/* 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_ */
/**
* void mtx_mrrr(double *a,double *b, double *eigvalues, int n)
*
* @brief This function finds the eigenvalues of matrix T (represented by diagonal a and offdiagonal b)
* using the multiple relatively robust representation algorithm.
* It's a wrapper around the stegr routine found in LAPACK.
* @param a, the diagonal of tridiagonal matrix T
* @param b, the off-diagonal of tridiagonal matrix T (size should n-1)
* @param (out)eigvalues, the eigenvalues of the matrix
* @param n, the dimensions of the square matrix T
*/
void mtx_mrrr(double *a,double *b, double *eigvalues, int n)
{
/* JOBZ (input) CHARACTER*1 */
/* = 'N': Compute eigenvalues only; */
/* = 'V': Compute eigenvalues and eigenvectors. */
char jobz = 'N';
/* 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. */
char range = 'A';
/* N (input) INTEGER */
/* The order of the matrix. N >= 0. */
integer n_prime = (integer)n;
/* D (input/output) REAL array, dimension (N) */
/* On entry, the N diagonal elements of the tridiagonal matrix */
/* T. On exit, D is overwritten. */
real *d = (real*)malloc(sizeof(real)*n);
convert_double_to_real_array(a,d,n);
/* E (input/output) REAL array, dimension (N) */
/* On entry, the (N-1) subdiagonal elements of the tridiagonal */
/* matrix T in elements 1 to N-1 of E. E(N) need not be set on */
/* input, but is used internally as workspace. */
/* On exit, E is overwritten. */
real *e = (real*)malloc(sizeof(real)*(n-1));
convert_double_to_real_array(b,e,n-1);
/* 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'. */
real *vl = NULL;
real *vu = NULL;
/* IL (input) INTEGER */
/* IU (input) INTEGER */
/* If RANGE='I', the indices (in ascending order) of the */
/* smallest and largest eigenvalues to be returned. */
/* 1 <= IL <= IU <= N, if N > 0. */
/* Not referenced if RANGE = 'A' or 'V'. */
integer il = 0;
integer iu = 0;
/* ABSTOL (input) REAL */
/* Unused. Was the absolute error tolerance for the */
/* eigenvalues/eigenvectors in previous versions. */
real *abstol = NULL;
/* 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. */
integer m = 0;
/* W (output) REAL array, dimension (N) */
/* The first M elements contain the selected eigenvalues in */
/* ascending order. */
real *w = (real*)malloc(sizeof(real)*n);
/* Z (output) REAL array, dimension (LDZ, max(1,M) ) */
/* If JOBZ = 'V', and if INFO = 0, then the first M columns of Z */
/* contain the orthonormal eigenvectors of the matrix T */
/* corresponding to the selected eigenvalues, with the i-th */
/* column of Z holding the eigenvector associated with W(i). */
/* If JOBZ = 'N', then Z is not referenced. */
/* Note: the user must ensure that at least max(1,M) columns are */
/* supplied in the array Z; if RANGE = 'V', the exact value of M */
/* is not known in advance and an upper bound must be used. */
/* Supplying N columns is always safe. */
real *z= NULL;
/* LDZ (input) INTEGER */
/* The leading dimension of the array Z. LDZ >= 1, and if */
/* JOBZ = 'V', then LDZ >= max(1,N). */
integer ldz = 1;
/* ISUPPZ (output) INTEGER ARRAY, dimension ( 2*max(1,M) ) */
/* The support of the eigenvectors in Z, i.e., the indices */
/* indicating the nonzero elements in Z. The i-th computed eigenvector */
/* is nonzero only in elements ISUPPZ( 2*i-1 ) through */
/* ISUPPZ( 2*i ). This is relevant in the case when the matrix */
/* is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0. */
integer *isuppz= NULL;
/* WORK (workspace/output) REAL array, dimension (LWORK) */
/* On exit, if INFO = 0, WORK(1) returns the optimal */
/* (and minimal) LWORK. */
real *work= (real*)malloc(sizeof(real)*12*n);
/* LWORK (input) INTEGER */
/* The dimension of the array WORK. LWORK >= max(1,18*N) */
/* if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'. */
/* If LWORK = -1, then a workspace query is assumed; the routine */
/* only calculates the optimal size of the WORK array, returns */
/* this value as the first entry of the WORK array, and no error */
/* message related to LWORK is issued by XERBLA. */
integer lwork= 12*n;
/* IWORK (workspace/output) INTEGER array, dimension (LIWORK) */
/* On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. */
integer *iwork = (integer*)malloc(sizeof(integer)*8*n);
/* LIWORK (input) INTEGER */
/* The dimension of the array IWORK. LIWORK >= max(1,10*N) */
/* if the eigenvectors are desired, and LIWORK >= max(1,8*N) */
/* if only the eigenvalues are to be computed. */
/* If LIWORK = -1, then a workspace query is assumed; the */
/* routine only calculates the optimal size of the IWORK array, */
/* returns this value as the first entry of the IWORK array, and */
/* no error message related to LIWORK is issued by XERBLA. */
integer liwork= 8*n;
/* INFO (output) INTEGER */
/* On exit, INFO */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* > 0: if INFO = 1X, internal error in SLARRE, */
/* if INFO = 2X, internal error in SLARRV. */
/* Here, the digit X = ABS( IINFO ) < 10, where IINFO is */
/* the nonzero error code returned by SLARRE or */
/* SLARRV, respectively. */
integer info = 0;
/*call the CLAPACK MRRR routine*/
sstegr_(&jobz, &range, &n_prime, d,
e, vl, vu, &il, &iu, abstol,
&m, w, z, &ldz, isuppz, work, &lwork, iwork, &liwork, &info);
/*copy back into the eigvalues array*/
convert_real_to_double_array(w,eigvalues,n);
/*free the memory*/
free(iwork);
free(work);
free(e);
free(d);
}
