releases will. See J. Barlow and J. 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) COMPLEX 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 A */
/* 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 ). */
/* ********* Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1 */
/* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK)) */
/* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. */
/* LWORK (input) INTEGER */
/* The length of the array WORK. LWORK >= max(1,2*N). */
/* For optimal efficiency, LWORK >= (NB+1)*N, */
/* where NB is the max of the blocksize for CHETRD and for */
/* CUNMTR as returned by ILAENV. */
/* If LWORK = -1, then a workspace query is assumed; the routine */
/* only calculates the optimal sizes of the WORK, RWORK and */
/* IWORK arrays, returns these values as the first entries of */
/* the WORK, RWORK and IWORK arrays, and no error message */
/* related to LWORK or LRWORK or LIWORK is issued by XERBLA. */
/* RWORK (workspace/output) REAL array, dimension (MAX(1,LRWORK)) */
/* On exit, if INFO = 0, RWORK(1) returns the optimal */
/* (and minimal) LRWORK. */
/* LRWORK (input) INTEGER */
/* The length of the array RWORK. LRWORK >= max(1,24*N). */
/* If LRWORK = -1, then a workspace query is assumed; the */
/* routine only calculates the optimal sizes of the WORK, RWORK */
/* and IWORK arrays, returns these values as the first entries */
/* of the WORK, RWORK and IWORK arrays, and no error message */
/* related to LWORK or LRWORK or LIWORK is issued by XERBLA. */
/* IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK)) */
/* On exit, if INFO = 0, IWORK(1) returns the optimal */
/* (and minimal) 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 sizes of the WORK, RWORK */
/* and IWORK arrays, returns these values as the first entries */
/* of the WORK, RWORK and IWORK arrays, and no error message */
/* related to LWORK or LRWORK or LIWORK is issued by XERBLA. */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* > 0: Internal error */
/* Further Details */
/* =============== */
/* Based on contributions by */
/* Inderjit Dhillon, IBM Almaden, USA */
/* Osni Marques, LBNL/NERSC, USA */
/* Ken Stanley, Computer Science Division, University of */
/* California at Berkeley, USA */
/* Jason Riedy, Computer Science Division, University of */
/* California at Berkeley, USA */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--w;
z_dim1 = *ldz;
z_offset = 1 + z_dim1;
z__ -= z_offset;
--isuppz;
--work;
--rwork;
--iwork;
/* Function Body */
ieeeok = ilaenv_(&c__10, "CHEEVR", "N", &c__1, &c__2, &c__3, &c__4);
lower = lsame_(uplo, "L");
wantz = lsame_(jobz, "V");
alleig = lsame_(range, "A");
valeig = lsame_(range, "V");
indeig = lsame_(range, "I");
lquery = *lwork == -1 || *lrwork == -1 || *liwork == -1;
/* Computing MAX */
i__1 = 1, i__2 = *n * 24;
lrwmin = max(i__1,i__2);
/* Computing MAX */
i__1 = 1, i__2 = *n * 10;
liwmin = max(i__1,i__2);
/* Computing MAX */
i__1 = 1, i__2 = *n << 1;
lwmin = max(i__1,i__2);
*info = 0;
if (! (wantz || lsame_(jobz, "N"))) {
*info = -1;
} else if (! (alleig || valeig || indeig)) {
*info = -2;
} else if (! (lower || lsame_(uplo, "U"))) {
*info = -3;
} else if (*n < 0) {
*info = -4;
} else if (*lda < max(1,*n)) {
*info = -6;
} else {
if (valeig) {
if (*n > 0 && *vu <= *vl) {
*info = -8;
}
} else if (indeig) {
if (*il < 1 || *il > max(1,*n)) {
*info = -9;
} else if (*iu < min(*n,*il) || *iu > *n) {
*info = -10;
}
}
}
if (*info == 0) {
if (*ldz < 1 || wantz && *ldz < *n) {
*info = -15;
}
}
if (*info == 0) {
nb = ilaenv_(&c__1, "CHETRD", uplo, n, &c_n1, &c_n1, &c_n1);
/* Computing MAX */
i__1 = nb, i__2 = ilaenv_(&c__1, "CUNMTR", uplo, n, &c_n1, &c_n1, &
c_n1);
nb = max(i__1,i__2);
/* Computing MAX */
i__1 = (nb + 1) * *n;
lwkopt = max(i__1,lwmin);
work[1].r = (real) lwkopt, work[1].i = 0.f;
rwork[1] = (real) lrwmin;
iwork[1] = liwmin;
if (*lwork < lwmin && ! lquery) {
*info = -18;
} else if (*lrwork < lrwmin && ! lquery) {
*info = -20;
} else if (*liwork < liwmin && ! lquery) {
*info = -22;
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CHEEVR", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
*m = 0;
if (*n == 0) {
work[1].r = 1.f, work[1].i = 0.f;
return 0;
}
if (*n == 1) {
work[1].r = 2.f, work[1].i = 0.f;
if (alleig || indeig) {
*m = 1;
i__1 = a_dim1 + 1;
w[1] = a[i__1].r;
} else {
i__1 = a_dim1 + 1;
i__2 = a_dim1 + 1;
if (*vl < a[i__1].r && *vu >= a[i__2].r) {
*m = 1;
i__1 = a_dim1 + 1;
w[1] = a[i__1].r;
}
}
if (wantz) {
i__1 = z_dim1 + 1;
z__[i__1].r = 1.f, z__[i__1].i = 0.f;
}
return 0;
}
/* 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);
/* Scale matrix to allowable range, if necessary. */
iscale = 0;
abstll = *abstol;
if (valeig) {
vll = *vl;
vuu = *vu;
}
anrm = clansy_("M", uplo, n, &a[a_offset], lda, &rwork[1]);
if (anrm > 0.f && anrm < rmin) {
iscale = 1;
sigma = rmin / anrm;
} else if (anrm > rmax) {
iscale = 1;
sigma = rmax / anrm;
}
if (iscale == 1) {
if (lower) {
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
i__2 = *n - j + 1;
csscal_(&i__2, &sigma, &a[j + j * a_dim1], &c__1);
/* L10: */
}
} else {
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
csscal_(&j, &sigma, &a[j * a_dim1 + 1], &c__1);
/* L20: */
}
}
if (*abstol > 0.f) {
abstll = *abstol * sigma;
}
if (valeig) {
vll = *vl * sigma;
vuu = *vu * sigma;
}
}
/* Initialize indices into workspaces. Note: The IWORK indices are */
/* used only if SSTERF or CSTEMR fail. */
/* WORK(INDTAU:INDTAU+N-1) stores the complex scalar factors of the */
/* elementary reflectors used in CHETRD. */
indtau = 1;
/* INDWK is the starting offset of the remaining complex workspace, */
/* and LLWORK is the remaining complex workspace size. */
indwk = indtau + *n;
llwork = *lwork - indwk + 1;
/* RWORK(INDRD:INDRD+N-1) stores the real tridiagonal's diagonal */
/* entries. */
indrd = 1;
/* RWORK(INDRE:INDRE+N-1) stores the off-diagonal entries of the */
/* tridiagonal matrix from CHETRD. */
indre = indrd + *n;
/* RWORK(INDRDD:INDRDD+N-1) is a copy of the diagonal entries over */
/* -written by CSTEMR (the SSTERF path copies the diagonal to W). */
indrdd = indre + *n;
/* RWORK(INDREE:INDREE+N-1) is a copy of the off-diagonal entries over */
/* -written while computing the eigenvalues in SSTERF and CSTEMR. */
indree = indrdd + *n;
/* INDRWK is the starting offset of the left-over real workspace, and */
/* LLRWORK is the remaining workspace size. */
indrwk = indree + *n;
llrwork = *lrwork - indrwk + 1;
/* IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in SSTEBZ and */
/* stores the block indices of each of the M<=N eigenvalues. */
indibl = 1;
/* IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in SSTEBZ and */
/* stores the starting and finishing indices of each block. */
indisp = indibl + *n;
/* IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors */
/* that corresponding to eigenvectors that fail to converge in */
/* SSTEIN. This information is discarded; if any fail, the driver */
/* returns INFO > 0. */
indifl = indisp + *n;
/* INDIWO is the offset of the remaining integer workspace. */
indiwo = indisp + *n;
/* Call CHETRD to reduce Hermitian matrix to tridiagonal form. */
chetrd_(uplo, n, &a[a_offset], lda, &rwork[indrd], &rwork[indre], &work[
indtau], &work[indwk], &llwork, &iinfo);
/* If all eigenvalues are desired */
/* then call SSTERF or CSTEMR and CUNMTR. */
test = FALSE_;
if (indeig) {
if (*il == 1 && *iu == *n) {
test = TRUE_;
}
}
if ((alleig || test) && ieeeok == 1) {
if (! wantz) {
scopy_(n, &rwork[indrd], &c__1, &w[1], &c__1);
i__1 = *n - 1;
scopy_(&i__1, &rwork[indre], &c__1, &rwork[indree], &c__1);
ssterf_(n, &w[1], &rwork[indree], info);
} else {
i__1 = *n - 1;
scopy_(&i__1, &rwork[indre], &c__1, &rwork[indree], &c__1);
scopy_(n, &rwork[indrd], &c__1, &rwork[indrdd], &c__1);
if (*abstol <= *n * 2.f * eps) {
tryrac = TRUE_;
} else {
tryrac = FALSE_;
}
cstemr_(jobz, "A", n, &rwork[indrdd], &rwork[indree], vl, vu, il,
iu, m, &w[1], &z__[z_offset], ldz, n, &isuppz[1], &tryrac,
&rwork[indrwk], &llrwork, &iwork[1], liwork, info);
/* Apply unitary matrix used in reduction to tridiagonal */
/* form to eigenvectors returned by CSTEIN. */
if (wantz && *info == 0) {
indwkn = indwk;
llwrkn = *lwork - indwkn + 1;
cunmtr_("L", uplo, "N", n, m, &a[a_offset], lda, &work[indtau]
, &z__[z_offset], ldz, &work[indwkn], &llwrkn, &iinfo);
}
}
if (*info == 0) {
*m = *n;
goto L30;
}
*info = 0;
}
/* Otherwise, call SSTEBZ and, if eigenvectors are desired, CSTEIN. */
/* Also call SSTEBZ and CSTEIN if CSTEMR fails. */
if (wantz) {
*(unsigned char *)order = 'B';
} else {
*(unsigned char *)order = 'E';
}
sstebz_(range, order, n, &vll, &vuu, il, iu, &abstll, &rwork[indrd], &
rwork[indre], m, &nsplit, &w[1], &iwork[indibl], &iwork[indisp], &
rwork[indrwk], &iwork[indiwo], info);
if (wantz) {
cstein_(n, &rwork[indrd], &rwork[indre], m, &w[1], &iwork[indibl], &
iwork[indisp], &z__[z_offset], ldz, &rwork[indrwk], &iwork[
indiwo], &iwork[indifl], info);
/* Apply unitary matrix used in reduction to tridiagonal */
/* form to eigenvectors returned by CSTEIN. */
indwkn = indwk;
llwrkn = *lwork - indwkn + 1;
cunmtr_("L", uplo, "N", n, m, &a[a_offset], lda, &work[indtau], &z__[
z_offset], ldz, &work[indwkn], &llwrkn, &iinfo);
}
/* If matrix was scaled, then rescale eigenvalues appropriately. */
L30:
if (iscale == 1) {
if (*info == 0) {
imax = *m;
} else {
imax = *info - 1;
}
r__1 = 1.f / sigma;
sscal_(&imax, &r__1, &w[1], &c__1);
}
/* If eigenvalues are not in order, then sort them, along with */
/* eigenvectors. */
if (wantz) {
