#include "blaswrap.h"
#include "f2c.h"
/* 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 )
{
/* -- LAPACK driver routine (version 3.1) --
Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
November 2006
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
=====================================================================
Test the input parameters.
Parameter adjustments */
/* Table of constant values */
static integer c__10 = 10;
static integer c__1 = 1;
static integer c__2 = 2;
static integer c__3 = 3;
static integer c__4 = 4;
static integer c_n1 = -1;
/* 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 */
static integer i__, j, nb, jj;
static real eps, vll, vuu, tmp1, anrm;
static integer imax;
static real rmin, rmax;
static logical test;
static integer itmp1, indrd, indre;
static real sigma;
extern logical lsame_(char *, char *);
static integer iinfo;
extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *);
static char order[1];
static integer indwk;
extern /* Subroutine */ int cswap_(integer *, complex *, integer *,
complex *, integer *);
static integer lwmin;
static logical lower;
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *);
static logical wantz, alleig, indeig;
static integer iscale, ieeeok, indibl, indrdd, indifl, indree;
static 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 *);
static real safmin;
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *, ftnlen, ftnlen);
extern /* Subroutine */ int xerbla_(char *, integer *);
static real abstll, bignum;
static integer indtau, indisp;
extern /* Subroutine */ int cstein_(integer *, real *, real *, integer *,
real *, integer *, integer *, complex *, integer *, real *,
integer *, integer *, integer *);
static 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 *);
static integer indrwk, liwmin;
static logical tryrac;
extern /* Subroutine */ int ssterf_(integer *, real *, real *, integer *);
static integer lrwmin, llwrkn, llwork, nsplit;
static 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 *);
static logical lquery;
static integer lwkopt, llrwork;
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, (
ftnlen)6, (ftnlen)1);
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, (ftnlen)6,
(ftnlen)1);
/* Computing MAX */
i__1 = nb, i__2 = ilaenv_(&c__1, "CUNMTR", uplo, n, &c_n1, &c_n1, &
c_n1, (ftnlen)6, (ftnlen)1);
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 * 1.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_ */