/* Subroutine */ int dlarre_(char *range, integer *n, doublereal *vl,
doublereal *vu, integer *il, integer *iu, doublereal *d__, doublereal
*e, doublereal *e2, doublereal *rtol1, doublereal *rtol2, doublereal *
spltol, integer *nsplit, integer *isplit, integer *m, doublereal *w,
doublereal *werr, doublereal *wgap, integer *iblock, integer *indexw,
doublereal *gers, doublereal *pivmin, doublereal *work, integer *
iwork, integer *info)
{
/* -- LAPACK auxiliary routine (version 3.1) --
Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
November 2006
Purpose
=======
To find the desired eigenvalues of a given real symmetric
tridiagonal matrix T, DLARRE sets any "small" off-diagonal
elements to zero, and for each unreduced block T_i, it finds
(a) a suitable shift at one end of the block's spectrum,
(b) the base representation, T_i - sigma_i I = L_i D_i L_i^T, and
(c) eigenvalues of each L_i D_i L_i^T.
The representations and eigenvalues found are then used by
DSTEMR to compute the eigenvectors of T.
The accuracy varies depending on whether bisection is used to
find a few eigenvalues or the dqds algorithm (subroutine DLASQ2) to
conpute all and then discard any unwanted one.
As an added benefit, DLARRE also outputs the n
Gerschgorin intervals for the matrices L_i D_i L_i^T.
Arguments
=========
RANGE (input) CHARACTER
= 'A': ("All") all eigenvalues will be found.
= 'V': ("Value") all eigenvalues in the half-open interval
(VL, VU] will be found.
= 'I': ("Index") the IL-th through IU-th eigenvalues (of the
entire matrix) will be found.
N (input) INTEGER
The order of the matrix. N > 0.
VL (input/output) DOUBLE PRECISION
VU (input/output) DOUBLE PRECISION
If RANGE='V', the lower and upper bounds for the eigenvalues.
Eigenvalues less than or equal to VL, or greater than VU,
will not be returned. VL < VU.
If RANGE='I' or ='A', DLARRE computes bounds on the desired
part of the spectrum.
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.
D (input/output) DOUBLE PRECISION array, dimension (N)
On entry, the N diagonal elements of the tridiagonal
matrix T.
On exit, the N diagonal elements of the diagonal
matrices D_i.
E (input/output) DOUBLE PRECISION array, dimension (N)
On entry, the first (N-1) entries contain the subdiagonal
elements of the tridiagonal matrix T; E(N) need not be set.
On exit, E contains the subdiagonal elements of the unit
bidiagonal matrices L_i. The entries E( ISPLIT( I ) ),
1 <= I <= NSPLIT, contain the base points sigma_i on output.
E2 (input/output) DOUBLE PRECISION array, dimension (N)
On entry, the first (N-1) entries contain the SQUARES of the
subdiagonal elements of the tridiagonal matrix T;
E2(N) need not be set.
On exit, the entries E2( ISPLIT( I ) ),
1 <= I <= NSPLIT, have been set to zero
RTOL1 (input) DOUBLE PRECISION
RTOL2 (input) DOUBLE PRECISION
Parameters for bisection.
An interval [LEFT,RIGHT] has converged if
RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
SPLTOL (input) DOUBLE PRECISION
The threshold for splitting.
NSPLIT (output) INTEGER
The number of blocks T splits into. 1 <= NSPLIT <= N.
ISPLIT (output) INTEGER array, dimension (N)
The splitting points, at which T breaks up into blocks.
The first block consists of rows/columns 1 to ISPLIT(1),
the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
etc., and the NSPLIT-th consists of rows/columns
ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
M (output) INTEGER
The total number of eigenvalues (of all L_i D_i L_i^T)
found.
W (output) DOUBLE PRECISION array, dimension (N)
The first M elements contain the eigenvalues. The
eigenvalues of each of the blocks, L_i D_i L_i^T, are
sorted in ascending order ( DLARRE may use the
remaining N-M elements as workspace).
WERR (output) DOUBLE PRECISION array, dimension (N)
The error bound on the corresponding eigenvalue in W.
WGAP (output) DOUBLE PRECISION array, dimension (N)
The separation from the right neighbor eigenvalue in W.
The gap is only with respect to the eigenvalues of the same block
as each block has its own representation tree.
Exception: at the right end of a block we store the left gap
IBLOCK (output) INTEGER array, dimension (N)
The indices of the blocks (submatrices) associated with the
corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue
W(i) belongs to the first block from the top, =2 if W(i)
belongs to the second block, etc.
INDEXW (output) INTEGER array, dimension (N)
The indices of the eigenvalues within each block (submatrix);
for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the
i-th eigenvalue W(i) is the 10-th eigenvalue in block 2
GERS (output) DOUBLE PRECISION array, dimension (2*N)
The N Gerschgorin intervals (the i-th Gerschgorin interval
is (GERS(2*i-1), GERS(2*i)).
PIVMIN (output) DOUBLE PRECISION
The minimum pivot in the Sturm sequence for T.
WORK (workspace) DOUBLE PRECISION array, dimension (6*N)
Workspace.
IWORK (workspace) INTEGER array, dimension (5*N)
Workspace.
INFO (output) INTEGER
= 0: successful exit
> 0: A problem occured in DLARRE.
< 0: One of the called subroutines signaled an internal problem.
Needs inspection of the corresponding parameter IINFO
for further information.
=-1: Problem in DLARRD.
= 2: No base representation could be found in MAXTRY iterations.
Increasing MAXTRY and recompilation might be a remedy.
=-3: Problem in DLARRB when computing the refined root
representation for DLASQ2.
=-4: Problem in DLARRB when preforming bisection on the
desired part of the spectrum.
=-5: Problem in DLASQ2.
=-6: Problem in DLASQ2.
Further Details
The base representations are required to suffer very little
element growth and consequently define all their eigenvalues to
high relative accuracy.
===============
Based on contributions by
Beresford Parlett, University of California, Berkeley, USA
Jim Demmel, University of California, Berkeley, USA
Inderjit Dhillon, University of Texas, Austin, USA
Osni Marques, LBNL/NERSC, USA
Christof Voemel, University of California, Berkeley, USA
=====================================================================
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
static integer c__2 = 2;
/* System generated locals */
integer i__1, i__2;
doublereal d__1, d__2, d__3;
/* Builtin functions */
double sqrt(doublereal), log(doublereal);
/* Local variables */
static integer i__, j;
static doublereal s1, s2;
static integer mb;
static doublereal gl;
static integer in, mm;
static doublereal gu;
static integer cnt;
static doublereal eps, tau, tmp, rtl;
static integer cnt1, cnt2;
static doublereal tmp1, eabs;
static integer iend, jblk;
static doublereal eold;
static integer indl;
static doublereal dmax__, emax;
static integer wend, idum, indu;
static doublereal rtol;
static integer iseed[4];
static doublereal avgap, sigma;
extern logical lsame_(char *, char *);
static integer iinfo;
extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *,
doublereal *, integer *);
static logical norep;
extern /* Subroutine */ int dlasq2_(integer *, doublereal *, integer *);
extern doublereal dlamch_(char *);
static integer ibegin;
static logical forceb;
static integer irange;
static doublereal sgndef;
extern /* Subroutine */ int dlarra_(integer *, doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *, integer *, integer *,
integer *), dlarrb_(integer *, doublereal *, doublereal *,
integer *, integer *, doublereal *, doublereal *, integer *,
doublereal *, doublereal *, doublereal *, doublereal *, integer *,
doublereal *, doublereal *, integer *, integer *), dlarrc_(char *
, integer *, doublereal *, doublereal *, doublereal *, doublereal
*, doublereal *, integer *, integer *, integer *, integer *);
static integer wbegin;
extern /* Subroutine */ int dlarrd_(char *, char *, integer *, doublereal
*, doublereal *, integer *, integer *, doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *, doublereal *, integer *
, integer *, integer *, doublereal *, doublereal *, doublereal *,
doublereal *, integer *, integer *, doublereal *, integer *,
integer *);
static doublereal safmin, spdiam;
extern /* Subroutine */ int dlarrk_(integer *, integer *, doublereal *,
doublereal *, doublereal *, doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *, integer *);
static logical usedqd;
static doublereal clwdth, isleft;
extern /* Subroutine */ int dlarnv_(integer *, integer *, integer *,
doublereal *);
static doublereal isrght, bsrtol, dpivot;
--iwork;
--work;
--gers;
--indexw;
--iblock;
--wgap;
--werr;
--w;
--isplit;
--e2;
--e;
--d__;
/* Function Body */
*info = 0;
/* Decode RANGE */
if (lsame_(range, "A")) {
irange = 1;
} else if (lsame_(range, "V")) {
irange = 3;
} else if (lsame_(range, "I")) {
irange = 2;
}
*m = 0;
/* Get machine constants */
safmin = dlamch_("S");
eps = dlamch_("P");
/* Set parameters */
rtl = sqrt(eps);
bsrtol = sqrt(eps);
/* Treat case of 1x1 matrix for quick return */
if (*n == 1) {
if (irange == 1 || irange == 3 && d__[1] > *vl && d__[1] <= *vu ||
irange == 2 && *il == 1 && *iu == 1) {
*m = 1;
w[1] = d__[1];
/* The computation error of the eigenvalue is zero */
werr[1] = 0.;
wgap[1] = 0.;
iblock[1] = 1;
indexw[1] = 1;
gers[1] = d__[1];
gers[2] = d__[1];
}
/* store the shift for the initial RRR, which is zero in this case */
e[1] = 0.;
return 0;
}
/* General case: tridiagonal matrix of order > 1
Init WERR, WGAP. Compute Gerschgorin intervals and spectral diameter.
Compute maximum off-diagonal entry and pivmin. */
gl = d__[1];
gu = d__[1];
eold = 0.;
emax = 0.;
e[*n] = 0.;
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
werr[i__] = 0.;
wgap[i__] = 0.;
eabs = (d__1 = e[i__], abs(d__1));
if (eabs >= emax) {
emax = eabs;
}
tmp1 = eabs + eold;
gers[(i__ << 1) - 1] = d__[i__] - tmp1;
/* Computing MIN */
d__1 = gl, d__2 = gers[(i__ << 1) - 1];
gl = min(d__1,d__2);
gers[i__ * 2] = d__[i__] + tmp1;
/* Computing MAX */
d__1 = gu, d__2 = gers[i__ * 2];
gu = max(d__1,d__2);
eold = eabs;
/* L5: */
}
/* The minimum pivot allowed in the Sturm sequence for T
Computing MAX
Computing 2nd power */
d__3 = emax;
d__1 = 1., d__2 = d__3 * d__3;
*pivmin = safmin * max(d__1,d__2);
/* Compute spectral diameter. The Gerschgorin bounds give an
estimate that is wrong by at most a factor of SQRT(2) */
spdiam = gu - gl;
/* Compute splitting points */
dlarra_(n, &d__[1], &e[1], &e2[1], spltol, &spdiam, nsplit, &isplit[1], &
iinfo);
/* Can force use of bisection instead of faster DQDS.