i__1 = *m - 1;
for (j = 1; j <= i__1; ++j) {
i__ = 0;
tmp1 = w[j];
i__2 = *m;
for (jj = j + 1; jj <= i__2; ++jj) {
if (w[jj] < tmp1) {
i__ = jj;
tmp1 = w[jj];
}
/* L40: */
}
if (i__ != 0) {
itmp1 = iwork[indibl + i__ - 1];
w[i__] = w[j];
iwork[indibl + i__ - 1] = iwork[indibl + j - 1];
w[j] = tmp1;
iwork[indibl + j - 1] = itmp1;
cswap_(n, &z__[i__ * z_dim1 + 1], &c__1, &z__[j * z_dim1 + 1],
&c__1);
}
/* L50: */
}
}
/* Set WORK(1) to optimal workspace size. */
work[1].r = (real) lwkopt, work[1].i = 0.f;
rwork[1] = (real) lrwmin;
iwork[1] = liwmin;
return 0;
/* End of CHEEVR */
} /* cheevr_ */
/* Subroutine */ int sstevr_(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, vll, vuu, tmp1;
static integer imax;
static real rmin, rmax, tnrm;
static integer itmp1;
static real sigma;
extern logical lsame_(char *, char *, ftnlen, ftnlen);
extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *);
static char order[1];
static integer lwmin;
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *), sswap_(integer *, real *, integer *, real *, integer *
);
static logical wantz, alleig, indeig;
static integer iscale, ieeeok, indibl, indifl;
static logical valeig;
extern doublereal slamch_(char *, ftnlen);
static real safmin;
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *, ftnlen, ftnlen);
extern /* Subroutine */ int xerbla_(char *, integer *, ftnlen);
static real bignum;
static integer indisp, indiwo, liwmin;
extern doublereal slanst_(char *, integer *, real *, real *, ftnlen);
extern /* Subroutine */ int sstein_(integer *, real *, real *, integer *,
real *, integer *, integer *, real *, integer *, real *, integer *
, integer *, integer *), ssterf_(integer *, real *, real *,
integer *), sstegr_(char *, char *, integer *, real *, real *,
real *, real *, integer *, integer *, real *, integer *, real *,
real *, integer *, integer *, real *, integer *, integer *,
integer *, integer *, ftnlen, ftnlen);
static integer nsplit;
extern /* Subroutine */ int sstebz_(char *, char *, integer *, real *,
real *, integer *, integer *, real *, real *, real *, integer *,
integer *, real *, integer *, integer *, real *, integer *,
integer *, ftnlen, ftnlen);
static real smlnum;
static logical lquery;
/* -- LAPACK driver routine (version 3.0) -- */
/* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., */
/* Courant Institute, Argonne National Lab, and Rice University */
/* March 20, 2000 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SSTEVR 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. */
/* Whenever possible, SSTEVR calls SSTEGR to compute the */
/* eigenspectrum using Relatively Robust Representations. SSTEGR */
/* computes eigenvalues 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 : SSTEVR calls SSTEGR when the full spectrum is requested */
/* on machines which conform to the ieee-754 floating point standard. */
/* SSTEVR calls SSTEBZ and SSTEIN on non-ieee machines and */
/* when partial spectrum requests are made. */
/* Normal execution of SSTEGR may create NaNs and infinities and */
/* hence may abort due to a floating point exception in environments */
/* which do not handle NaNs and infinities in the ieee standard default */
/* manner. */
/* 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. */
/* ********* For RANGE = 'V' or 'I' and IU - IL < N - 1, SSTEBZ and */
/* ********* SSTEIN are called */
/* 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 */
/* A. */
/* On exit, D may be multiplied by a constant factor chosen */
/* to avoid over/underflow in computing the eigenvalues. */
/* E (input/output) REAL array, dimension (N) */
/* On entry, the (n-1) subdiagonal elements of the tridiagonal */
/* matrix A in elements 1 to N-1 of E; E(N) need not be set. */
/* On exit, E may be multiplied by a constant factor chosen */
/* to avoid over/underflow in computing the eigenvalues. */
/* 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. */
/* An approximate eigenvalue is accepted as converged */
/* when it is determined to lie in an interval [a,b] */
/* of width less than or equal to */
/* ABSTOL + EPS * max( |a|,|b| ) , */
/* where EPS is the machine precision. If ABSTOL is less than */
/* or equal to zero, then EPS*|T| will be used in its place, */
/* where |T| is the 1-norm of the tridiagonal matrix obtained */
/* by reducing A to tridiagonal form. */
/* See "Computing Small Singular Values of Bidiagonal Matrices */
/* with Guaranteed High Relative Accuracy," by Demmel and */
/* Kahan, LAPACK Working Note #3. */
/* If high relative accuracy is important, set ABSTOL to */
/* SLAMCH( 'Safe minimum' ). Doing so will guarantee that */
/* eigenvalues are computed to high relative accuracy when */
/* possible in future releases. The current code does not */
/* make any guarantees about high relative accuracy, but */
/* future releases will. See J. Barlow and J. 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 A */
/* corresponding to the selected eigenvalues, with the i-th */
/* column of Z holding the eigenvector associated with W(i). */
/* 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 ). */
/* ********* Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1 */
/* 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 >= 20*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 (and */
/* minimal) LIWORK. */
/* LIWORK (input) INTEGER */
/* The dimension of the array IWORK. LIWORK >= 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: Internal error */
/* Further Details */
/* =============== */
/* Based on contributions by */
/* Inderjit Dhillon, IBM Almaden, USA */
/* Osni Marques, LBNL/NERSC, USA */
/* Ken Stanley, Computer Science Division, University of */
/* California at 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 */
ieeeok = ilaenv_(&c__10, "SSTEVR", "N", &c__1, &c__2, &c__3, &c__4, (
ftnlen)6, (ftnlen)1);
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 * 20;
liwmin = *n * 10;
*info = 0;
if (! (wantz || lsame_(jobz, "N", (ftnlen)1, (ftnlen)1))) {
*info = -1;
} else if (! (alleig || valeig || indeig)) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else {
if (valeig) {
if (*n > 0 && *vu <= *vl) {
*info = -7;
}
} else if (indeig) {
if (*il < 1 || *il > max(1,*n)) {
*info = -8;
} else if (*iu < min(*n,*il) || *iu > *n) {
*info = -9;
}
}
}
if (*info == 0) {
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_("SSTEVR", &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. */
iscale = 0;
vll = *vl;
vuu = *vu;
tnrm = slanst_("M", n, &d__[1], &e[1], (ftnlen)1);
if (tnrm > 0.f && tnrm < rmin) {
iscale = 1;
sigma = rmin / tnrm;
} else if (tnrm > rmax) {
iscale = 1;
sigma = rmax / tnrm;
}
if (iscale == 1) {
sscal_(n, &sigma, &d__[1], &c__1);
i__1 = *n - 1;
sscal_(&i__1, &sigma, &e[1], &c__1);
if (valeig) {
vll = *vl * sigma;
vuu = *vu * sigma;
}
}
/* If all eigenvalues are desired, then */
/* call SSTERF or SSTEGR. If this fails for some eigenvalue, then */
/* try SSTEBZ. */
if ((alleig || indeig && *il == 1 && *iu == *n) && ieeeok == 1) {
i__1 = *n - 1;
scopy_(&i__1, &e[1], &c__1, &work[1], &c__1);
if (! wantz) {
scopy_(n, &d__[1], &c__1, &w[1], &c__1);
ssterf_(n, &w[1], &work[1], info);
} else {
scopy_(n, &d__[1], &c__1, &work[*n + 1], &c__1);
i__1 = *lwork - (*n << 1);
sstegr_(jobz, "A", n, &work[*n + 1], &work[1], vl, vu, il, iu,
abstol, m, &w[1], &z__[z_offset], ldz, &isuppz[1], &work[(
*n << 1) + 1], &i__1, &iwork[1], liwork, info, (ftnlen)1,
(ftnlen)1);
}
if (*info == 0) {
*m = *n;
goto L10;
}
*info = 0;
}
/* Otherwise, call SSTEBZ and, if eigenvectors are desired, SSTEIN. */
if (wantz) {
*(unsigned char *)order = 'B';
} else {
*(unsigned char *)order = 'E';
}
indibl = 1;
indisp = indibl + *n;
indifl = indisp + *n;
indiwo = indifl + *n;
sstebz_(range, order, n, &vll, &vuu, il, iu, abstol, &d__[1], &e[1], m, &
nsplit, &w[1], &iwork[indibl], &iwork[indisp], &work[1], &iwork[
indiwo], info, (ftnlen)1, (ftnlen)1);
if (wantz) {
sstein_(n, &d__[1], &e[1], m, &w[1], &iwork[indibl], &iwork[indisp], &
z__[z_offset], ldz, &work[1], &iwork[indiwo], &iwork[indifl],
info);
}
/* If matrix was scaled, then rescale eigenvalues appropriately. */
L10:
if (iscale == 1) {
if (*info == 0) {
imax = *m;
} else {
imax = *info - 1;
}
r__1 = 1.f / sigma;
sscal_(&imax, &r__1, &w[1], &c__1);
}
/* If eigenvalues are not in order, then sort them, along with */
/* eigenvectors. */
if (wantz) {
i__1 = *m - 1;
for (j = 1; j <= i__1; ++j) {
i__ = 0;
tmp1 = w[j];
i__2 = *m;
for (jj = j + 1; jj <= i__2; ++jj) {
if (w[jj] < tmp1) {
i__ = jj;
tmp1 = w[jj];
}
/* L20: */
}
if (i__ != 0) {
itmp1 = iwork[i__];
w[i__] = w[j];
iwork[i__] = iwork[j];
w[j] = tmp1;
iwork[j] = itmp1;
sswap_(n, &z__[i__ * z_dim1 + 1], &c__1, &z__[j * z_dim1 + 1],
&c__1);
}
/* L30: */
}
}
/* Causes problems with tests 19 & 20: */
/* IF (wantz .and. INDEIG ) Z( 1,1) = Z(1,1) / 1.002 + .002 */
work[1] = (real) lwmin;
iwork[1] = liwmin;
return 0;
/* End of SSTEVR */
} /* sstevr_ */
/* Subroutine */ int ssyevx_(char *jobz, char *range, char *uplo, integer *n,
real *a, integer *lda, real *vl, real *vu, integer *il, integer *iu,
real *abstol, integer *m, real *w, real *z__, integer *ldz, real *
work, integer *lwork, integer *iwork, integer *ifail, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, z_dim1, z_offset, i__1, i__2;
real r__1, r__2;
/* Local variables */
integer i__, j, nb, jj;
real eps, vll, vuu, tmp1;
integer indd, inde;
real anrm;
integer imax;
real rmin, rmax;
logical test;
integer itmp1, indee;
real sigma;
integer iinfo;
char order[1];
logical lower;
logical wantz, alleig, indeig;