Option left in the code for future multisection work. */
forceb = FALSE_;
if (irange == 1 && ! forceb) {
/* Set interval [VL,VU] that contains all eigenvalues */
*vl = gl;
*vu = gu;
} else {
/* We call DLARRD to find crude approximations to the eigenvalues
in the desired range. In case IRANGE = INDRNG, we also obtain the
interval (VL,VU] that contains all the wanted eigenvalues.
An interval [LEFT,RIGHT] has converged if
RIGHT-LEFT.LT.RTOL*MAX(ABS(LEFT),ABS(RIGHT))
DLARRD needs a WORK of size 4*N, IWORK of size 3*N */
dlarrd_(range, "B", n, vl, vu, il, iu, &gers[1], &bsrtol, &d__[1], &e[
1], &e2[1], pivmin, nsplit, &isplit[1], &mm, &w[1], &werr[1],
vl, vu, &iblock[1], &indexw[1], &work[1], &iwork[1], &iinfo);
if (iinfo != 0) {
*info = -1;
return 0;
}
/* Make sure that the entries M+1 to N in W, WERR, IBLOCK, INDEXW are 0 */
i__1 = *n;
for (i__ = mm + 1; i__ <= i__1; ++i__) {
w[i__] = 0.;
werr[i__] = 0.;
iblock[i__] = 0;
indexw[i__] = 0;
/* L14: */
}
}
/* **
Loop over unreduced blocks */
ibegin = 1;
wbegin = 1;
i__1 = *nsplit;
for (jblk = 1; jblk <= i__1; ++jblk) {
iend = isplit[jblk];
in = iend - ibegin + 1;
/* 1 X 1 block */
if (in == 1) {
if (irange == 1 || irange == 3 && d__[ibegin] > *vl && d__[ibegin]
<= *vu || irange == 2 && iblock[wbegin] == jblk) {
++(*m);
w[*m] = d__[ibegin];
werr[*m] = 0.;
/* The gap for a single block doesn't matter for the later
algorithm and is assigned an arbitrary large value */
wgap[*m] = 0.;
iblock[*m] = jblk;
indexw[*m] = 1;
++wbegin;
}
/* E( IEND ) holds the shift for the initial RRR */
e[iend] = 0.;
ibegin = iend + 1;
goto L170;
}
/* Blocks of size larger than 1x1
E( IEND ) will hold the shift for the initial RRR, for now set it =0 */
e[iend] = 0.;
/* Find local outer bounds GL,GU for the block */
gl = d__[ibegin];
gu = d__[ibegin];
i__2 = iend;
for (i__ = ibegin; i__ <= i__2; ++i__) {
/* Computing MIN */
d__1 = gers[(i__ << 1) - 1];
gl = min(d__1,gl);
/* Computing MAX */
d__1 = gers[i__ * 2];
gu = max(d__1,gu);
/* L15: */
}
spdiam = gu - gl;
if (! (irange == 1 && ! forceb)) {
/* Count the number of eigenvalues in the current block. */
mb = 0;
i__2 = mm;
for (i__ = wbegin; i__ <= i__2; ++i__) {
if (iblock[i__] == jblk) {
++mb;
} else {
goto L21;
}
/* L20: */
}
L21:
if (mb == 0) {
/* No eigenvalue in the current block lies in the desired range
E( IEND ) holds the shift for the initial RRR */
e[iend] = 0.;
ibegin = iend + 1;
goto L170;
} else {
/* Decide whether dqds or bisection is more efficient */
usedqd = (doublereal) mb > in * .5 && ! forceb;
wend = wbegin + mb - 1;
/* Calculate gaps for the current block
In later stages, when representations for individual
eigenvalues are different, we use SIGMA = E( IEND ). */
sigma = 0.;
i__2 = wend - 1;
for (i__ = wbegin; i__ <= i__2; ++i__) {
/* Computing MAX */
d__1 = 0., d__2 = w[i__ + 1] - werr[i__ + 1] - (w[i__] +
werr[i__]);
wgap[i__] = max(d__1,d__2);
/* L30: */
}
/* Computing MAX */
d__1 = 0., d__2 = *vu - sigma - (w[wend] + werr[wend]);
wgap[wend] = max(d__1,d__2);
/* Find local index of the first and last desired evalue. */
indl = indexw[wbegin];
indu = indexw[wend];
}
}
if (irange == 1 && ! forceb || usedqd) {
/* Case of DQDS
Find approximations to the extremal eigenvalues of the block */
dlarrk_(&in, &c__1, &gl, &gu, &d__[ibegin], &e2[ibegin], pivmin, &
rtl, &tmp, &tmp1, &iinfo);
if (iinfo != 0) {
*info = -1;
return 0;
}
/* Computing MAX */
d__2 = gl, d__3 = tmp - tmp1 - eps * 100. * (d__1 = tmp - tmp1,
abs(d__1));
isleft = max(d__2,d__3);
dlarrk_(&in, &in, &gl, &gu, &d__[ibegin], &e2[ibegin], pivmin, &
rtl, &tmp, &tmp1, &iinfo);
if (iinfo != 0) {
*info = -1;
return 0;
}
/* Computing MIN */
d__2 = gu, d__3 = tmp + tmp1 + eps * 100. * (d__1 = tmp + tmp1,
abs(d__1));
isrght = min(d__2,d__3);
/* Improve the estimate of the spectral diameter */
spdiam = isrght - isleft;
} else {
/* Case of bisection
Find approximations to the wanted extremal eigenvalues
Computing MAX */
d__2 = gl, d__3 = w[wbegin] - werr[wbegin] - eps * 100. * (d__1 =
w[wbegin] - werr[wbegin], abs(d__1));
isleft = max(d__2,d__3);
/* Computing MIN */
d__2 = gu, d__3 = w[wend] + werr[wend] + eps * 100. * (d__1 = w[
wend] + werr[wend], abs(d__1));
isrght = min(d__2,d__3);
}
/* Decide whether the base representation for the current block
L_JBLK D_JBLK L_JBLK^T = T_JBLK - sigma_JBLK I
should be on the left or the right end of the current block.
The strategy is to shift to the end which is "more populated"
Furthermore, decide whether to use DQDS for the computation of
the eigenvalue approximations at the end of DLARRE or bisection.
dqds is chosen if all eigenvalues are desired or the number of
eigenvalues to be computed is large compared to the blocksize. */
if (irange == 1 && ! forceb) {
/* If all the eigenvalues have to be computed, we use dqd */
usedqd = TRUE_;
/* INDL is the local index of the first eigenvalue to compute */
indl = 1;
indu = in;
/* MB = number of eigenvalues to compute */
mb = in;
wend = wbegin + mb - 1;
/* Define 1/4 and 3/4 points of the spectrum */
s1 = isleft + spdiam * .25;
s2 = isrght - spdiam * .25;
} else {
/* DLARRD has computed IBLOCK and INDEXW for each eigenvalue
approximation.
choose sigma */
if (usedqd) {
s1 = isleft + spdiam * .25;
s2 = isrght - spdiam * .25;
} else {
tmp = min(isrght,*vu) - max(isleft,*vl);
s1 = max(isleft,*vl) + tmp * .25;
s2 = min(isrght,*vu) - tmp * .25;
}
}
/* Compute the negcount at the 1/4 and 3/4 points */
if (mb > 1) {
dlarrc_("T", &in, &s1, &s2, &d__[ibegin], &e[ibegin], pivmin, &
cnt, &cnt1, &cnt2, &iinfo);
}
if (mb == 1) {
sigma = gl;
sgndef = 1.;
} else if (cnt1 - indl >= indu - cnt2) {
if (irange == 1 && ! forceb) {
sigma = max(isleft,gl);
} else if (usedqd) {
/* use Gerschgorin bound as shift to get pos def matrix
for dqds */
sigma = isleft;
} else {
/* use approximation of the first desired eigenvalue of the
block as shift */
sigma = max(isleft,*vl);
}
sgndef = 1.;
} else {
if (irange == 1 && ! forceb) {
sigma = min(isrght,gu);
} else if (usedqd) {
/* use Gerschgorin bound as shift to get neg def matrix
for dqds */
sigma = isrght;
} else {
/* use approximation of the first desired eigenvalue of the
block as shift */
sigma = min(isrght,*vu);
}
sgndef = -1.;
}
/* An initial SIGMA has been chosen that will be used for computing
T - SIGMA I = L D L^T
Define the increment TAU of the shift in case the initial shift
needs to be refined to obtain a factorization with not too much
element growth. */
if (usedqd) {
/* The initial SIGMA was to the outer end of the spectrum
the matrix is definite and we need not retreat. */
tau = spdiam * eps * *n + *pivmin * 2.;
} else {
if (mb > 1) {
clwdth = w[wend] + werr[wend] - w[wbegin] - werr[wbegin];
avgap = (d__1 = clwdth / (doublereal) (wend - wbegin), abs(
d__1));
if (sgndef == 1.) {
/* Computing MAX */
d__1 = wgap[wbegin];
tau = max(d__1,avgap) * .5;
/* Computing MAX */
d__1 = tau, d__2 = werr[wbegin];
tau = max(d__1,d__2);
} else {
/* Computing MAX */
d__1 = wgap[wend - 1];
tau = max(d__1,avgap) * .5;
/* Computing MAX */
d__1 = tau, d__2 = werr[wend];
tau = max(d__1,d__2);
}
} else {
tau = werr[wbegin];
}
}
for (idum = 1; idum <= 6; ++idum) {
/* Compute L D L^T factorization of tridiagonal matrix T - sigma I.
Store D in WORK(1:IN), L in WORK(IN+1:2*IN), and reciprocals of
pivots in WORK(2*IN+1:3*IN) */
dpivot = d__[ibegin] - sigma;
work[1] = dpivot;
dmax__ = abs(work[1]);
j = ibegin;
i__2 = in - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
work[(in << 1) + i__] = 1. / work[i__];
tmp = e[j] * work[(in << 1) + i__];
work[in + i__] = tmp;
dpivot = d__[j + 1] - sigma - tmp * e[j];
work[i__ + 1] = dpivot;
/* Computing MAX */
d__1 = dmax__, d__2 = abs(dpivot);
dmax__ = max(d__1,d__2);
++j;
/* L70: */
}
/* check for element growth */
if (dmax__ > spdiam * 64.) {
norep = TRUE_;
} else {
norep = FALSE_;
}
if (usedqd && ! norep) {
/* Ensure the definiteness of the representation
All entries of D (of L D L^T) must have the same sign */
i__2 = in;
for (i__ = 1; i__ <= i__2; ++i__) {
tmp = sgndef * work[i__];
if (tmp < 0.) {
norep = TRUE_;
}
/* L71: */
}
}
if (norep) {
/* Note that in the case of IRANGE=ALLRNG, we use the Gerschgorin
shift which makes the matrix definite. So we should end up
here really only in the case of IRANGE = VALRNG or INDRNG. */
if (idum == 5) {
if (sgndef == 1.) {
/* The fudged Gerschgorin shift should succeed */
sigma = gl - spdiam * 2. * eps * *n - *pivmin * 4.;
} else {
sigma = gu + spdiam * 2. * eps * *n + *pivmin * 4.;
}
} else {
sigma -= sgndef * tau;
tau *= 2.;
}
} else {
/* an initial RRR is found */
goto L83;
}
/* L80: */
}
/* if the program reaches this point, no base representation could be
found in MAXTRY iterations. */
*info = 2;
return 0;
L83:
/* At this point, we have found an initial base representation
T - SIGMA I = L D L^T with not too much element growth.
Store the shift. */
e[iend] = sigma;
/* Store D and L. */
dcopy_(&in, &work[1], &c__1, &d__[ibegin], &c__1);
i__2 = in - 1;
dcopy_(&i__2, &work[in + 1], &c__1, &e[ibegin], &c__1);
if (mb > 1) {
/* Perturb each entry of the base representation by a small
(but random) relative amount to overcome difficulties with
glued matrices. */
for (i__ = 1; i__ <= 4; ++i__) {
iseed[i__ - 1] = 1;
/* L122: */
}
i__2 = (in << 1) - 1;
dlarnv_(&c__2, iseed, &i__2, &work[1]);
i__2 = in - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
d__[ibegin + i__ - 1] *= eps * 8. * work[i__] + 1.;
e[ibegin + i__ - 1] *= eps * 8. * work[in + i__] + 1.;
/* L125: */
}
d__[iend] *= eps * 4. * work[in] + 1.;
}
/* Don't update the Gerschgorin intervals because keeping track
of the updates would be too much work in DLARRV.
We update W instead and use it to locate the proper Gerschgorin
intervals.
Compute the required eigenvalues of L D L' by bisection or dqds */
if (! usedqd) {
/* If DLARRD has been used, shift the eigenvalue approximations
according to their representation. This is necessary for
a uniform DLARRV since dqds computes eigenvalues of the
shifted representation. In DLARRV, W will always hold the
UNshifted eigenvalue approximation. */
i__2 = wend;
for (j = wbegin; j <= i__2; ++j) {
w[j] -= sigma;
werr[j] += (d__1 = w[j], abs(d__1)) * eps;
/* L134: */
}
/* call DLARRB to reduce eigenvalue error of the approximations
from DLARRD */
i__2 = iend - 1;
for (i__ = ibegin; i__ <= i__2; ++i__) {
/* Computing 2nd power */
d__1 = e[i__];
work[i__] = d__[i__] * (d__1 * d__1);
/* L135: */
}
/* use bisection to find EV from INDL to INDU */
i__2 = indl - 1;
dlarrb_(&in, &d__[ibegin], &work[ibegin], &indl, &indu, rtol1,
rtol2, &i__2, &w[wbegin], &wgap[wbegin], &werr[wbegin], &
work[(*n << 1) + 1], &iwork[1], pivmin, &spdiam, &in, &
iinfo);
if (iinfo != 0) {
*info = -4;
return 0;
}
/* DLARRB computes all gaps correctly except for the last one
Record distance to VU/GU
Computing MAX */
d__1 = 0., d__2 = *vu - sigma - (w[wend] + werr[wend]);
wgap[wend] = max(d__1,d__2);
i__2 = indu;
for (i__ = indl; i__ <= i__2; ++i__) {
++(*m);
iblock[*m] = jblk;
indexw[*m] = i__;
/* L138: */
}
} else {
/* Call dqds to get all eigs (and then possibly delete unwanted
eigenvalues).
Note that dqds finds the eigenvalues of the L D L^T representation
of T to high relative accuracy. High relative accuracy
might be lost when the shift of the RRR is subtracted to obtain
the eigenvalues of T. However, T is not guaranteed to define its
eigenvalues to high relative accuracy anyway.
Set RTOL to the order of the tolerance used in DLASQ2
This is an ESTIMATED error, the worst case bound is 4*N*EPS
which is usually too large and requires unnecessary work to be
done by bisection when computing the eigenvectors */
rtol = log((doublereal) in) * 4. * eps;
j = ibegin;
i__2 = in - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
work[(i__ << 1) - 1] = (d__1 = d__[j], abs(d__1));
work[i__ * 2] = e[j] * e[j] * work[(i__ << 1) - 1];
++j;
/* L140: */
}
work[(in << 1) - 1] = (d__1 = d__[iend], abs(d__1));
work[in * 2] = 0.;
dlasq2_(&in, &work[1], &iinfo);
if (iinfo != 0) {
/* If IINFO = -5 then an index is part of a tight cluster
and should be changed. The index is in IWORK(1) and the
gap is in WORK(N+1) */
*info = -5;
return 0;
} else {
/* Test that all eigenvalues are positive as expected */
i__2 = in;
for (i__ = 1; i__ <= i__2; ++i__) {
if (work[i__] < 0.) {
*info = -6;
return 0;
}
/* L149: */
}
}
if (sgndef > 0.) {
i__2 = indu;
for (i__ = indl; i__ <= i__2; ++i__) {
++(*m);
w[*m] = work[in - i__ + 1];
iblock[*m] = jblk;
indexw[*m] = i__;
/* L150: */
}
} else {
i__2 = indu;
for (i__ = indl; i__ <= i__2; ++i__) {
++(*m);
w[*m] = -work[i__];
iblock[*m] = jblk;
indexw[*m] = i__;
/* L160: */
}
}
i__2 = *m;
for (i__ = *m - mb + 1; i__ <= i__2; ++i__) {
/* the value of RTOL below should be the tolerance in DLASQ2 */
werr[i__] = rtol * (d__1 = w[i__], abs(d__1));
/* L165: */
}
i__2 = *m - 1;
for (i__ = *m - mb + 1; i__ <= i__2; ++i__) {
/* compute the right gap between the intervals
Computing MAX */
d__1 = 0., d__2 = w[i__ + 1] - werr[i__ + 1] - (w[i__] + werr[
i__]);
wgap[i__] = max(d__1,d__2);
/* L166: */
}
/* Computing MAX */
d__1 = 0., d__2 = *vu - sigma - (w[*m] + werr[*m]);
wgap[*m] = max(d__1,d__2);
}
/* proceed with next block */
ibegin = iend + 1;
wbegin = wend + 1;
L170:
;
}
return 0;
/* end of DLARRE */
} /* dlarre_ */
/* Subroutine */
int zstemr_(char *jobz, char *range, integer *n, doublereal * d__, doublereal *e, doublereal *vl, doublereal *vu, integer *il, integer *iu, integer *m, doublereal *w, doublecomplex *z__, integer * ldz, integer *nzc, integer *isuppz, logical *tryrac, doublereal *work, integer *lwork, integer *iwork, integer *liwork, integer *info)
{
/* System generated locals */
integer z_dim1, z_offset, i__1, i__2;
doublereal d__1, d__2;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
integer i__, j;
doublereal r1, r2;
integer jj;
doublereal cs;
integer in;
doublereal sn, wl, wu;
integer iil, iiu;
doublereal eps, tmp;
integer indd, iend, jblk, wend;
doublereal rmin, rmax;
integer itmp;
doublereal tnrm;
extern /* Subroutine */
int dlae2_(doublereal *, doublereal *, doublereal *, doublereal *, doublereal *);
integer inde2, itmp2;
doublereal rtol1, rtol2;
extern /* Subroutine */
int dscal_(integer *, doublereal *, doublereal *, integer *);
doublereal scale;
integer indgp;
extern logical lsame_(char *, char *);
integer iinfo, iindw, ilast;
extern /* Subroutine */
int dcopy_(integer *, doublereal *, integer *, doublereal *, integer *);
integer lwmin;
logical wantz;
extern /* Subroutine */
int zswap_(integer *, doublecomplex *, integer *, doublecomplex *, integer *), dlaev2_(doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *);
extern doublereal dlamch_(char *);
logical alleig;
integer ibegin;
logical indeig;
integer iindbl;
logical valeig;
extern /* Subroutine */
int dlarrc_(char *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, integer *, integer *, integer *), dlarre_(char *, integer *, doublereal *, doublereal *, integer *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, integer *, integer *, doublereal *, doublereal *, doublereal *, integer *, integer *, doublereal *, doublereal *, doublereal *, integer *, integer *);
integer wbegin;
doublereal safmin;
extern /* Subroutine */
int dlarrj_(integer *, doublereal *, doublereal *, integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, integer *, doublereal *, doublereal *, integer *), xerbla_(char *, integer *);
doublereal bignum;
integer inderr, iindwk, indgrs, offset;
extern doublereal dlanst_(char *, integer *, doublereal *, doublereal *);
extern /* Subroutine */
int dlarrr_(integer *, doublereal *, doublereal *, integer *), dlasrt_(char *, integer *, doublereal *, integer *);
doublereal thresh;
integer iinspl, indwrk, ifirst, liwmin, nzcmin;
doublereal pivmin;
integer nsplit;
doublereal smlnum;
extern /* Subroutine */
int zlarrv_(integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, integer *, integer *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, integer *, doublereal *, doublecomplex *, integer *, integer *, doublereal *, integer *, integer *);
logical lquery, zquery;
/* -- LAPACK computational routine (version 3.5.0) -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