integer iscale, indibl;
logical valeig;
real safmin;
real abstll, bignum;
integer indtau, indisp, indiwo, indwkn;
integer indwrk, lwkmin;
integer llwrkn, llwork, nsplit;
real smlnum;
integer lwkopt;
logical lquery;
/* -- LAPACK driver routine (version 3.2) -- */
/* November 2006 */
/* Purpose */
/* ======= */
/* SSYEVX computes selected eigenvalues and, optionally, eigenvectors */
/* of a real symmetric matrix A. Eigenvalues and eigenvectors can be */
/* selected by specifying either a range of values or a range of indices */
/* for the desired eigenvalues. */
/* 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. */
/* UPLO (input) CHARACTER*1 */
/* = 'U': Upper triangle of A is stored; */
/* = 'L': Lower triangle of A is stored. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* A (input/output) REAL array, dimension (LDA, N) */
/* On entry, the symmetric matrix A. If UPLO = 'U', the */
/* leading N-by-N upper triangular part of A contains the */
/* upper triangular part of the matrix A. If UPLO = 'L', */
/* the leading N-by-N lower triangular part of A contains */
/* the lower triangular part of the matrix A. */
/* On exit, the lower triangle (if UPLO='L') or the upper */
/* triangle (if UPLO='U') of A, including the diagonal, is */
/* destroyed. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,N). */
/* 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. */
/* An approximate eigenvalue is accepted as converged */
/* when it is determined to lie in an interval [a,b] */
/* of width less than or equal to */
/* ABSTOL + EPS * max( |a|,|b| ) , */
/* where EPS is the machine precision. If ABSTOL is less than */
/* or equal to zero, then EPS*|T| will be used in its place, */
/* where |T| is the 1-norm of the tridiagonal matrix obtained */
/* by reducing A to tridiagonal form. */
/* Eigenvalues will be computed most accurately when ABSTOL is */
/* set to twice the underflow threshold 2*SLAMCH('S'), not zero. */
/* If this routine returns with INFO>0, indicating that some */
/* eigenvectors did not converge, try setting ABSTOL to */
/* 2*SLAMCH('S'). */
/* See "Computing Small Singular Values of Bidiagonal Matrices */
/* with Guaranteed High Relative Accuracy," by Demmel and */
/* Kahan, LAPACK Working Note #3. */
/* 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) */
/* On normal exit, 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 A */
/* corresponding to the selected eigenvalues, with the i-th */
/* column of Z holding the eigenvector associated with W(i). */
/* If an eigenvector fails to converge, then that column of Z */
/* contains the latest approximation to the eigenvector, and the */
/* index of the eigenvector is returned in IFAIL. */
/* 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). */
/* WORK (workspace/output) REAL array, dimension (MAX(1,LWORK)) */
/* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. */
/* LWORK (input) INTEGER */
/* The length of the array WORK. LWORK >= 1, when N <= 1; */
/* otherwise 8*N. */
/* For optimal efficiency, LWORK >= (NB+3)*N, */
/* where NB is the max of the blocksize for SSYTRD and SORMTR */
/* returned by ILAENV. */
/* 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) INTEGER array, dimension (5*N) */
/* IFAIL (output) INTEGER array, dimension (N) */
/* If JOBZ = 'V', then if INFO = 0, the first M elements of */
/* IFAIL are zero. If INFO > 0, then IFAIL contains the */
/* indices of the eigenvectors that failed to converge. */
/* If JOBZ = 'N', then IFAIL is not referenced. */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* > 0: if INFO = i, then i eigenvectors failed to converge. */
/* Their indices are stored in array IFAIL. */
/* ===================================================================== */
/* Test the input parameters. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--w;
z_dim1 = *ldz;
z_offset = 1 + z_dim1;
z__ -= z_offset;
--work;
--iwork;
--ifail;
/* Function Body */
lower = lsame_(uplo, "L");
wantz = lsame_(jobz, "V");
alleig = lsame_(range, "A");
valeig = lsame_(range, "V");
indeig = lsame_(range, "I");
lquery = *lwork == -1;
*info = 0;
if (! (wantz || lsame_(jobz, "N"))) {
*info = -1;
} else if (! (alleig || valeig || indeig)) {
*info = -2;
} else if (! (lower || lsame_(uplo, "U"))) {
*info = -3;
} else if (*n < 0) {
*info = -4;
} else if (*lda < max(1,*n)) {
*info = -6;
} else {
if (valeig) {
if (*n > 0 && *vu <= *vl) {
*info = -8;
}
} else if (indeig) {
if (*il < 1 || *il > max(1,*n)) {
*info = -9;
} else if (*iu < min(*n,*il) || *iu > *n) {
*info = -10;
}
}
}
if (*info == 0) {
if (*ldz < 1 || wantz && *ldz < *n) {
*info = -15;
}
}
if (*info == 0) {
if (*n <= 1) {
lwkmin = 1;
work[1] = (real) lwkmin;
} else {
lwkmin = *n << 3;
nb = ilaenv_(&c__1, "SSYTRD", uplo, n, &c_n1, &c_n1, &c_n1);
/* Computing MAX */
i__1 = nb, i__2 = ilaenv_(&c__1, "SORMTR", uplo, n, &c_n1, &c_n1,
&c_n1);
nb = max(i__1,i__2);
/* Computing MAX */
i__1 = lwkmin, i__2 = (nb + 3) * *n;
lwkopt = max(i__1,i__2);
work[1] = (real) lwkopt;
}
if (*lwork < lwkmin && ! lquery) {
*info = -17;
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SSYEVX", &i__1);
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] = a[a_dim1 + 1];
} else {
if (*vl < a[a_dim1 + 1] && *vu >= a[a_dim1 + 1]) {
*m = 1;
w[1] = a[a_dim1 + 1];
}
}
if (wantz) {
z__[z_dim1 + 1] = 1.f;
}
return 0;
}
/* 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);
/* Scale matrix to allowable range, if necessary. */
iscale = 0;
abstll = *abstol;
if (valeig) {
vll = *vl;
vuu = *vu;
}
anrm = slansy_("M", uplo, n, &a[a_offset], lda, &work[1]);
if (anrm > 0.f && anrm < rmin) {
iscale = 1;
sigma = rmin / anrm;
} else if (anrm > rmax) {
iscale = 1;
sigma = rmax / anrm;
}
if (iscale == 1) {
if (lower) {
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
i__2 = *n - j + 1;
sscal_(&i__2, &sigma, &a[j + j * a_dim1], &c__1);
}
} else {
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
sscal_(&j, &sigma, &a[j * a_dim1 + 1], &c__1);
}
}
if (*abstol > 0.f) {
abstll = *abstol * sigma;
}
if (valeig) {
vll = *vl * sigma;
vuu = *vu * sigma;
}
}
/* Call SSYTRD to reduce symmetric matrix to tridiagonal form. */
indtau = 1;
inde = indtau + *n;
indd = inde + *n;
indwrk = indd + *n;
llwork = *lwork - indwrk + 1;
ssytrd_(uplo, n, &a[a_offset], lda, &work[indd], &work[inde], &work[
indtau], &work[indwrk], &llwork, &iinfo);
/* If all eigenvalues are desired and ABSTOL is less than or equal to */
/* zero, then call SSTERF or SORGTR and SSTEQR. If this fails for */
/* some eigenvalue, then try SSTEBZ. */
test = FALSE_;
if (indeig) {
if (*il == 1 && *iu == *n) {
test = TRUE_;
}
}
if ((alleig || test) && *abstol <= 0.f) {
scopy_(n, &work[indd], &c__1, &w[1], &c__1);
indee = indwrk + (*n << 1);
if (! wantz) {
i__1 = *n - 1;
scopy_(&i__1, &work[inde], &c__1, &work[indee], &c__1);
ssterf_(n, &w[1], &work[indee], info);
} else {
slacpy_("A", n, n, &a[a_offset], lda, &z__[z_offset], ldz);
sorgtr_(uplo, n, &z__[z_offset], ldz, &work[indtau], &work[indwrk]
, &llwork, &iinfo);
i__1 = *n - 1;
scopy_(&i__1, &work[inde], &c__1, &work[indee], &c__1);
ssteqr_(jobz, n, &w[1], &work[indee], &z__[z_offset], ldz, &work[
indwrk], info);
if (*info == 0) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
ifail[i__] = 0;
}
}
}
if (*info == 0) {
*m = *n;
goto L40;
}
*info = 0;
}
/* Otherwise, call SSTEBZ and, if eigenvectors are desired, SSTEIN. */
if (wantz) {
*(unsigned char *)order = 'B';
} else {
*(unsigned char *)order = 'E';
}
indibl = 1;
indisp = indibl + *n;
indiwo = indisp + *n;
sstebz_(range, order, n, &vll, &vuu, il, iu, &abstll, &work[indd], &work[
inde], m, &nsplit, &w[1], &iwork[indibl], &iwork[indisp], &work[
indwrk], &iwork[indiwo], info);
if (wantz) {
sstein_(n, &work[indd], &work[inde], m, &w[1], &iwork[indibl], &iwork[
indisp], &z__[z_offset], ldz, &work[indwrk], &iwork[indiwo], &
ifail[1], info);
/* Apply orthogonal matrix used in reduction to tridiagonal */
/* form to eigenvectors returned by SSTEIN. */
indwkn = inde;
llwrkn = *lwork - indwkn + 1;
sormtr_("L", uplo, "N", n, m, &a[a_offset], lda, &work[indtau], &z__[
z_offset], ldz, &work[indwkn], &llwrkn, &iinfo);
}
/* If matrix was scaled, then rescale eigenvalues appropriately. */
L40:
if (iscale == 1) {
if (*info == 0) {
imax = *m;
} else {
imax = *info - 1;
}
r__1 = 1.f / sigma;
sscal_(&imax, &r__1, &w[1], &c__1);
}
/* If eigenvalues are not in order, then sort them, along with */
/* eigenvectors. */
if (wantz) {
i__1 = *m - 1;
for (j = 1; j <= i__1; ++j) {
i__ = 0;
tmp1 = w[j];
i__2 = *m;
for (jj = j + 1; jj <= i__2; ++jj) {
if (w[jj] < tmp1) {
i__ = jj;
tmp1 = w[jj];
}
}
if (i__ != 0) {
itmp1 = iwork[indibl + i__ - 1];
w[i__] = w[j];
iwork[indibl + i__ - 1] = iwork[indibl + j - 1];
w[j] = tmp1;
iwork[indibl + j - 1] = itmp1;
sswap_(n, &z__[i__ * z_dim1 + 1], &c__1, &z__[j * z_dim1 + 1],
&c__1);
if (*info != 0) {
itmp1 = ifail[i__];
ifail[i__] = ifail[j];
ifail[j] = itmp1;
}
}
}
}
/* Set WORK(1) to optimal workspace size. */
work[1] = (real) lwkopt;
return 0;
/* End of SSYEVX */
} /* ssyevx_ */
/* Subroutine */ int chpevx_(char *jobz, char *range, char *uplo, integer *n,
complex *ap, real *vl, real *vu, integer *il, integer *iu, real *
abstol, integer *m, real *w, complex *z__, integer *ldz, complex *
work, real *rwork, integer *iwork, integer *ifail, integer *info)
{
/* -- LAPACK driver routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
June 30, 1999
Purpose
=======
CHPEVX computes selected eigenvalues and, optionally, eigenvectors
of a complex Hermitian matrix A in packed storage.