/* November 2013 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--d__;
--e;
--w;
z_dim1 = *ldz;
z_offset = 1 + z_dim1;
z__ -= z_offset;
--isuppz;
--work;
--iwork;
/* Function Body */
wantz = lsame_(jobz, "V");
alleig = lsame_(range, "A");
valeig = lsame_(range, "V");
indeig = lsame_(range, "I");
lquery = *lwork == -1 || *liwork == -1;
zquery = *nzc == -1;
/* DSTEMR needs WORK of size 6*N, IWORK of size 3*N. */
/* In addition, DLARRE needs WORK of size 6*N, IWORK of size 5*N. */
/* Furthermore, ZLARRV needs WORK of size 12*N, IWORK of size 7*N. */
if (wantz)
{
lwmin = *n * 18;
liwmin = *n * 10;
}
else
{
/* need less workspace if only the eigenvalues are wanted */
lwmin = *n * 12;
liwmin = *n << 3;
}
wl = 0.;
wu = 0.;
iil = 0;
iiu = 0;
nsplit = 0;
if (valeig)
{
/* We do not reference VL, VU in the cases RANGE = 'I','A' */
/* The interval (WL, WU] contains all the wanted eigenvalues. */
/* It is either given by the user or computed in DLARRE. */
wl = *vl;
wu = *vu;
}
else if (indeig)
{
/* We do not reference IL, IU in the cases RANGE = 'V','A' */
iil = *il;
iiu = *iu;
}
*info = 0;
if (! (wantz || lsame_(jobz, "N")))
{
*info = -1;
}
else if (! (alleig || valeig || indeig))
{
*info = -2;
}
else if (*n < 0)
{
*info = -3;
}
else if (valeig && *n > 0 && wu <= wl)
{
*info = -7;
}
else if (indeig && (iil < 1 || iil > *n))
{
*info = -8;
}
else if (indeig && (iiu < iil || iiu > *n))
{
*info = -9;
}
else if (*ldz < 1 || wantz && *ldz < *n)
{
*info = -13;
}
else if (*lwork < lwmin && ! lquery)
{
*info = -17;
}
else if (*liwork < liwmin && ! lquery)
{
*info = -19;
}
/* Get machine constants. */
safmin = dlamch_("Safe minimum");
eps = dlamch_("Precision");
smlnum = safmin / eps;
bignum = 1. / smlnum;
rmin = sqrt(smlnum);
/* Computing MIN */
d__1 = sqrt(bignum);
d__2 = 1. / sqrt(sqrt(safmin)); // , expr subst
rmax = min(d__1,d__2);
if (*info == 0)
{
work[1] = (doublereal) lwmin;
iwork[1] = liwmin;
if (wantz && alleig)
{
nzcmin = *n;
}
else if (wantz && valeig)
{
dlarrc_("T", n, vl, vu, &d__[1], &e[1], &safmin, &nzcmin, &itmp, & itmp2, info);
}
else if (wantz && indeig)
{
nzcmin = iiu - iil + 1;
}
else
{
/* WANTZ .EQ. FALSE. */
nzcmin = 0;
}
if (zquery && *info == 0)
{
i__1 = z_dim1 + 1;
z__[i__1].r = (doublereal) nzcmin;
z__[i__1].i = 0.; // , expr subst
}
else if (*nzc < nzcmin && ! zquery)
{
*info = -14;
}
}
if (*info != 0)
{
i__1 = -(*info);
xerbla_("ZSTEMR", &i__1);
return 0;
}
else if (lquery || zquery)
{
return 0;
}
/* Handle N = 0, 1, and 2 cases immediately */
*m = 0;
if (*n == 0)
{
return 0;
}
if (*n == 1)
{
if (alleig || indeig)
{
*m = 1;
w[1] = d__[1];
}
else
{
if (wl < d__[1] && wu >= d__[1])
{
*m = 1;
w[1] = d__[1];
}
}
if (wantz && ! zquery)
{
i__1 = z_dim1 + 1;
z__[i__1].r = 1.;
z__[i__1].i = 0.; // , expr subst
isuppz[1] = 1;
isuppz[2] = 1;
}
return 0;
}
if (*n == 2)
{
if (! wantz)
{
dlae2_(&d__[1], &e[1], &d__[2], &r1, &r2);
}
else if (wantz && ! zquery)
{
dlaev2_(&d__[1], &e[1], &d__[2], &r1, &r2, &cs, &sn);
}
if (alleig || valeig && r2 > wl && r2 <= wu || indeig && iil == 1)
{
++(*m);
w[*m] = r2;
if (wantz && ! zquery)
{
i__1 = *m * z_dim1 + 1;
d__1 = -sn;
z__[i__1].r = d__1;
z__[i__1].i = 0.; // , expr subst
i__1 = *m * z_dim1 + 2;
z__[i__1].r = cs;
z__[i__1].i = 0.; // , expr subst
/* Note: At most one of SN and CS can be zero. */
if (sn != 0.)
{
if (cs != 0.)
{
isuppz[(*m << 1) - 1] = 1;
isuppz[(*m << 1) - 1] = 2;
}
else
{
isuppz[(*m << 1) - 1] = 1;
isuppz[(*m << 1) - 1] = 1;
}
}
else
{
isuppz[(*m << 1) - 1] = 2;
isuppz[*m * 2] = 2;
}
}
}
if (alleig || valeig && r1 > wl && r1 <= wu || indeig && iiu == 2)
{
++(*m);
w[*m] = r1;
if (wantz && ! zquery)
{
i__1 = *m * z_dim1 + 1;
z__[i__1].r = cs;
z__[i__1].i = 0.; // , expr subst
i__1 = *m * z_dim1 + 2;
z__[i__1].r = sn;
z__[i__1].i = 0.; // , expr subst
/* Note: At most one of SN and CS can be zero. */
if (sn != 0.)
{
if (cs != 0.)
{
isuppz[(*m << 1) - 1] = 1;
isuppz[(*m << 1) - 1] = 2;
}
else
{
isuppz[(*m << 1) - 1] = 1;
isuppz[(*m << 1) - 1] = 1;
}
}
else
{
isuppz[(*m << 1) - 1] = 2;
isuppz[*m * 2] = 2;
}
}
}
}
else
{
/* Continue with general N */
indgrs = 1;
inderr = (*n << 1) + 1;
indgp = *n * 3 + 1;
indd = (*n << 2) + 1;
inde2 = *n * 5 + 1;
indwrk = *n * 6 + 1;
iinspl = 1;
iindbl = *n + 1;
iindw = (*n << 1) + 1;
iindwk = *n * 3 + 1;
/* Scale matrix to allowable range, if necessary. */
/* The allowable range is related to the PIVMIN parameter;
see the */
/* comments in DLARRD. The preference for scaling small values */
/* up is heuristic;
we expect users' matrices not to be close to the */
/* RMAX threshold. */
scale = 1.;
tnrm = dlanst_("M", n, &d__[1], &e[1]);
if (tnrm > 0. && tnrm < rmin)
{
scale = rmin / tnrm;
}
else if (tnrm > rmax)
{
scale = rmax / tnrm;
}
if (scale != 1.)
{
dscal_(n, &scale, &d__[1], &c__1);
i__1 = *n - 1;
dscal_(&i__1, &scale, &e[1], &c__1);
tnrm *= scale;
if (valeig)
{
/* If eigenvalues in interval have to be found, */
/* scale (WL, WU] accordingly */
wl *= scale;
wu *= scale;
}
}
/* Compute the desired eigenvalues of the tridiagonal after splitting */
/* into smaller subblocks if the corresponding off-diagonal elements */
/* are small */
/* THRESH is the splitting parameter for DLARRE */
/* A negative THRESH forces the old splitting criterion based on the */
/* size of the off-diagonal. A positive THRESH switches to splitting */
/* which preserves relative accuracy. */
if (*tryrac)
{
/* Test whether the matrix warrants the more expensive relative approach. */
dlarrr_(n, &d__[1], &e[1], &iinfo);
}
else
{
/* The user does not care about relative accurately eigenvalues */
iinfo = -1;
}
/* Set the splitting criterion */
if (iinfo == 0)
{
thresh = eps;
}
else
{
thresh = -eps;
/* relative accuracy is desired but T does not guarantee it */
*tryrac = FALSE_;
}
if (*tryrac)
{
/* Copy original diagonal, needed to guarantee relative accuracy */
dcopy_(n, &d__[1], &c__1, &work[indd], &c__1);
}
/* Store the squares of the offdiagonal values of T */
i__1 = *n - 1;
for (j = 1;
j <= i__1;
++j)
{
/* Computing 2nd power */
d__1 = e[j];
work[inde2 + j - 1] = d__1 * d__1;
/* L5: */
}
/* Set the tolerance parameters for bisection */
if (! wantz)
{
/* DLARRE computes the eigenvalues to full precision. */
rtol1 = eps * 4.;
rtol2 = eps * 4.;
}
else
{
/* DLARRE computes the eigenvalues to less than full precision. */
/* ZLARRV will refine the eigenvalue approximations, and we only */
/* need less accurate initial bisection in DLARRE. */
/* Note: these settings do only affect the subset case and DLARRE */
rtol1 = sqrt(eps);
/* Computing MAX */
d__1 = sqrt(eps) * .005;
d__2 = eps * 4.; // , expr subst
rtol2 = max(d__1,d__2);
}
dlarre_(range, n, &wl, &wu, &iil, &iiu, &d__[1], &e[1], &work[inde2], &rtol1, &rtol2, &thresh, &nsplit, &iwork[iinspl], m, &w[1], & work[inderr], &work[indgp], &iwork[iindbl], &iwork[iindw], & work[indgrs], &pivmin, &work[indwrk], &iwork[iindwk], &iinfo);
if (iinfo != 0)
{
*info = f2c_abs(iinfo) + 10;
return 0;
}
/* Note that if RANGE .NE. 'V', DLARRE computes bounds on the desired */
/* part of the spectrum. All desired eigenvalues are contained in */
/* (WL,WU] */
if (wantz)
{
/* Compute the desired eigenvectors corresponding to the computed */
/* eigenvalues */
zlarrv_(n, &wl, &wu, &d__[1], &e[1], &pivmin, &iwork[iinspl], m, & c__1, m, &c_b18, &rtol1, &rtol2, &w[1], &work[inderr], & work[indgp], &iwork[iindbl], &iwork[iindw], &work[indgrs], &z__[z_offset], ldz, &isuppz[1], &work[indwrk], &iwork[ iindwk], &iinfo);
if (iinfo != 0)
{
*info = f2c_abs(iinfo) + 20;
return 0;
}
}
else
{
/* DLARRE computes eigenvalues of the (shifted) root representation */
/* ZLARRV returns the eigenvalues of the unshifted matrix. */
/* However, if the eigenvectors are not desired by the user, we need */
/* to apply the corresponding shifts from DLARRE to obtain the */
/* eigenvalues of the original matrix. */
i__1 = *m;
for (j = 1;
j <= i__1;
++j)
{
itmp = iwork[iindbl + j - 1];
w[j] += e[iwork[iinspl + itmp - 1]];
/* L20: */
}
}
if (*tryrac)
{
/* Refine computed eigenvalues so that they are relatively accurate */
/* with respect to the original matrix T. */
ibegin = 1;
wbegin = 1;
i__1 = iwork[iindbl + *m - 1];
for (jblk = 1;
jblk <= i__1;
++jblk)
{
iend = iwork[iinspl + jblk - 1];
in = iend - ibegin + 1;
wend = wbegin - 1;
/* check if any eigenvalues have to be refined in this block */
L36:
if (wend < *m)
{
if (iwork[iindbl + wend] == jblk)
{
++wend;
goto L36;
}
}
if (wend < wbegin)
{
ibegin = iend + 1;
goto L39;
}
offset = iwork[iindw + wbegin - 1] - 1;
ifirst = iwork[iindw + wbegin - 1];
ilast = iwork[iindw + wend - 1];
rtol2 = eps * 4.;
dlarrj_(&in, &work[indd + ibegin - 1], &work[inde2 + ibegin - 1], &ifirst, &ilast, &rtol2, &offset, &w[wbegin], & work[inderr + wbegin - 1], &work[indwrk], &iwork[ iindwk], &pivmin, &tnrm, &iinfo);
ibegin = iend + 1;
wbegin = wend + 1;
L39:
;
}
}
/* If matrix was scaled, then rescale eigenvalues appropriately. */
if (scale != 1.)