Eigenvalues/vectors can be selected by specifying either a range of
values or a range of indices for the desired eigenvalues.
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.
UPLO (input) CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N (input) INTEGER
The order of the matrix A. N >= 0.
AP (input/output) COMPLEX array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the Hermitian matrix
A, packed columnwise in a linear array. The j-th column of A
is stored in the array AP as follows:
if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
On exit, AP is overwritten by values generated during the
reduction to tridiagonal form. If UPLO = 'U', the diagonal
and first superdiagonal of the tridiagonal matrix T overwrite
the corresponding elements of A, and if UPLO = 'L', the
diagonal and first subdiagonal of T overwrite the
corresponding elements of A.
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.
An approximate eigenvalue is accepted as converged
when it is determined to lie in an interval [a,b]
of width less than or equal to
ABSTOL + EPS * max( |a|,|b| ) ,
where EPS is the machine precision. If ABSTOL is less than
or equal to zero, then EPS*|T| will be used in its place,
where |T| is the 1-norm of the tridiagonal matrix obtained
by reducing AP to tridiagonal form.
Eigenvalues will be computed most accurately when ABSTOL is
set to twice the underflow threshold 2*SLAMCH('S'), not zero.
If this routine returns with INFO>0, indicating that some
eigenvectors did not converge, try setting ABSTOL to
2*SLAMCH('S').
See "Computing Small Singular Values of Bidiagonal Matrices
with Guaranteed High Relative Accuracy," by Demmel and
Kahan, LAPACK Working Note #3.
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)
If INFO = 0, the selected eigenvalues in ascending order.
Z (output) COMPLEX 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 A
corresponding to the selected eigenvalues, with the i-th
column of Z holding the eigenvector associated with W(i).
If an eigenvector fails to converge, then that column of Z
contains the latest approximation to the eigenvector, and
the index of the eigenvector is returned in IFAIL.
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).
WORK (workspace) COMPLEX array, dimension (2*N)
RWORK (workspace) REAL array, dimension (7*N)
IWORK (workspace) INTEGER array, dimension (5*N)
IFAIL (output) INTEGER array, dimension (N)
If JOBZ = 'V', then if INFO = 0, the first M elements of
IFAIL are zero. If INFO > 0, then IFAIL contains the
indices of the eigenvectors that failed to converge.
If JOBZ = 'N', then IFAIL is not referenced.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, then i eigenvectors failed to converge.
Their indices are stored in array IFAIL.
=====================================================================
Test the input parameters.
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
/* 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 indd, inde;
static real anrm;
static integer imax;
static real rmin, rmax;
static integer itmp1, i__, j, indee;
static real sigma;
extern logical lsame_(char *, char *);
static integer iinfo;
extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *);
static char order[1];
extern /* Subroutine */ int cswap_(integer *, complex *, integer *,
complex *, integer *), scopy_(integer *, real *, integer *, real *
, integer *);
static logical wantz;
static integer jj;
static logical alleig, indeig;
static integer iscale, indibl;
extern doublereal clanhp_(char *, char *, integer *, complex *, real *);
static logical valeig;
extern doublereal slamch_(char *);
extern /* Subroutine */ int csscal_(integer *, real *, complex *, integer
*);
static real safmin;
extern /* Subroutine */ int xerbla_(char *, integer *);
static real abstll, bignum;
static integer indiwk, indisp, indtau;
extern /* Subroutine */ int chptrd_(char *, integer *, complex *, real *,
real *, complex *, integer *), cstein_(integer *, real *,
real *, integer *, real *, integer *, integer *, complex *,
integer *, real *, integer *, integer *, integer *);
static integer indrwk, indwrk;
extern /* Subroutine */ int csteqr_(char *, integer *, real *, real *,
complex *, integer *, real *, integer *), cupgtr_(char *,
integer *, complex *, complex *, complex *, integer *, complex *,
integer *), ssterf_(integer *, real *, real *, integer *);
static integer nsplit;
extern /* Subroutine */ int cupmtr_(char *, char *, char *, integer *,
integer *, complex *, complex *, complex *, integer *, complex *,
integer *);
static real smlnum;
extern /* Subroutine */ int sstebz_(char *, char *, integer *, real *,
real *, integer *, integer *, real *, real *, real *, integer *,
integer *, real *, integer *, integer *, real *, integer *,
integer *);
static real eps, vll, vuu, tmp1;
#define z___subscr(a_1,a_2) (a_2)*z_dim1 + a_1
#define z___ref(a_1,a_2) z__[z___subscr(a_1,a_2)]
--ap;
--w;
z_dim1 = *ldz;
z_offset = 1 + z_dim1 * 1;
z__ -= z_offset;
--work;
--rwork;
--iwork;
--ifail;
/* Function Body */
wantz = lsame_(jobz, "V");
alleig = lsame_(range, "A");
valeig = lsame_(range, "V");
indeig = lsame_(range, "I");
*info = 0;
if (! (wantz || lsame_(jobz, "N"))) {
*info = -1;
} else if (! (alleig || valeig || indeig)) {
*info = -2;
} else if (! (lsame_(uplo, "L") || lsame_(uplo,
"U"))) {
*info = -3;
} else if (*n < 0) {
*info = -4;
} else {
if (valeig) {
if (*n > 0 && *vu <= *vl) {
*info = -7;
}
} else if (indeig) {
if (*il < 1 || *il > max(1,*n)) {
*info = -8;
} else if (*iu < min(*n,*il) || *iu > *n) {
*info = -9;
}
}
}
if (*info == 0) {
if (*ldz < 1 || wantz && *ldz < *n) {
*info = -14;
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CHPEVX", &i__1);
return 0;
}
/* Quick return if possible */
*m = 0;
if (*n == 0) {
return 0;
}
if (*n == 1) {
if (alleig || indeig) {
*m = 1;
w[1] = ap[1].r;
} else {
if (*vl < ap[1].r && *vu >= ap[1].r) {
*m = 1;
w[1] = ap[1].r;
}
}
if (wantz) {
i__1 = z___subscr(1, 1);
z__[i__1].r = 1.f, z__[i__1].i = 0.f;
}
return 0;
}
/* 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);
/* Scale matrix to allowable range, if necessary. */
iscale = 0;
abstll = *abstol;
if (valeig) {
vll = *vl;
vuu = *vu;
} else {
vll = 0.f;
vuu = 0.f;
}
anrm = clanhp_("M", uplo, n, &ap[1], &rwork[1]);
if (anrm > 0.f && anrm < rmin) {
iscale = 1;
sigma = rmin / anrm;
} else if (anrm > rmax) {
iscale = 1;
sigma = rmax / anrm;
}
if (iscale == 1) {
i__1 = *n * (*n + 1) / 2;
csscal_(&i__1, &sigma, &ap[1], &c__1);
if (*abstol > 0.f) {
abstll = *abstol * sigma;
}
if (valeig) {
vll = *vl * sigma;
vuu = *vu * sigma;
}
}
/* Call CHPTRD to reduce Hermitian packed matrix to tridiagonal form. */
indd = 1;
inde = indd + *n;
indrwk = inde + *n;
indtau = 1;
indwrk = indtau + *n;
chptrd_(uplo, n, &ap[1], &rwork[indd], &rwork[inde], &work[indtau], &
iinfo);
/* If all eigenvalues are desired and ABSTOL is less than or equal
to zero, then call SSTERF or CUPGTR and CSTEQR. If this fails
for some eigenvalue, then try SSTEBZ. */
if ((alleig || indeig && *il == 1 && *iu == *n) && *abstol <= 0.f) {
scopy_(n, &rwork[indd], &c__1, &w[1], &c__1);
indee = indrwk + (*n << 1);
if (! wantz) {
i__1 = *n - 1;
scopy_(&i__1, &rwork[inde], &c__1, &rwork[indee], &c__1);
ssterf_(n, &w[1], &rwork[indee], info);
} else {
cupgtr_(uplo, n, &ap[1], &work[indtau], &z__[z_offset], ldz, &
work[indwrk], &iinfo);
i__1 = *n - 1;
scopy_(&i__1, &rwork[inde], &c__1, &rwork[indee], &c__1);
csteqr_(jobz, n, &w[1], &rwork[indee], &z__[z_offset], ldz, &
rwork[indrwk], info);
if (*info == 0) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
ifail[i__] = 0;
/* L10: */
}
}
}
if (*info == 0) {
*m = *n;
goto L20;
}
*info = 0;
}
/* Otherwise, call SSTEBZ and, if eigenvectors are desired, CSTEIN. */
if (wantz) {
*(unsigned char *)order = 'B';
} else {
*(unsigned char *)order = 'E';
}
indibl = 1;
indisp = indibl + *n;
indiwk = indisp + *n;
sstebz_(range, order, n, &vll, &vuu, il, iu, &abstll, &rwork[indd], &
rwork[inde], m, &nsplit, &w[1], &iwork[indibl], &iwork[indisp], &
rwork[indrwk], &iwork[indiwk], info);
if (wantz) {
cstein_(n, &rwork[indd], &rwork[inde], m, &w[1], &iwork[indibl], &
iwork[indisp], &z__[z_offset], ldz, &rwork[indrwk], &iwork[
indiwk], &ifail[1], info);
/* Apply unitary matrix used in reduction to tridiagonal
form to eigenvectors returned by CSTEIN. */
indwrk = indtau + *n;
cupmtr_("L", uplo, "N", n, m, &ap[1], &work[indtau], &z__[z_offset],
ldz, &work[indwrk], info);
}
/* If matrix was scaled, then rescale eigenvalues appropriately. */
L20:
if (iscale == 1) {
if (*info == 0) {
imax = *m;
} else {
imax = *info - 1;
}
r__1 = 1.f / sigma;
sscal_(&imax, &r__1, &w[1], &c__1);
}
/* If eigenvalues are not in order, then sort them, along with
eigenvectors. */
if (wantz) {
i__1 = *m - 1;
for (j = 1; j <= i__1; ++j) {
i__ = 0;
tmp1 = w[j];
i__2 = *m;
for (jj = j + 1; jj <= i__2; ++jj) {
if (w[jj] < tmp1) {
i__ = jj;
tmp1 = w[jj];
}
/* L30: */
}
if (i__ != 0) {
itmp1 = iwork[indibl + i__ - 1];
w[i__] = w[j];
iwork[indibl + i__ - 1] = iwork[indibl + j - 1];
w[j] = tmp1;
iwork[indibl + j - 1] = itmp1;
cswap_(n, &z___ref(1, i__), &c__1, &z___ref(1, j), &c__1);
if (*info != 0) {
itmp1 = ifail[i__];
ifail[i__] = ifail[j];
ifail[j] = itmp1;
}
}
/* L40: */
}
}
return 0;
/* End of CHPEVX */
} /* chpevx_ */
/* Subroutine */