{
d__1 = 1. / scale;
dscal_(m, &d__1, &w[1], &c__1);
}
}
/* If eigenvalues are not in increasing order, then sort them, */
/* possibly along with eigenvectors. */
if (nsplit > 1 || *n == 2)
{
if (! wantz)
{
dlasrt_("I", m, &w[1], &iinfo);
if (iinfo != 0)
{
*info = 3;
return 0;
}
}
else
{
i__1 = *m - 1;
for (j = 1;
j <= i__1;
++j)
{
i__ = 0;
tmp = w[j];
i__2 = *m;
for (jj = j + 1;
jj <= i__2;
++jj)
{
if (w[jj] < tmp)
{
i__ = jj;
tmp = w[jj];
}
/* L50: */
}
if (i__ != 0)
{
w[i__] = w[j];
w[j] = tmp;
if (wantz)
{
zswap_(n, &z__[i__ * z_dim1 + 1], &c__1, &z__[j * z_dim1 + 1], &c__1);
itmp = isuppz[(i__ << 1) - 1];
isuppz[(i__ << 1) - 1] = isuppz[(j << 1) - 1];
isuppz[(j << 1) - 1] = itmp;
itmp = isuppz[i__ * 2];
isuppz[i__ * 2] = isuppz[j * 2];
isuppz[j * 2] = itmp;
}
}
/* L60: */
}
}
}
work[1] = (doublereal) lwmin;
iwork[1] = liwmin;
return 0;
/* End of ZSTEMR */
}
/* Subroutine */ int dstemr_(char *jobz, char *range, integer *n, doublereal *
d__, doublereal *e, doublereal *vl, doublereal *vu, integer *il,
integer *iu, integer *m, doublereal *w, doublereal *z__, integer *ldz,
integer *nzc, integer *isuppz, logical *tryrac, doublereal *work,
integer *lwork, integer *iwork, integer *liwork, integer *info)
{
/* System generated locals */
integer z_dim1, z_offset, i__1, i__2;
doublereal d__1, d__2;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
integer i__, j;
doublereal r1, r2;
integer jj;
doublereal cs;
integer in;
doublereal sn, wl, wu;
integer iil, iiu;
doublereal eps, tmp;
integer indd, iend, jblk, wend;
doublereal rmin, rmax;
integer itmp;
doublereal tnrm;
extern /* Subroutine */ int dlae2_(doublereal *, doublereal *, doublereal
*, doublereal *, doublereal *);
integer inde2, itmp2;
doublereal rtol1, rtol2;
extern /* Subroutine */ int dscal_(integer *, doublereal *, doublereal *,
integer *);
doublereal scale;
integer indgp;
extern logical lsame_(char *, char *);
integer iinfo, iindw, ilast;
extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *,
doublereal *, integer *), dswap_(integer *, doublereal *, integer
*, doublereal *, integer *);
integer lwmin;
logical wantz;
extern /* Subroutine */ int dlaev2_(doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *, doublereal *,
doublereal *);
extern doublereal dlamch_(char *);
logical alleig;
integer ibegin;
logical indeig;
integer iindbl;
logical valeig;
extern /* Subroutine */ int dlarrc_(char *, integer *, doublereal *,
doublereal *, doublereal *, doublereal *, doublereal *, integer *,
integer *, integer *, integer *), dlarre_(char *,
integer *, doublereal *, doublereal *, integer *, integer *,
doublereal *, doublereal *, doublereal *, doublereal *,
doublereal *, doublereal *, integer *, integer *, integer *,
doublereal *, doublereal *, doublereal *, integer *, integer *,
doublereal *, doublereal *, doublereal *, integer *, integer *);
integer wbegin;
doublereal safmin;
extern /* Subroutine */ int dlarrj_(integer *, doublereal *, doublereal *,
integer *, integer *, doublereal *, integer *, doublereal *,
doublereal *, doublereal *, integer *, doublereal *, doublereal *,
integer *), xerbla_(char *, integer *);
doublereal bignum;
integer inderr, iindwk, indgrs, offset;
extern doublereal dlanst_(char *, integer *, doublereal *, doublereal *);
extern /* Subroutine */ int dlarrr_(integer *, doublereal *, doublereal *,
integer *), dlarrv_(integer *, doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *, integer *, integer *,
integer *, integer *, doublereal *, doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *, integer *, integer *,
doublereal *, doublereal *, integer *, integer *, doublereal *,
integer *, integer *), dlasrt_(char *, integer *, doublereal *,
integer *);
doublereal thresh;
integer iinspl, ifirst, indwrk, liwmin, nzcmin;
doublereal pivmin;
integer nsplit;
doublereal smlnum;
logical lquery, zquery;
/* -- LAPACK computational routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* DSTEMR computes selected eigenvalues and, optionally, eigenvectors */
/* of a real symmetric tridiagonal matrix T. Any such unreduced matrix has */
/* a well defined set of pairwise different real eigenvalues, the corresponding */
/* real eigenvectors are pairwise orthogonal. */
/* The spectrum may be computed either completely or partially by specifying */
/* either an interval (VL,VU] or a range of indices IL:IU for the desired */
/* eigenvalues. */
/* Depending on the number of desired eigenvalues, these are computed either */
/* by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are */
/* computed by the use of various suitable L D L^T factorizations near clusters */
/* of close eigenvalues (referred to as RRRs, Relatively Robust */
/* Representations). An informal sketch of the algorithm follows. */
/* For each unreduced block (submatrix) of T, */
/* (a) Compute T - sigma I = L D L^T, so that L and D */
/* define all the wanted eigenvalues to high relative accuracy. */
/* This means that small relative changes in the entries of D and L */
/* cause only small relative changes in the eigenvalues and */
/* eigenvectors. The standard (unfactored) representation of the */
/* tridiagonal matrix T does not have this property in general. */
/* (b) Compute the eigenvalues to suitable accuracy. */
/* If the eigenvectors are desired, the algorithm attains full */
/* accuracy of the computed eigenvalues only right before */
/* the corresponding vectors have to be computed, see steps c) and d). */
/* (c) For each cluster of close eigenvalues, select a new */
/* shift close to the cluster, find a new factorization, and refine */
/* the shifted eigenvalues to suitable accuracy. */
/* (d) For each eigenvalue with a large enough relative separation compute */
/* the corresponding eigenvector by forming a rank revealing twisted */
/* factorization. Go back to (c) for any clusters that remain. */
/* For more details, see: */
/* - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations */
/* to compute orthogonal eigenvectors of symmetric tridiagonal matrices," */
/* Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004. */
/* - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and */
/* Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25, */
/* 2004. Also LAPACK Working Note 154. */
/* - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric */
/* tridiagonal eigenvalue/eigenvector problem", */
/* Computer Science Division Technical Report No. UCB/CSD-97-971, */
/* UC Berkeley, May 1997. */
/* Notes: */
/* 1.DSTEMR works only on machines which follow IEEE-754 */
/* floating-point standard in their handling of infinities and NaNs. */
/* This permits the use of efficient inner loops avoiding a check for */
/* zero divisors. */
/* Arguments */
/* ========= */
/* JOBZ (input) CHARACTER*1 */
/* = 'N': Compute eigenvalues only; */
/* = 'V': Compute eigenvalues and eigenvectors. */
/* RANGE (input) CHARACTER*1 */
/* = 'A': all eigenvalues will be found. */
/* = 'V': all eigenvalues in the half-open interval (VL,VU] */
/* will be found. */
/* = 'I': the IL-th through IU-th eigenvalues will be found. */
/* N (input) INTEGER */
/* The order of the matrix. N >= 0. */
/* D (input/output) DOUBLE PRECISION array, dimension (N) */
/* On entry, the N diagonal elements of the tridiagonal matrix */
/* T. On exit, D is overwritten. */
/* E (input/output) DOUBLE PRECISION array, dimension (N) */
/* On entry, the (N-1) subdiagonal elements of the tridiagonal */
/* matrix T in elements 1 to N-1 of E. E(N) need not be set on */
/* input, but is used internally as workspace. */
/* On exit, E is overwritten. */
/* VL (input) DOUBLE PRECISION */
/* VU (input) DOUBLE PRECISION */
/* If RANGE='V', the lower and upper bounds of the interval to */
/* be searched for eigenvalues. VL < VU. */
/* Not referenced if RANGE = 'A' or 'I'. */
/* IL (input) INTEGER */
/* IU (input) INTEGER */
/* If RANGE='I', the indices (in ascending order) of the */
/* smallest and largest eigenvalues to be returned. */
/* 1 <= IL <= IU <= N, if N > 0. */
/* Not referenced if RANGE = 'A' or 'V'. */
/* M (output) INTEGER */
/* The total number of eigenvalues found. 0 <= M <= N. */