int chbgvx_(char *jobz, char *range, char *uplo, integer *n, integer *ka, integer *kb, complex *ab, integer *ldab, complex *bb, integer *ldbb, complex *q, integer *ldq, real *vl, real *vu, integer * il, integer *iu, real *abstol, integer *m, real *w, complex *z__, integer *ldz, complex *work, real *rwork, integer *iwork, integer * ifail, integer *info)
{
/* System generated locals */
integer ab_dim1, ab_offset, bb_dim1, bb_offset, q_dim1, q_offset, z_dim1, z_offset, i__1, i__2;
/* Local variables */
integer i__, j, jj;
real tmp1;
integer indd, inde;
char vect[1];
logical test;
integer itmp1, indee;
extern logical lsame_(char *, char *);
extern /* Subroutine */
int cgemv_(char *, integer *, integer *, complex * , complex *, integer *, complex *, integer *, complex *, complex * , integer *);
integer iinfo;
char order[1];
extern /* Subroutine */
int ccopy_(integer *, complex *, integer *, complex *, integer *), cswap_(integer *, complex *, integer *, complex *, integer *);
logical upper;
extern /* Subroutine */
int scopy_(integer *, real *, integer *, real *, integer *);
logical wantz, alleig, indeig;
integer indibl;
extern /* Subroutine */
int chbtrd_(char *, char *, integer *, integer *, complex *, integer *, real *, real *, complex *, integer *, complex *, integer *);
logical valeig;
extern /* Subroutine */
int chbgst_(char *, char *, integer *, integer *, integer *, complex *, integer *, complex *, integer *, complex *, integer *, complex *, real *, integer *), clacpy_( char *, integer *, integer *, complex *, integer *, complex *, integer *), xerbla_(char *, integer *), cpbstf_( char *, integer *, integer *, complex *, integer *, integer *);
integer indiwk, indisp;
extern /* Subroutine */
int cstein_(integer *, real *, real *, integer *, real *, integer *, integer *, complex *, integer *, real *, integer *, integer *, integer *);
integer indrwk, indwrk;
extern /* Subroutine */
int csteqr_(char *, integer *, real *, real *, complex *, integer *, real *, integer *), ssterf_(integer *, real *, real *, integer *);
integer nsplit;
extern /* Subroutine */
int sstebz_(char *, char *, integer *, real *, real *, integer *, integer *, real *, real *, real *, integer *, integer *, real *, integer *, integer *, real *, integer *, integer *);
/* -- LAPACK driver routine (version 3.4.0) -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
/* November 2011 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
ab_dim1 = *ldab;
ab_offset = 1 + ab_dim1;
ab -= ab_offset;
bb_dim1 = *ldbb;
bb_offset = 1 + bb_dim1;
bb -= bb_offset;
q_dim1 = *ldq;
q_offset = 1 + q_dim1;
q -= q_offset;
--w;
z_dim1 = *ldz;
z_offset = 1 + z_dim1;
z__ -= z_offset;
--work;
--rwork;
--iwork;
--ifail;
/* Function Body */
wantz = lsame_(jobz, "V");
upper = lsame_(uplo, "U");
alleig = lsame_(range, "A");
valeig = lsame_(range, "V");
indeig = lsame_(range, "I");
*info = 0;
if (! (wantz || lsame_(jobz, "N")))
{
*info = -1;
}
else if (! (alleig || valeig || indeig))
{
*info = -2;
}
else if (! (upper || lsame_(uplo, "L")))
{
*info = -3;
}
else if (*n < 0)
{
*info = -4;
}
else if (*ka < 0)
{
*info = -5;
}
else if (*kb < 0 || *kb > *ka)
{
*info = -6;
}
else if (*ldab < *ka + 1)
{
*info = -8;
}
else if (*ldbb < *kb + 1)
{
*info = -10;
}
else if (*ldq < 1 || wantz && *ldq < *n)
{
*info = -12;
}
else
{
if (valeig)
{
if (*n > 0 && *vu <= *vl)
{
*info = -14;
}
}
else if (indeig)
{
if (*il < 1 || *il > max(1,*n))
{
*info = -15;
}
else if (*iu < min(*n,*il) || *iu > *n)
{
*info = -16;
}
}
}
if (*info == 0)
{
if (*ldz < 1 || wantz && *ldz < *n)
{
*info = -21;
}
}
if (*info != 0)
{
i__1 = -(*info);
xerbla_("CHBGVX", &i__1);
return 0;
}
/* Quick return if possible */
*m = 0;
if (*n == 0)
{
return 0;
}
/* Form a split Cholesky factorization of B. */
cpbstf_(uplo, n, kb, &bb[bb_offset], ldbb, info);
if (*info != 0)
{
*info = *n + *info;
return 0;
}
/* Transform problem to standard eigenvalue problem. */
chbgst_(jobz, uplo, n, ka, kb, &ab[ab_offset], ldab, &bb[bb_offset], ldbb, &q[q_offset], ldq, &work[1], &rwork[1], &iinfo);
/* Solve the standard eigenvalue problem. */
/* Reduce Hermitian band matrix to tridiagonal form. */
indd = 1;
inde = indd + *n;
indrwk = inde + *n;
indwrk = 1;
if (wantz)
{
*(unsigned char *)vect = 'U';
}
else
{
*(unsigned char *)vect = 'N';
}
chbtrd_(vect, uplo, n, ka, &ab[ab_offset], ldab, &rwork[indd], &rwork[ inde], &q[q_offset], ldq, &work[indwrk], &iinfo);
/* If all eigenvalues are desired and ABSTOL is less than or equal */
/* to zero, then call SSTERF or CSTEQR. If this fails for some */
/* eigenvalue, then try SSTEBZ. */
test = FALSE_;
if (indeig)
{
if (*il == 1 && *iu == *n)
{
test = TRUE_;
}
}
if ((alleig || test) && *abstol <= 0.f)
{
scopy_(n, &rwork[indd], &c__1, &w[1], &c__1);
indee = indrwk + (*n << 1);
i__1 = *n - 1;
scopy_(&i__1, &rwork[inde], &c__1, &rwork[indee], &c__1);
if (! wantz)
{
ssterf_(n, &w[1], &rwork[indee], info);
}
else
{
clacpy_("A", n, n, &q[q_offset], ldq, &z__[z_offset], ldz);
csteqr_(jobz, n, &w[1], &rwork[indee], &z__[z_offset], ldz, & rwork[indrwk], info);
if (*info == 0)
{
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
ifail[i__] = 0;
/* L10: */
}
}
}
if (*info == 0)
{
*m = *n;
goto L30;
}
*info = 0;
}
/* Otherwise, call SSTEBZ and, if eigenvectors are desired, */
/* call CSTEIN. */
if (wantz)
{
*(unsigned char *)order = 'B';
}
else
{
*(unsigned char *)order = 'E';
}
indibl = 1;
indisp = indibl + *n;
indiwk = indisp + *n;
sstebz_(range, order, n, vl, vu, il, iu, abstol, &rwork[indd], &rwork[ inde], m, &nsplit, &w[1], &iwork[indibl], &iwork[indisp], &rwork[ indrwk], &iwork[indiwk], info);
if (wantz)
{
cstein_(n, &rwork[indd], &rwork[inde], m, &w[1], &iwork[indibl], & iwork[indisp], &z__[z_offset], ldz, &rwork[indrwk], &iwork[ indiwk], &ifail[1], info);
/* Apply unitary matrix used in reduction to tridiagonal */
/* form to eigenvectors returned by CSTEIN. */
i__1 = *m;
for (j = 1;
j <= i__1;
++j)
{
ccopy_(n, &z__[j * z_dim1 + 1], &c__1, &work[1], &c__1);
cgemv_("N", n, n, &c_b2, &q[q_offset], ldq, &work[1], &c__1, & c_b1, &z__[j * z_dim1 + 1], &c__1);
/* L20: */
}
}
L30: /* If eigenvalues are not in order, then sort them, along with */
/* eigenvectors. */
if (wantz)
{
i__1 = *m - 1;
for (j = 1;
j <= i__1;
++j)
{
i__ = 0;
tmp1 = w[j];
i__2 = *m;
for (jj = j + 1;
jj <= i__2;
++jj)
{
if (w[jj] < tmp1)
{
i__ = jj;
tmp1 = w[jj];
}
/* L40: */
}
if (i__ != 0)
{
itmp1 = iwork[indibl + i__ - 1];
w[i__] = w[j];
iwork[indibl + i__ - 1] = iwork[indibl + j - 1];
w[j] = tmp1;
iwork[indibl + j - 1] = itmp1;
cswap_(n, &z__[i__ * z_dim1 + 1], &c__1, &z__[j * z_dim1 + 1], &c__1);
if (*info != 0)
{
itmp1 = ifail[i__];
ifail[i__] = ifail[j];
ifail[j] = itmp1;
}
}
/* L50: */
}
}
return 0;
/* End of CHBGVX */
}
/* Subroutine */ int chbgvx_(char *jobz, char *range, char *uplo, integer *n,
integer *ka, integer *kb, complex *ab, integer *ldab, complex *bb,
integer *ldbb, complex *q, integer *ldq, real *vl, real *vu, integer *
il, integer *iu, real *abstol, integer *m, real *w, complex *z__,
integer *ldz, complex *work, real *rwork, integer *iwork, integer *
ifail, integer *info)
{
/* System generated locals */
integer ab_dim1, ab_offset, bb_dim1, bb_offset, q_dim1, q_offset, z_dim1,
z_offset, i__1, i__2;
/* Local variables */
integer i__, j, jj;
real tmp1;
integer indd, inde;
char vect[1];
logical test;
integer itmp1, indee;
extern logical lsame_(char *, char *);
extern /* Subroutine */ int cgemv_(char *, integer *, integer *, complex *
, complex *, integer *, complex *, integer *, complex *, complex *
, integer *);
integer iinfo;
char order[1];
extern /* Subroutine */ int ccopy_(integer *, complex *, integer *,
complex *, integer *), cswap_(integer *, complex *, integer *,
complex *, integer *);
logical upper;
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *);
logical wantz, alleig, indeig;
integer indibl;
extern /* Subroutine */ int chbtrd_(char *, char *, integer *, integer *,
complex *, integer *, real *, real *, complex *, integer *,
complex *, integer *);
logical valeig;
extern /* Subroutine */ int chbgst_(char *, char *, integer *, integer *,
integer *, complex *, integer *, complex *, integer *, complex *,
integer *, complex *, real *, integer *), clacpy_(
char *, integer *, integer *, complex *, integer *, complex *,
integer *), xerbla_(char *, integer *), cpbstf_(
char *, integer *, integer *, complex *, integer *, integer *);
integer indiwk, indisp;
extern /* Subroutine */ int cstein_(integer *, real *, real *, integer *,
real *, integer *, integer *, complex *, integer *, real *,
integer *, integer *, integer *);
integer indrwk, indwrk;
extern /* Subroutine */ int csteqr_(char *, integer *, real *, real *,
complex *, integer *, real *, integer *), ssterf_(integer
*, real *, real *, integer *);
integer nsplit;
extern /* Subroutine */ int sstebz_(char *, char *, integer *, real *,
real *, integer *, integer *, real *, real *, real *, integer *,
integer *, real *, integer *, integer *, real *, integer *,
integer *);
/* -- LAPACK driver routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CHBGVX computes all the eigenvalues, and optionally, the eigenvectors */
/* of a complex generalized Hermitian-definite banded eigenproblem, of */
/* the form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian */
/* and banded, and B is also positive definite. Eigenvalues and */
/* eigenvectors can be selected by specifying either all eigenvalues, */
/* a range of values or a range of indices for the desired eigenvalues. */