/* If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. */
/* W (output) DOUBLE PRECISION array, dimension (N) */
/* The first M elements contain the selected eigenvalues in */
/* ascending order. */
/* Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M) ) */
/* If JOBZ = 'V', and if INFO = 0, then the first M columns of Z */
/* contain the orthonormal eigenvectors of the matrix T */
/* corresponding to the selected eigenvalues, with the i-th */
/* column of Z holding the eigenvector associated with W(i). */
/* If JOBZ = 'N', then Z is not referenced. */
/* Note: the user must ensure that at least max(1,M) columns are */
/* supplied in the array Z; if RANGE = 'V', the exact value of M */
/* is not known in advance and can be computed with a workspace */
/* query by setting NZC = -1, see below. */
/* LDZ (input) INTEGER */
/* The leading dimension of the array Z. LDZ >= 1, and if */
/* JOBZ = 'V', then LDZ >= max(1,N). */
/* NZC (input) INTEGER */
/* The number of eigenvectors to be held in the array Z. */
/* If RANGE = 'A', then NZC >= max(1,N). */
/* If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU]. */
/* If RANGE = 'I', then NZC >= IU-IL+1. */
/* If NZC = -1, then a workspace query is assumed; the */
/* routine calculates the number of columns of the array Z that */
/* are needed to hold the eigenvectors. */
/* This value is returned as the first entry of the Z array, and */
/* no error message related to NZC is issued by XERBLA. */
/* ISUPPZ (output) INTEGER ARRAY, dimension ( 2*max(1,M) ) */
/* The support of the eigenvectors in Z, i.e., the indices */
/* indicating the nonzero elements in Z. The i-th computed eigenvector */
/* is nonzero only in elements ISUPPZ( 2*i-1 ) through */
/* ISUPPZ( 2*i ). This is relevant in the case when the matrix */
/* is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0. */
/* TRYRAC (input/output) LOGICAL */
/* If TRYRAC.EQ..TRUE., indicates that the code should check whether */
/* the tridiagonal matrix defines its eigenvalues to high relative */
/* accuracy. If so, the code uses relative-accuracy preserving */
/* algorithms that might be (a bit) slower depending on the matrix. */
/* If the matrix does not define its eigenvalues to high relative */
/* accuracy, the code can uses possibly faster algorithms. */
/* If TRYRAC.EQ..FALSE., the code is not required to guarantee */
/* relatively accurate eigenvalues and can use the fastest possible */
/* techniques. */
/* On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix */
/* does not define its eigenvalues to high relative accuracy. */
/* WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) */
/* On exit, if INFO = 0, WORK(1) returns the optimal */
/* (and minimal) LWORK. */
/* LWORK (input) INTEGER */
/* The dimension of the array WORK. LWORK >= max(1,18*N) */
/* if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'. */
/* If LWORK = -1, then a workspace query is assumed; the routine */
/* only calculates the optimal size of the WORK array, returns */
/* this value as the first entry of the WORK array, and no error */
/* message related to LWORK is issued by XERBLA. */
/* IWORK (workspace/output) INTEGER array, dimension (LIWORK) */
/* On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. */
/* LIWORK (input) INTEGER */
/* The dimension of the array IWORK. LIWORK >= max(1,10*N) */
/* if the eigenvectors are desired, and LIWORK >= max(1,8*N) */
/* if only the eigenvalues are to be computed. */
/* If LIWORK = -1, then a workspace query is assumed; the */
/* routine only calculates the optimal size of the IWORK array, */
/* returns this value as the first entry of the IWORK array, and */
/* no error message related to LIWORK is issued by XERBLA. */
/* INFO (output) INTEGER */
/* On exit, INFO */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* > 0: if INFO = 1X, internal error in DLARRE, */
/* if INFO = 2X, internal error in DLARRV. */
/* Here, the digit X = ABS( IINFO ) < 10, where IINFO is */
/* the nonzero error code returned by DLARRE or */
/* DLARRV, respectively. */
/* Further Details */
/* =============== */
/* Based on contributions by */
/* Beresford Parlett, University of California, Berkeley, USA */
/* Jim Demmel, University of California, Berkeley, USA */
/* Inderjit Dhillon, University of Texas, Austin, USA */
/* Osni Marques, LBNL/NERSC, USA */
/* Christof Voemel, University of California, Berkeley, USA */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--d__;
--e;
--w;
z_dim1 = *ldz;
z_offset = 1 + z_dim1;
z__ -= z_offset;
--isuppz;
--work;
--iwork;
/* Function Body */
wantz = lsame_(jobz, "V");
alleig = lsame_(range, "A");
valeig = lsame_(range, "V");
indeig = lsame_(range, "I");
lquery = *lwork == -1 || *liwork == -1;
zquery = *nzc == -1;
/* DSTEMR needs WORK of size 6*N, IWORK of size 3*N. */
/* In addition, DLARRE needs WORK of size 6*N, IWORK of size 5*N. */
/* Furthermore, DLARRV needs WORK of size 12*N, IWORK of size 7*N. */
if (wantz) {
lwmin = *n * 18;
liwmin = *n * 10;
} else {
/* need less workspace if only the eigenvalues are wanted */
lwmin = *n * 12;
liwmin = *n << 3;
}
wl = 0.;
wu = 0.;
iil = 0;
iiu = 0;
if (valeig) {
/* We do not reference VL, VU in the cases RANGE = 'I','A' */
/* The interval (WL, WU] contains all the wanted eigenvalues. */
/* It is either given by the user or computed in DLARRE. */
wl = *vl;
wu = *vu;
} else if (indeig) {
/* We do not reference IL, IU in the cases RANGE = 'V','A' */
iil = *il;
iiu = *iu;
}
*info = 0;
if (! (wantz || lsame_(jobz, "N"))) {
*info = -1;
} else if (! (alleig || valeig || indeig)) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else if (valeig && *n > 0 && wu <= wl) {
*info = -7;
} else if (indeig && (iil < 1 || iil > *n)) {
*info = -8;
} else if (indeig && (iiu < iil || iiu > *n)) {
*info = -9;
} else if (*ldz < 1 || wantz && *ldz < *n) {
*info = -13;
} else if (*lwork < lwmin && ! lquery) {
*info = -17;
} else if (*liwork < liwmin && ! lquery) {
*info = -19;
}
/* Get machine constants. */
safmin = dlamch_("Safe minimum");
eps = dlamch_("Precision");
smlnum = safmin / eps;
bignum = 1. / smlnum;
rmin = sqrt(smlnum);
/* Computing MIN */
d__1 = sqrt(bignum), d__2 = 1. / sqrt(sqrt(safmin));
rmax = min(d__1,d__2);
if (*info == 0) {
work[1] = (doublereal) lwmin;
iwork[1] = liwmin;
if (wantz && alleig) {
nzcmin = *n;
} else if (wantz && valeig) {
dlarrc_("T", n, vl, vu, &d__[1], &e[1], &safmin, &nzcmin, &itmp, &
itmp2, info);
} else if (wantz && indeig) {
nzcmin = iiu - iil + 1;
} else {
/* WANTZ .EQ. FALSE. */
nzcmin = 0;
}
if (zquery && *info == 0) {
z__[z_dim1 + 1] = (doublereal) nzcmin;
} else if (*nzc < nzcmin && ! zquery) {
*info = -14;
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DSTEMR", &i__1);
return 0;
} else if (lquery || zquery) {
return 0;
}
/* Handle N = 0, 1, and 2 cases immediately */
*m = 0;
if (*n == 0) {
return 0;
}
if (*n == 1) {
if (alleig || indeig) {
*m = 1;
w[1] = d__[1];
} else {
if (wl < d__[1] && wu >= d__[1]) {
*m = 1;
w[1] = d__[1];
}
}
if (wantz && ! zquery) {
z__[z_dim1 + 1] = 1.;
isuppz[1] = 1;
isuppz[2] = 1;
}
return 0;
}
if (*n == 2) {
if (! wantz) {
dlae2_(&d__[1], &e[1], &d__[2], &r1, &r2);
} else if (wantz && ! zquery) {
dlaev2_(&d__[1], &e[1], &d__[2], &r1, &r2, &cs, &sn);
}
if (alleig || valeig && r2 > wl && r2 <= wu || indeig && iil == 1) {
++(*m);
w[*m] = r2;
if (wantz && ! zquery) {
z__[*m * z_dim1 + 1] = -sn;
z__[*m * z_dim1 + 2] = cs;
/* Note: At most one of SN and CS can be zero. */
if (sn != 0.) {
if (cs != 0.) {
isuppz[(*m << 1) - 1] = 1;
isuppz[(*m << 1) - 1] = 2;
} else {
isuppz[(*m << 1) - 1] = 1;
isuppz[(*m << 1) - 1] = 1;
}
} else {
isuppz[(*m << 1) - 1] = 2;
isuppz[*m * 2] = 2;
}
}
}
if (alleig || valeig && r1 > wl && r1 <= wu || indeig && iiu == 2) {
++(*m);
w[*m] = r1;
if (wantz && ! zquery) {
z__[*m * z_dim1 + 1] = cs;
z__[*m * z_dim1 + 2] = sn;
/* Note: At most one of SN and CS can be zero. */
if (sn != 0.) {
if (cs != 0.) {
isuppz[(*m << 1) - 1] = 1;
isuppz[(*m << 1) - 1] = 2;
} else {
isuppz[(*m << 1) - 1] = 1;
isuppz[(*m << 1) - 1] = 1;
}
} else {
isuppz[(*m << 1) - 1] = 2;
isuppz[*m * 2] = 2;
}
}
}
return 0;
}
/* Continue with general N */
indgrs = 1;
inderr = (*n << 1) + 1;
indgp = *n * 3 + 1;
indd = (*n << 2) + 1;
inde2 = *n * 5 + 1;
indwrk = *n * 6 + 1;
iinspl = 1;
iindbl = *n + 1;
iindw = (*n << 1) + 1;
iindwk = *n * 3 + 1;
/* Scale matrix to allowable range, if necessary. */
/* The allowable range is related to the PIVMIN parameter; see the */
/* comments in DLARRD. The preference for scaling small values */
/* up is heuristic; we expect users' matrices not to be close to the */
/* RMAX threshold. */
scale = 1.;
tnrm = dlanst_("M", n, &d__[1], &e[1]);
if (tnrm > 0. && tnrm < rmin) {
scale = rmin / tnrm;
} else if (tnrm > rmax) {
scale = rmax / tnrm;
}
if (scale != 1.) {
dscal_(n, &scale, &d__[1], &c__1);
i__1 = *n - 1;
dscal_(&i__1, &scale, &e[1], &c__1);
tnrm *= scale;
if (valeig) {
/* If eigenvalues in interval have to be found, */
/* scale (WL, WU] accordingly */
wl *= scale;
wu *= scale;
}
}
/* Compute the desired eigenvalues of the tridiagonal after splitting */
/* into smaller subblocks if the corresponding off-diagonal elements */
/* are small */
/* THRESH is the splitting parameter for DLARRE */
/* A negative THRESH forces the old splitting criterion based on the */
/* size of the off-diagonal. A positive THRESH switches to splitting */
/* which preserves relative accuracy. */
if (*tryrac) {
/* Test whether the matrix warrants the more expensive relative approach. */
dlarrr_(n, &d__[1], &e[1], &iinfo);
} else {
/* The user does not care about relative accurately eigenvalues */
iinfo = -1;
}
/* Set the splitting criterion */
if (iinfo == 0) {
thresh = eps;
} else {
thresh = -eps;
/* relative accuracy is desired but T does not guarantee it */
*tryrac = FALSE_;
}
if (*tryrac) {
/* Copy original diagonal, needed to guarantee relative accuracy */
dcopy_(n, &d__[1], &c__1, &work[indd], &c__1);
}
/* Store the squares of the offdiagonal values of T */
i__1 = *n - 1;
for (j = 1; j <= i__1; ++j) {
/* Computing 2nd power */
d__1 = e[j];
work[inde2 + j - 1] = d__1 * d__1;
/* L5: */
}
/* Set the tolerance parameters for bisection */
if (! wantz) {
/* DLARRE computes the eigenvalues to full precision. */
rtol1 = eps * 4.;
rtol2 = eps * 4.;
} else {
/* DLARRE computes the eigenvalues to less than full precision. */
/* DLARRV will refine the eigenvalue approximations, and we can */
/* need less accurate initial bisection in DLARRE. */
/* Note: these settings do only affect the subset case and DLARRE */
rtol1 = sqrt(eps);
/* Computing MAX */
d__1 = sqrt(eps) * .005, d__2 = eps * 4.;
rtol2 = max(d__1,d__2);
}
dlarre_(range, n, &wl, &wu, &iil, &iiu, &d__[1], &e[1], &work[inde2], &
rtol1, &rtol2, &thresh, &nsplit, &iwork[iinspl], m, &w[1], &work[
inderr], &work[indgp], &iwork[iindbl], &iwork[iindw], &work[
indgrs], &pivmin, &work[indwrk], &iwork[iindwk], &iinfo);
if (iinfo != 0) {
*info = abs(iinfo) + 10;
return 0;
}
/* Note that if RANGE .NE. 'V', DLARRE computes bounds on the desired */
/* part of the spectrum. All desired eigenvalues are contained in */
/* (WL,WU] */
if (wantz) {
/* Compute the desired eigenvectors corresponding to the computed */
/* eigenvalues */
dlarrv_(n, &wl, &wu, &d__[1], &e[1], &pivmin, &iwork[iinspl], m, &
c__1, m, &c_b18, &rtol1, &rtol2, &w[1], &work[inderr], &work[
indgp], &iwork[iindbl], &iwork[iindw], &work[indgrs], &z__[
z_offset], ldz, &isuppz[1], &work[indwrk], &iwork[iindwk], &
iinfo);
if (iinfo != 0) {
*info = abs(iinfo) + 20;
return 0;
}
} else {
/* DLARRE computes eigenvalues of the (shifted) root representation */
/* DLARRV returns the eigenvalues of the unshifted matrix. */
/* However, if the eigenvectors are not desired by the user, we need */
/* to apply the corresponding shifts from DLARRE to obtain the */
/* eigenvalues of the original matrix. */
i__1 = *m;
for (j = 1; j <= i__1; ++j) {
itmp = iwork[iindbl + j - 1];
w[j] += e[iwork[iinspl + itmp - 1]];
/* L20: */
}
}
if (*tryrac) {
/* Refine computed eigenvalues so that they are relatively accurate */
/* with respect to the original matrix T. */
ibegin = 1;
wbegin = 1;
i__1 = iwork[iindbl + *m - 1];
for (jblk = 1; jblk <= i__1; ++jblk) {
iend = iwork[iinspl + jblk - 1];
in = iend - ibegin + 1;
wend = wbegin - 1;
/* check if any eigenvalues have to be refined in this block */
L36:
if (wend < *m) {
if (iwork[iindbl + wend] == jblk) {
++wend;
goto L36;
}
}
if (wend < wbegin) {
ibegin = iend + 1;
goto L39;
}
offset = iwork[iindw + wbegin - 1] - 1;
ifirst = iwork[iindw + wbegin - 1];
ilast = iwork[iindw + wend - 1];
rtol2 = eps * 4.;
dlarrj_(&in, &work[indd + ibegin - 1], &work[inde2 + ibegin - 1],
&ifirst, &ilast, &rtol2, &offset, &w[wbegin], &work[
inderr + wbegin - 1], &work[indwrk], &iwork[iindwk], &
pivmin, &tnrm, &iinfo);
ibegin = iend + 1;
wbegin = wend + 1;
L39:
;
}
}
/* If matrix was scaled, then rescale eigenvalues appropriately. */
if (scale != 1.) {
d__1 = 1. / scale;
dscal_(m, &d__1, &w[1], &c__1);
}
/* If eigenvalues are not in increasing order, then sort them, */
/* possibly along with eigenvectors. */
if (nsplit > 1) {
if (! wantz) {
dlasrt_("I", m, &w[1], &iinfo);
if (iinfo != 0) {
*info = 3;
return 0;
}
} else {
i__1 = *m - 1;
for (j = 1; j <= i__1; ++j) {
i__ = 0;
tmp = w[j];
i__2 = *m;
for (jj = j + 1; jj <= i__2; ++jj) {
if (w[jj] < tmp) {
i__ = jj;
tmp = w[jj];
}
/* L50: */
}
if (i__ != 0) {
w[i__] = w[j];
w[j] = tmp;
if (wantz) {
dswap_(n, &z__[i__ * z_dim1 + 1], &c__1, &z__[j *
z_dim1 + 1], &c__1);
itmp = isuppz[(i__ << 1) - 1];
isuppz[(i__ << 1) - 1] = isuppz[(j << 1) - 1];
isuppz[(j << 1) - 1] = itmp;
itmp = isuppz[i__ * 2];
isuppz[i__ * 2] = isuppz[j * 2];
isuppz[j * 2] = itmp;
}
}
/* L60: */
}
}
}
work[1] = (doublereal) lwmin;
iwork[1] = liwmin;
return 0;
/* End of DSTEMR */
} /* dstemr_ */
/* Subroutine */
int dlarre_(char *range, integer *n, doublereal *vl, doublereal *vu, integer *il, integer *iu, doublereal *d__, doublereal *e, doublereal *e2, doublereal *rtol1, doublereal *rtol2, doublereal * spltol, integer *nsplit, integer *isplit, integer *m, doublereal *w, doublereal *werr, doublereal *wgap, integer *iblock, integer *indexw, doublereal *gers, doublereal *pivmin, doublereal *work, integer * iwork, integer *info)
{
/* System generated locals */
integer i__1, i__2;
doublereal d__1, d__2, d__3;
/* Builtin functions */
double sqrt(doublereal), log(doublereal);
/* Local variables */
integer i__, j;
doublereal s1, s2;
integer mb;
doublereal gl;
integer in, mm;
doublereal gu;
integer cnt;
doublereal eps, tau, tmp, rtl;
integer cnt1, cnt2;
doublereal tmp1, eabs;
integer iend, jblk;
doublereal eold;
integer indl;
doublereal dmax__, emax;
integer wend, idum, indu;
doublereal rtol;
integer iseed[4];
doublereal avgap, sigma;
extern logical lsame_(char *, char *);
integer iinfo;
extern /* Subroutine */
int dcopy_(integer *, doublereal *, integer *, doublereal *, integer *);
logical norep;
extern /* Subroutine */
int dlasq2_(integer *, doublereal *, integer *);
extern doublereal dlamch_(char *);
integer ibegin;
logical forceb;
integer irange;
doublereal sgndef;
extern /* Subroutine */
int dlarra_(integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, integer *, integer *), dlarrb_(integer *, doublereal *, doublereal *, integer *, integer *, doublereal *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, doublereal *, doublereal *, integer *, integer *), dlarrc_(char * , integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, integer *, integer *, integer *);
integer wbegin;
extern /* Subroutine */
int dlarrd_(char *, char *, integer *, doublereal *, doublereal *, integer *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, integer * , integer *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, integer *, doublereal *, integer *, integer *);
doublereal safmin, spdiam;
extern /* Subroutine */
int dlarrk_(integer *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, integer *);
logical usedqd;
doublereal clwdth, isleft;
extern /* Subroutine */
int dlarnv_(integer *, integer *, integer *, doublereal *);
doublereal isrght, bsrtol, dpivot;
/* -- LAPACK auxiliary routine (version 3.4.2) -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
/* September 2012 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Parameter adjustments */