/* 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. */
/* UPLO (input) CHARACTER*1 */
/* = 'U': Upper triangles of A and B are stored; */
/* = 'L': Lower triangles of A and B are stored. */
/* N (input) INTEGER */
/* The order of the matrices A and B. N >= 0. */
/* KA (input) INTEGER */
/* The number of superdiagonals of the matrix A if UPLO = 'U', */
/* or the number of subdiagonals if UPLO = 'L'. KA >= 0. */
/* KB (input) INTEGER */
/* The number of superdiagonals of the matrix B if UPLO = 'U', */
/* or the number of subdiagonals if UPLO = 'L'. KB >= 0. */
/* AB (input/output) COMPLEX array, dimension (LDAB, N) */
/* On entry, the upper or lower triangle of the Hermitian band */
/* matrix A, stored in the first ka+1 rows of the array. The */
/* j-th column of A is stored in the j-th column of the array AB */
/* as follows: */
/* if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; */
/* if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). */
/* On exit, the contents of AB are destroyed. */
/* LDAB (input) INTEGER */
/* The leading dimension of the array AB. LDAB >= KA+1. */
/* BB (input/output) COMPLEX array, dimension (LDBB, N) */
/* On entry, the upper or lower triangle of the Hermitian band */
/* matrix B, stored in the first kb+1 rows of the array. The */
/* j-th column of B is stored in the j-th column of the array BB */
/* as follows: */
/* if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j; */
/* if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb). */
/* On exit, the factor S from the split Cholesky factorization */
/* B = S**H*S, as returned by CPBSTF. */
/* LDBB (input) INTEGER */
/* The leading dimension of the array BB. LDBB >= KB+1. */
/* Q (output) COMPLEX array, dimension (LDQ, N) */
/* If JOBZ = 'V', the n-by-n matrix used in the reduction of */
/* A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x, */
/* and consequently C to tridiagonal form. */
/* If JOBZ = 'N', the array Q is not referenced. */
/* LDQ (input) INTEGER */
/* The leading dimension of the array Q. If JOBZ = 'N', */
/* LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N). */
/* 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. */
/* An approximate eigenvalue is accepted as converged */
/* when it is determined to lie in an interval [a,b] */
/* of width less than or equal to */
/* ABSTOL + EPS * max( |a|,|b| ) , */
/* where EPS is the machine precision. If ABSTOL is less than */
/* or equal to zero, then EPS*|T| will be used in its place, */
/* where |T| is the 1-norm of the tridiagonal matrix obtained */
/* by reducing AP to tridiagonal form. */
/* Eigenvalues will be computed most accurately when ABSTOL is */
/* set to twice the underflow threshold 2*SLAMCH('S'), not zero. */
/* If this routine returns with INFO>0, indicating that some */
/* eigenvectors did not converge, try setting ABSTOL to */
/* 2*SLAMCH('S'). */
/* 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) */
/* If INFO = 0, the eigenvalues in ascending order. */
/* Z (output) COMPLEX array, dimension (LDZ, N) */
/* If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of */
/* eigenvectors, with the i-th column of Z holding the */
/* eigenvector associated with W(i). The eigenvectors are */
/* normalized so that Z**H*B*Z = I. */
/* If JOBZ = 'N', then Z is not referenced. */
/* LDZ (input) INTEGER */
/* The leading dimension of the array Z. LDZ >= 1, and if */
/* JOBZ = 'V', LDZ >= N. */
/* WORK (workspace) COMPLEX array, dimension (N) */
/* RWORK (workspace) REAL array, dimension (7*N) */
/* IWORK (workspace) INTEGER array, dimension (5*N) */
/* IFAIL (output) INTEGER array, dimension (N) */
/* If JOBZ = 'V', then if INFO = 0, the first M elements of */
/* IFAIL are zero. If INFO > 0, then IFAIL contains the */
/* indices of the eigenvectors that failed to converge. */
/* If JOBZ = 'N', then IFAIL is not referenced. */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* > 0: if INFO = i, and i is: */
/* <= N: then i eigenvectors failed to converge. Their */
/* indices are stored in array IFAIL. */
/* > N: if INFO = N + i, for 1 <= i <= N, then CPBSTF */
/* returned INFO = i: B is not positive definite. */
/* The factorization of B could not be completed and */
/* no eigenvalues or eigenvectors were computed. */
/* Further Details */
/* =============== */
/* Based on contributions by */
/* Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
ab_dim1 = *ldab;
ab_offset = 1 + ab_dim1;
ab -= ab_offset;
bb_dim1 = *ldbb;
bb_offset = 1 + bb_dim1;
bb -= bb_offset;
q_dim1 = *ldq;
q_offset = 1 + q_dim1;
q -= q_offset;
--w;
z_dim1 = *ldz;
z_offset = 1 + z_dim1;
z__ -= z_offset;
--work;
--rwork;
--iwork;
--ifail;
/* Function Body */
wantz = lsame_(jobz, "V");
upper = lsame_(uplo, "U");
alleig = lsame_(range, "A");
valeig = lsame_(range, "V");
indeig = lsame_(range, "I");
*info = 0;
if (! (wantz || lsame_(jobz, "N"))) {
*info = -1;
} else if (! (alleig || valeig || indeig)) {
*info = -2;
} else if (! (upper || lsame_(uplo, "L"))) {
*info = -3;
} else if (*n < 0) {
*info = -4;
} else if (*ka < 0) {
*info = -5;
} else if (*kb < 0 || *kb > *ka) {
*info = -6;
} else if (*ldab < *ka + 1) {
*info = -8;
} else if (*ldbb < *kb + 1) {
*info = -10;
} else if (*ldq < 1 || wantz && *ldq < *n) {
*info = -12;
} else {
if (valeig) {
if (*n > 0 && *vu <= *vl) {
*info = -14;
}
} else if (indeig) {
if (*il < 1 || *il > max(1,*n)) {
*info = -15;
} else if (*iu < min(*n,*il) || *iu > *n) {
*info = -16;
}
}
}
if (*info == 0) {
if (*ldz < 1 || wantz && *ldz < *n) {
*info = -21;
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CHBGVX", &i__1);
return 0;
}
/* Quick return if possible */
*m = 0;
if (*n == 0) {
return 0;
}
/* Form a split Cholesky factorization of B. */
cpbstf_(uplo, n, kb, &bb[bb_offset], ldbb, info);
if (*info != 0) {
*info = *n + *info;
return 0;
}
/* Transform problem to standard eigenvalue problem. */
chbgst_(jobz, uplo, n, ka, kb, &ab[ab_offset], ldab, &bb[bb_offset], ldbb,
&q[q_offset], ldq, &work[1], &rwork[1], &iinfo);
/* Solve the standard eigenvalue problem. */
/* Reduce Hermitian band matrix to tridiagonal form. */
indd = 1;
inde = indd + *n;
indrwk = inde + *n;
indwrk = 1;
if (wantz) {
*(unsigned char *)vect = 'U';
} else {
*(unsigned char *)vect = 'N';
}
chbtrd_(vect, uplo, n, ka, &ab[ab_offset], ldab, &rwork[indd], &rwork[
inde], &q[q_offset], ldq, &work[indwrk], &iinfo);
/* If all eigenvalues are desired and ABSTOL is less than or equal */
/* to zero, then call SSTERF or CSTEQR. If this fails for some */
/* eigenvalue, then try SSTEBZ. */
test = FALSE_;
if (indeig) {
if (*il == 1 && *iu == *n) {
test = TRUE_;
}
}
if ((alleig || test) && *abstol <= 0.f) {
scopy_(n, &rwork[indd], &c__1, &w[1], &c__1);
indee = indrwk + (*n << 1);
i__1 = *n - 1;
scopy_(&i__1, &rwork[inde], &c__1, &rwork[indee], &c__1);
if (! wantz) {
ssterf_(n, &w[1], &rwork[indee], info);
} else {
clacpy_("A", n, n, &q[q_offset], ldq, &z__[z_offset], ldz);
csteqr_(jobz, n, &w[1], &rwork[indee], &z__[z_offset], ldz, &
rwork[indrwk], info);
if (*info == 0) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
ifail[i__] = 0;
/* L10: */
}
}
}
if (*info == 0) {
*m = *n;
goto L30;
}
*info = 0;
}
/* Otherwise, call SSTEBZ and, if eigenvectors are desired, */
/* call CSTEIN. */
if (wantz) {
*(unsigned char *)order = 'B';
} else {
*(unsigned char *)order = 'E';
}
indibl = 1;
indisp = indibl + *n;
indiwk = indisp + *n;
sstebz_(range, order, n, vl, vu, il, iu, abstol, &rwork[indd], &rwork[
inde], m, &nsplit, &w[1], &iwork[indibl], &iwork[indisp], &rwork[
indrwk], &iwork[indiwk], info);
if (wantz) {
cstein_(n, &rwork[indd], &rwork[inde], m, &w[1], &iwork[indibl], &
iwork[indisp], &z__[z_offset], ldz, &rwork[indrwk], &iwork[
indiwk], &ifail[1], info);
/* Apply unitary matrix used in reduction to tridiagonal */
/* form to eigenvectors returned by CSTEIN. */
i__1 = *m;
for (j = 1; j <= i__1; ++j) {
ccopy_(n, &z__[j * z_dim1 + 1], &c__1, &work[1], &c__1);
cgemv_("N", n, n, &c_b2, &q[q_offset], ldq, &work[1], &c__1, &
c_b1, &z__[j * z_dim1 + 1], &c__1);
/* L20: */
}
}
L30:
/* If eigenvalues are not in order, then sort them, along with */
/* eigenvectors. */
if (wantz) {
i__1 = *m - 1;
for (j = 1; j <= i__1; ++j) {
i__ = 0;
tmp1 = w[j];
i__2 = *m;
for (jj = j + 1; jj <= i__2; ++jj) {
if (w[jj] < tmp1) {
i__ = jj;
tmp1 = w[jj];
}
/* L40: */
}
if (i__ != 0) {
itmp1 = iwork[indibl + i__ - 1];
w[i__] = w[j];
iwork[indibl + i__ - 1] = iwork[indibl + j - 1];
w[j] = tmp1;
iwork[indibl + j - 1] = itmp1;
cswap_(n, &z__[i__ * z_dim1 + 1], &c__1, &z__[j * z_dim1 + 1],
&c__1);
if (*info != 0) {
itmp1 = ifail[i__];
ifail[i__] = ifail[j];
ifail[j] = itmp1;
}
}
/* L50: */
}
}
return 0;
/* End of CHBGVX */
} /* chbgvx_ */
/* Subroutine */ int ssbevx_(char *jobz, char *range, char *uplo, integer *n,
integer *kd, real *ab, integer *ldab, real *q, integer *ldq, real *vl,
real *vu, integer *il, integer *iu, real *abstol, integer *m, real *
w, real *z, integer *ldz, real *work, integer *iwork, integer *ifail,
integer *info)
{
/* -- LAPACK driver 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
=======
SSBEVX computes selected eigenvalues and, optionally, eigenvectors
of a real symmetric band matrix A. Eigenvalues and eigenvectors can
be selected by specifying either a range of values or a range of
indices for the desired eigenvalues.
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.
UPLO (input) CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N (input) INTEGER
The order of the matrix A. N >= 0.
KD (input) INTEGER
The number of superdiagonals of the matrix A if UPLO = 'U',
or the number of subdiagonals if UPLO = 'L'. KD >= 0.
AB (input/output) REAL array, dimension (LDAB, N)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first KD+1 rows of the array. The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
On exit, AB is overwritten by values generated during the
reduction to tridiagonal form. If UPLO = 'U', the first
superdiagonal and the diagonal of the tridiagonal matrix T
are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
the diagonal and first subdiagonal of T are returned in the
first two rows of AB.
LDAB (input) INTEGER
The leading dimension of the array AB. LDAB >= KD + 1.
Q (output) REAL array, dimension (LDQ, N)
If JOBZ = 'V', the N-by-N orthogonal matrix used in the
reduction to tridiagonal form.
If JOBZ = 'N', the array Q is not referenced.
LDQ (input) INTEGER
The leading dimension of the array Q. If JOBZ = 'V', then
LDQ >= max(1,N).
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.
An approximate eigenvalue is accepted as converged
when it is determined to lie in an interval [a,b]
of width less than or equal to
ABSTOL + EPS * max( |a|,|b| ) ,
where EPS is the machine precision. If ABSTOL is less than
or equal to zero, then EPS*|T| will be used in its place,
where |T| is the 1-norm of the tridiagonal matrix obtained
by reducing AB to tridiagonal form.
Eigenvalues will be computed most accurately when ABSTOL is
set to twice the underflow threshold 2*SLAMCH('S'), not zero.
If this routine returns with INFO>0, indicating that some
eigenvectors did not converge, try setting ABSTOL to
2*SLAMCH('S').
See "Computing Small Singular Values of Bidiagonal Matrices
with Guaranteed High Relative Accuracy," by Demmel and
Kahan, LAPACK Working Note #3.
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 A
corresponding to the selected eigenvalues, with the i-th
column of Z holding the eigenvector associated with W(i).
If an eigenvector fails to converge, then that column of Z
contains the latest approximation to the eigenvector, and the
index of the eigenvector is returned in IFAIL.
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).
WORK (workspace) REAL array, dimension (7*N)
IWORK (workspace) INTEGER array, dimension (5*N)
IFAIL (output) INTEGER array, dimension (N)
If JOBZ = 'V', then if INFO = 0, the first M elements of
IFAIL are zero. If INFO > 0, then IFAIL contains the
indices of the eigenvectors that failed to converge.
If JOBZ = 'N', then IFAIL is not referenced.
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = i, then i eigenvectors failed to converge.
Their indices are stored in array IFAIL.
=====================================================================
Test the input parameters.
Parameter adjustments
Function Body */
/* Table of constant values */
static real c_b14 = 1.f;
static integer c__1 = 1;
static real c_b34 = 0.f;
/* System generated locals */
integer ab_dim1, ab_offset, q_dim1, q_offset, z_dim1, z_offset, i__1,
i__2;
real r__1, r__2;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
static integer indd, inde;
static real anrm;
static integer imax;
static real rmin, rmax;
static integer itmp1, i, j, indee;
static real sigma;
extern logical lsame_(char *, char *);
static integer iinfo;
extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *);
static char order[1];
extern /* Subroutine */ int sgemv_(char *, integer *, integer *, real *,
real *, integer *, real *, integer *, real *, real *, integer *);
static logical lower;
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *), sswap_(integer *, real *, integer *, real *, integer *
);
static logical wantz;
static integer jj;
static logical alleig, indeig;
static integer iscale, indibl;
static logical valeig;
extern doublereal slamch_(char *);
static real safmin;
extern /* Subroutine */ int xerbla_(char *, integer *);
static real abstll, bignum;
extern doublereal slansb_(char *, char *, integer *, integer *, real *,
integer *, real *);
extern /* Subroutine */ int slascl_(char *, integer *, integer *, real *,
real *, integer *, integer *, real *, integer *, integer *);
static integer indisp, indiwo;
extern /* Subroutine */ int slacpy_(char *, integer *, integer *, real *,
integer *, real *, integer *);
static integer indwrk;
extern /* Subroutine */ int ssbtrd_(char *, char *, integer *, integer *,
real *, integer *, real *, real *, real *, integer *, real *,
integer *), sstein_(integer *, real *, real *,
integer *, real *, integer *, integer *, real *, integer *, real *
, integer *, integer *, integer *), ssterf_(integer *, real *,
real *, integer *);
static integer nsplit;
static real smlnum;
extern /* Subroutine */ int sstebz_(char *, char *, integer *, real *,
real *, integer *, integer *, real *, real *, real *, integer *,
integer *, real *, integer *, integer *, real *, integer *,
integer *), ssteqr_(char *, integer *, real *,
real *, real *, integer *, real *, integer *);
static real eps, vll, vuu, tmp1;
#define W(I) w[(I)-1]
#define WORK(I) work[(I)-1]
#define IWORK(I) iwork[(I)-1]
#define IFAIL(I) ifail[(I)-1]
#define AB(I,J) ab[(I)-1 + ((J)-1)* ( *ldab)]
#define Q(I,J) q[(I)-1 + ((J)-1)* ( *ldq)]
#define Z(I,J) z[(I)-1 + ((J)-1)* ( *ldz)]
wantz = lsame_(jobz, "V");
alleig = lsame_(range, "A");
valeig = lsame_(range, "V");
indeig = lsame_(range, "I");
lower = lsame_(uplo, "L");
*info = 0;
if (! (wantz || lsame_(jobz, "N"))) {
*info = -1;
} else if (! (alleig || valeig || indeig)) {
*info = -2;
} else if (! (lower || lsame_(uplo, "U"))) {
*info = -3;
} else if (*n < 0) {
*info = -4;
} else if (*kd < 0) {
*info = -5;
} else if (*ldab < *kd + 1) {
*info = -7;
} else if (*ldq < *n) {
*info = -9;
} else if (valeig && *n > 0 && *vu <= *vl) {
*info = -11;
} else if (indeig && *il < 1) {
*info = -12;
} else if (indeig && (*iu < min(*n,*il) || *iu > *n)) {
*info = -13;
} else if (*ldz < 1 || wantz && *ldz < *n) {
*info = -18;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SSBEVX", &i__1);
return 0;
}
/* Quick return if possible */
*m = 0;
if (*n == 0) {
return 0;
}
if (*n == 1) {
if (alleig || indeig) {
*m = 1;
W(1) = AB(1,1);
} else {
if (*vl < AB(1,1) && *vu >= AB(1,1)) {
*m = 1;
W(1) = AB(1,1);
}
}
if (wantz) {
Z(1,1) = 1.f;
}
return 0;
}
/* 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);
/* Scale matrix to allowable range, if necessary. */
iscale = 0;
abstll = *abstol;
if (valeig) {
vll = *vl;
vuu = *vu;
}
anrm = slansb_("M", uplo, n, kd, &AB(1,1), ldab, &WORK(1));
if (anrm > 0.f && anrm < rmin) {
iscale = 1;
sigma = rmin / anrm;
} else if (anrm > rmax) {
iscale = 1;
sigma = rmax / anrm;
}
if (iscale == 1) {
if (lower) {
slascl_("B", kd, kd, &c_b14, &sigma, n, n, &AB(1,1), ldab,
info);
} else {
slascl_("Q", kd, kd, &c_b14, &sigma, n, n, &AB(1,1), ldab,
info);
}
if (*abstol > 0.f) {
abstll = *abstol * sigma;
}
if (valeig) {
vll = *vl * sigma;
vuu = *vu * sigma;
}
}
/* Call SSBTRD to reduce symmetric band matrix to tridiagonal form. */
indd = 1;
inde = indd + *n;
indwrk = inde + *n;
ssbtrd_(jobz, uplo, n, kd, &AB(1,1), ldab, &WORK(indd), &WORK(inde),
&Q(1,1), ldq, &WORK(indwrk), &iinfo);
/* If all eigenvalues are desired and ABSTOL is less than or equal
to zero, then call SSTERF or SSTEQR. If this fails for some
eigenvalue, then try SSTEBZ. */
if ((alleig || indeig && *il == 1 && *iu == *n) && *abstol <= 0.f) {
scopy_(n, &WORK(indd), &c__1, &W(1), &c__1);
indee = indwrk + (*n << 1);
if (! wantz) {
i__1 = *n - 1;
scopy_(&i__1, &WORK(inde), &c__1, &WORK(indee), &c__1);
ssterf_(n, &W(1), &WORK(indee), info);
} else {
slacpy_("A", n, n, &Q(1,1), ldq, &Z(1,1), ldz);
i__1 = *n - 1;
scopy_(&i__1, &WORK(inde), &c__1, &WORK(indee), &c__1);
ssteqr_(jobz, n, &W(1), &WORK(indee), &Z(1,1), ldz, &WORK(
indwrk), info);
if (*info == 0) {
i__1 = *n;
for (i = 1; i <= *n; ++i) {
IFAIL(i) = 0;
/* L10: */
}
}
}
if (*info == 0) {
*m = *n;
goto L30;
}
*info = 0;
}
/* Otherwise, call SSTEBZ and, if eigenvectors are desired, SSTEIN. */
if (wantz) {
*(unsigned char *)order = 'B';
} else {
*(unsigned char *)order = 'E';
}
indibl = 1;
indisp = indibl + *n;
indiwo = indisp + *n;
sstebz_(range, order, n, &vll, &vuu, il, iu, &abstll, &WORK(indd), &WORK(
inde), m, &nsplit, &W(1), &IWORK(indibl), &IWORK(indisp), &WORK(
indwrk), &IWORK(indiwo), info);
if (wantz) {
sstein_(n, &WORK(indd), &WORK(inde), m, &W(1), &IWORK(indibl), &IWORK(
indisp), &Z(1,1), ldz, &WORK(indwrk), &IWORK(indiwo), &
IFAIL(1), info);
/* Apply orthogonal matrix used in reduction to tridiagonal
form to eigenvectors returned by SSTEIN. */
i__1 = *m;
for (j = 1; j <= *m; ++j) {
scopy_(n, &Z(1,j), &c__1, &WORK(1), &c__1);
sgemv_("N", n, n, &c_b14, &Q(1,1), ldq, &WORK(1), &c__1, &
c_b34, &Z(1,j), &c__1);
/* L20: */
}
}
/* If matrix was scaled, then rescale eigenvalues appropriately. */
L30:
if (iscale == 1) {
if (*info == 0) {
imax = *m;
} else {
imax = *info - 1;
}
r__1 = 1.f / sigma;
sscal_(&imax, &r__1, &W(1), &c__1);
}
/* If eigenvalues are not in order, then sort them, along with
eigenvectors. */
if (wantz) {
i__1 = *m - 1;
for (j = 1; j <= *m-1; ++j) {
i = 0;
tmp1 = W(j);
i__2 = *m;
for (jj = j + 1; jj <= *m; ++jj) {
if (W(jj) < tmp1) {
i = jj;
tmp1 = W(jj);
}
/* L40: */
}
if (i != 0) {
itmp1 = IWORK(indibl + i - 1);
W(i) = W(j);
IWORK(indibl + i - 1) = IWORK(indibl + j - 1);
W(j) = tmp1;
IWORK(indibl + j - 1) = itmp1;
sswap_(n, &Z(1,i), &c__1, &Z(1,j), &
c__1);
if (*info != 0) {
itmp1 = IFAIL(i);
IFAIL(i) = IFAIL(j);
IFAIL(j) = itmp1;
}
}
/* L50: */
}
}
return 0;
/* End of SSBEVX */
} /* ssbevx_ */
/* Subroutine */ int sstevx_(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, real *work, integer *
iwork, integer *ifail, integer *info)
{
/* -- LAPACK driver 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
=======
SSTEVX computes selected eigenvalues and, optionally, eigenvectors
of a real symmetric tridiagonal matrix A. Eigenvalues and
eigenvectors can be selected by specifying either a range of values
or a range of indices for the desired eigenvalues.
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
A.
On exit, D may be multiplied by a constant factor chosen
to avoid over/underflow in computing the eigenvalues.
E (input/output) REAL array, dimension (N)
On entry, the (n-1) subdiagonal elements of the tridiagonal
matrix A in elements 1 to N-1 of E; E(N) need not be set.
On exit, E may be multiplied by a constant factor chosen
to avoid over/underflow in computing the eigenvalues.
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.
An approximate eigenvalue is accepted as converged
when it is determined to lie in an interval [a,b]
of width less than or equal to
ABSTOL + EPS * max( |a|,|b| ) ,
where EPS is the machine precision. If ABSTOL is less
than or equal to zero, then EPS*|T| will be used in
its place, where |T| is the 1-norm of the tridiagonal
matrix.
Eigenvalues will be computed most accurately when ABSTOL is
set to twice the underflow threshold 2*SLAMCH('S'), not zero.
If this routine returns with INFO>0, indicating that some
eigenvectors did not converge, try setting ABSTOL to
2*SLAMCH('S').
See "Computing Small Singular Values of Bidiagonal Matrices
with Guaranteed High Relative Accuracy," by Demmel and
Kahan, LAPACK Working Note #3.
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 A
corresponding to the selected eigenvalues, with the i-th
column of Z holding the eigenvector associated with W(i).
If an eigenvector fails to converge (INFO > 0), then that
column of Z contains the latest approximation to the
eigenvector, and the index of the eigenvector is returned
in IFAIL. 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).
WORK (workspace) REAL array, dimension (5*N)
IWORK (workspace) INTEGER array, dimension (5*N)
IFAIL (output) INTEGER array, dimension (N)
If JOBZ = 'V', then if INFO = 0, the first M elements of
IFAIL are zero. If INFO > 0, then IFAIL contains the
indices of the eigenvectors that failed to converge.
If JOBZ = 'N', then IFAIL is not referenced.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, then i eigenvectors failed to converge.
Their indices are stored in array IFAIL.
=====================================================================
Test the input parameters.
Parameter adjustments
Function Body */
/* Table of constant values */
static integer c__1 = 1;
/* 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 imax;
static real rmin, rmax, tnrm;
static integer itmp1, i, j;
static real sigma;
extern logical lsame_(char *, char *);
extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *);
static char order[1];
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *), sswap_(integer *, real *, integer *, real *, integer *
);
static logical wantz;
static integer jj;
static logical alleig, indeig;
static integer iscale, indibl;
static logical valeig;
extern doublereal slamch_(char *);
static real safmin;
extern /* Subroutine */ int xerbla_(char *, integer *);
static real bignum;
static integer indisp, indiwo, indwrk;
extern doublereal slanst_(char *, integer *, real *, real *);
extern /* Subroutine */ int sstein_(integer *, real *, real *, integer *,
real *, integer *, integer *, real *, integer *, real *, integer *
, integer *, integer *), ssterf_(integer *, real *, real *,
integer *);
static integer nsplit;
extern /* Subroutine */ int sstebz_(char *, char *, integer *, real *,
real *, integer *, integer *, real *, real *, real *, integer *,
integer *, real *, integer *, integer *, real *, integer *,
integer *);
static real smlnum;
extern /* Subroutine */ int ssteqr_(char *, integer *, real *, real *,
real *, integer *, real *, integer *);
static real eps, vll, vuu, tmp1;
#define D(I) d[(I)-1]
#define E(I) e[(I)-1]
#define W(I) w[(I)-1]
#define WORK(I) work[(I)-1]
#define IWORK(I) iwork[(I)-1]
#define IFAIL(I) ifail[(I)-1]
#define Z(I,J) z[(I)-1 + ((J)-1)* ( *ldz)]
wantz = lsame_(jobz, "V");
alleig = lsame_(range, "A");
valeig = lsame_(range, "V");
indeig = lsame_(range, "I");
*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 && *vu <= *vl) {
*info = -7;
} else if (indeig && *il < 1) {
*info = -8;
} else if (indeig && (*iu < min(*n,*il) || *iu > *n)) {
*info = -9;
} else if (*ldz < 1 || wantz && *ldz < *n) {
*info = -14;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SSTEVX", &i__1);
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(1,1) = 1.f;
}
return 0;
}
/* 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);
/* Scale matrix to allowable range, if necessary. */
iscale = 0;
if (valeig) {
vll = *vl;
vuu = *vu;
}
tnrm = slanst_("M", n, &D(1), &E(1));
if (tnrm > 0.f && tnrm < rmin) {
iscale = 1;
sigma = rmin / tnrm;
} else if (tnrm > rmax) {
iscale = 1;
sigma = rmax / tnrm;
}
if (iscale == 1) {
sscal_(n, &sigma, &D(1), &c__1);
i__1 = *n - 1;
sscal_(&i__1, &sigma, &E(1), &c__1);
if (valeig) {
vll = *vl * sigma;
vuu = *vu * sigma;
}
}
/* If all eigenvalues are desired and ABSTOL is less than zero, then
call SSTERF or SSTEQR. If this fails for some eigenvalue, then
try SSTEBZ. */
if ((alleig || indeig && *il == 1 && *iu == *n) && *abstol <= 0.f) {
scopy_(n, &D(1), &c__1, &W(1), &c__1);
i__1 = *n - 1;
scopy_(&i__1, &E(1), &c__1, &WORK(1), &c__1);
indwrk = *n + 1;
if (! wantz) {
ssterf_(n, &W(1), &WORK(1), info);
} else {
ssteqr_("I", n, &W(1), &WORK(1), &Z(1,1), ldz, &WORK(indwrk),
info);
if (*info == 0) {
i__1 = *n;
for (i = 1; i <= *n; ++i) {
IFAIL(i) = 0;
/* L10: */
}
}
}
if (*info == 0) {
*m = *n;
goto L20;
}
*info = 0;
}
/* Otherwise, call SSTEBZ and, if eigenvectors are desired, SSTEIN. */
if (wantz) {
*(unsigned char *)order = 'B';
} else {
*(unsigned char *)order = 'E';
}
indwrk = 1;
indibl = 1;
indisp = indibl + *n;
indiwo = indisp + *n;
sstebz_(range, order, n, &vll, &vuu, il, iu, abstol, &D(1), &E(1), m, &
nsplit, &W(1), &IWORK(indibl), &IWORK(indisp), &WORK(indwrk), &
IWORK(indiwo), info);
if (wantz) {
sstein_(n, &D(1), &E(1), m, &W(1), &IWORK(indibl), &IWORK(indisp), &Z(1,1), ldz, &WORK(indwrk), &IWORK(indiwo), &IFAIL(1),
info);
}
/* If matrix was scaled, then rescale eigenvalues appropriately. */
L20:
if (iscale == 1) {
if (*info == 0) {
imax = *m;
} else {
imax = *info - 1;
}
r__1 = 1.f / sigma;
sscal_(&imax, &r__1, &W(1), &c__1);
}
/* If eigenvalues are not in order, then sort them, along with
eigenvectors. */
if (wantz) {
i__1 = *m - 1;
for (j = 1; j <= *m-1; ++j) {
i = 0;
tmp1 = W(j);
i__2 = *m;
for (jj = j + 1; jj <= *m; ++jj) {
if (W(jj) < tmp1) {
i = jj;
tmp1 = W(jj);
}
/* L30: */
}
if (i != 0) {
itmp1 = IWORK(indibl + i - 1);
W(i) = W(j);
IWORK(indibl + i - 1) = IWORK(indibl + j - 1);
W(j) = tmp1;
IWORK(indibl + j - 1) = itmp1;
sswap_(n, &Z(1,i), &c__1, &Z(1,j), &
c__1);
if (*info != 0) {
itmp1 = IFAIL(i);
IFAIL(i) = IFAIL(j);
IFAIL(j) = itmp1;
}
}
/* L40: */
}
}
return 0;
/* End of SSTEVX */
} /* sstevx_ */