--iwork;
--work;
--gers;
--indexw;
--iblock;
--wgap;
--werr;
--w;
--isplit;
--e2;
--e;
--d__;
/* Function Body */
*info = 0;
/* Decode RANGE */
irange = 0;
if (lsame_(range, "A"))
{
irange = 1;
}
else if (lsame_(range, "V"))
{
irange = 3;
}
else if (lsame_(range, "I"))
{
irange = 2;
}
*m = 0;
/* Get machine constants */
safmin = dlamch_("S");
eps = dlamch_("P");
/* Set parameters */
rtl = sqrt(eps);
bsrtol = sqrt(eps);
/* Treat case of 1x1 matrix for quick return */
if (*n == 1)
{
if (irange == 1 || irange == 3 && d__[1] > *vl && d__[1] <= *vu || irange == 2 && *il == 1 && *iu == 1)
{
*m = 1;
w[1] = d__[1];
/* The computation error of the eigenvalue is zero */
werr[1] = 0.;
wgap[1] = 0.;
iblock[1] = 1;
indexw[1] = 1;
gers[1] = d__[1];
gers[2] = d__[1];
}
/* store the shift for the initial RRR, which is zero in this case */
e[1] = 0.;
return 0;
}
/* General case: tridiagonal matrix of order > 1 */
/* Init WERR, WGAP. Compute Gerschgorin intervals and spectral diameter. */
/* Compute maximum off-diagonal entry and pivmin. */
gl = d__[1];
gu = d__[1];
eold = 0.;
emax = 0.;
e[*n] = 0.;
i__1 = *n;
for (i__ = 1;
i__ <= i__1;
++i__)
{
werr[i__] = 0.;
wgap[i__] = 0.;
eabs = (d__1 = e[i__], abs(d__1));
if (eabs >= emax)
{
emax = eabs;
}
tmp1 = eabs + eold;
gers[(i__ << 1) - 1] = d__[i__] - tmp1;
/* Computing MIN */
d__1 = gl;
d__2 = gers[(i__ << 1) - 1]; // , expr subst
gl = min(d__1,d__2);
gers[i__ * 2] = d__[i__] + tmp1;
/* Computing MAX */
d__1 = gu;
d__2 = gers[i__ * 2]; // , expr subst
gu = max(d__1,d__2);
eold = eabs;
/* L5: */
}
/* The minimum pivot allowed in the Sturm sequence for T */
/* Computing MAX */
/* Computing 2nd power */
d__3 = emax;
d__1 = 1.;
d__2 = d__3 * d__3; // , expr subst
*pivmin = safmin * max(d__1,d__2);
/* Compute spectral diameter. The Gerschgorin bounds give an */
/* estimate that is wrong by at most a factor of SQRT(2) */
spdiam = gu - gl;
/* Compute splitting points */
dlarra_(n, &d__[1], &e[1], &e2[1], spltol, &spdiam, nsplit, &isplit[1], & iinfo);
/* Can force use of bisection instead of faster DQDS. */
/* Option left in the code for future multisection work. */
forceb = FALSE_;
/* Initialize USEDQD, DQDS should be used for ALLRNG unless someone */
/* explicitly wants bisection. */
usedqd = irange == 1 && ! forceb;
if (irange == 1 && ! forceb)
{
/* Set interval [VL,VU] that contains all eigenvalues */
*vl = gl;
*vu = gu;
}
else
{
/* We call DLARRD to find crude approximations to the eigenvalues */
/* in the desired range. In case IRANGE = INDRNG, we also obtain the */
/* interval (VL,VU] that contains all the wanted eigenvalues. */
/* An interval [LEFT,RIGHT] has converged if */
/* RIGHT-LEFT.LT.RTOL*MAX(ABS(LEFT),ABS(RIGHT)) */
/* DLARRD needs a WORK of size 4*N, IWORK of size 3*N */
dlarrd_(range, "B", n, vl, vu, il, iu, &gers[1], &bsrtol, &d__[1], &e[ 1], &e2[1], pivmin, nsplit, &isplit[1], &mm, &w[1], &werr[1], vl, vu, &iblock[1], &indexw[1], &work[1], &iwork[1], &iinfo);
if (iinfo != 0)
{
*info = -1;
return 0;
}
/* Make sure that the entries M+1 to N in W, WERR, IBLOCK, INDEXW are 0 */
i__1 = *n;
for (i__ = mm + 1;
i__ <= i__1;
++i__)
{
w[i__] = 0.;
werr[i__] = 0.;
iblock[i__] = 0;
indexw[i__] = 0;
/* L14: */
}
}
/* ** */
/* Loop over unreduced blocks */
ibegin = 1;
wbegin = 1;
i__1 = *nsplit;
for (jblk = 1;
jblk <= i__1;
++jblk)
{
iend = isplit[jblk];
in = iend - ibegin + 1;
/* 1 X 1 block */
if (in == 1)
{
if (irange == 1 || irange == 3 && d__[ibegin] > *vl && d__[ibegin] <= *vu || irange == 2 && iblock[wbegin] == jblk)
{
++(*m);
w[*m] = d__[ibegin];
werr[*m] = 0.;
/* The gap for a single block doesn't matter for the later */
/* algorithm and is assigned an arbitrary large value */
wgap[*m] = 0.;
iblock[*m] = jblk;
indexw[*m] = 1;
++wbegin;
}
/* E( IEND ) holds the shift for the initial RRR */
e[iend] = 0.;
ibegin = iend + 1;
goto L170;
}
/* Blocks of size larger than 1x1 */
/* E( IEND ) will hold the shift for the initial RRR, for now set it =0 */
e[iend] = 0.;
/* Find local outer bounds GL,GU for the block */
gl = d__[ibegin];
gu = d__[ibegin];
i__2 = iend;
for (i__ = ibegin;
i__ <= i__2;
++i__)
{
/* Computing MIN */
d__1 = gers[(i__ << 1) - 1];
gl = min(d__1,gl);
/* Computing MAX */
d__1 = gers[i__ * 2];
gu = max(d__1,gu);
/* L15: */
}
spdiam = gu - gl;
if (! (irange == 1 && ! forceb))
{
/* Count the number of eigenvalues in the current block. */
mb = 0;
i__2 = mm;
for (i__ = wbegin;
i__ <= i__2;
++i__)
{
if (iblock[i__] == jblk)
{
++mb;
}
else
{
goto L21;
}
/* L20: */
}
L21:
if (mb == 0)
{
/* No eigenvalue in the current block lies in the desired range */
/* E( IEND ) holds the shift for the initial RRR */
e[iend] = 0.;
ibegin = iend + 1;
goto L170;
}
else
{
/* Decide whether dqds or bisection is more efficient */
usedqd = (doublereal) mb > in * .5 && ! forceb;
wend = wbegin + mb - 1;
/* Calculate gaps for the current block */
/* In later stages, when representations for individual */
/* eigenvalues are different, we use SIGMA = E( IEND ). */
sigma = 0.;
i__2 = wend - 1;
for (i__ = wbegin;
i__ <= i__2;
++i__)
{
/* Computing MAX */
d__1 = 0.;
d__2 = w[i__ + 1] - werr[i__ + 1] - (w[i__] + werr[i__]); // , expr subst
wgap[i__] = max(d__1,d__2);
/* L30: */
}
/* Computing MAX */
d__1 = 0.;
d__2 = *vu - sigma - (w[wend] + werr[wend]); // , expr subst
wgap[wend] = max(d__1,d__2);
/* Find local index of the first and last desired evalue. */
indl = indexw[wbegin];
indu = indexw[wend];
}
}
if (irange == 1 && ! forceb || usedqd)
{
/* Case of DQDS */
/* Find approximations to the extremal eigenvalues of the block */
dlarrk_(&in, &c__1, &gl, &gu, &d__[ibegin], &e2[ibegin], pivmin, & rtl, &tmp, &tmp1, &iinfo);
if (iinfo != 0)
{
*info = -1;
return 0;
}
/* Computing MAX */
d__2 = gl;
d__3 = tmp - tmp1 - eps * 100. * (d__1 = tmp - tmp1, abs(d__1)); // , expr subst
isleft = max(d__2,d__3);
dlarrk_(&in, &in, &gl, &gu, &d__[ibegin], &e2[ibegin], pivmin, & rtl, &tmp, &tmp1, &iinfo);
if (iinfo != 0)
{
*info = -1;
return 0;
}
/* Computing MIN */
d__2 = gu;
d__3 = tmp + tmp1 + eps * 100. * (d__1 = tmp + tmp1, abs(d__1)); // , expr subst
isrght = min(d__2,d__3);
/* Improve the estimate of the spectral diameter */
spdiam = isrght - isleft;
}
else
{
/* Case of bisection */
/* Find approximations to the wanted extremal eigenvalues */
/* Computing MAX */
d__2 = gl;
d__3 = w[wbegin] - werr[wbegin] - eps * 100. * (d__1 = w[wbegin] - werr[wbegin], abs(d__1)); // , expr subst
isleft = max(d__2,d__3);
/* Computing MIN */
d__2 = gu;
d__3 = w[wend] + werr[wend] + eps * 100. * (d__1 = w[ wend] + werr[wend], abs(d__1)); // , expr subst
isrght = min(d__2,d__3);
}
/* Decide whether the base representation for the current block */
/* L_JBLK D_JBLK L_JBLK^T = T_JBLK - sigma_JBLK I */
/* should be on the left or the right end of the current block. */
/* The strategy is to shift to the end which is "more populated" */
/* Furthermore, decide whether to use DQDS for the computation of */
/* the eigenvalue approximations at the end of DLARRE or bisection. */
/* dqds is chosen if all eigenvalues are desired or the number of */
/* eigenvalues to be computed is large compared to the blocksize. */
if (irange == 1 && ! forceb)
{
/* If all the eigenvalues have to be computed, we use dqd */
usedqd = TRUE_;
/* INDL is the local index of the first eigenvalue to compute */
indl = 1;
indu = in;
/* MB = number of eigenvalues to compute */
mb = in;
wend = wbegin + mb - 1;
/* Define 1/4 and 3/4 points of the spectrum */
s1 = isleft + spdiam * .25;
s2 = isrght - spdiam * .25;
}
else
{
/* DLARRD has computed IBLOCK and INDEXW for each eigenvalue */
/* approximation. */
/* choose sigma */
if (usedqd)
{
s1 = isleft + spdiam * .25;
s2 = isrght - spdiam * .25;
}
else
{
tmp = min(isrght,*vu) - max(isleft,*vl);
s1 = max(isleft,*vl) + tmp * .25;
s2 = min(isrght,*vu) - tmp * .25;
}
}
/* Compute the negcount at the 1/4 and 3/4 points */
if (mb > 1)
{
dlarrc_("T", &in, &s1, &s2, &d__[ibegin], &e[ibegin], pivmin, & cnt, &cnt1, &cnt2, &iinfo);
}
if (mb == 1)
{
sigma = gl;
sgndef = 1.;
}
else if (cnt1 - indl >= indu - cnt2)
{
if (irange == 1 && ! forceb)
{
sigma = max(isleft,gl);
}
else if (usedqd)
{
/* use Gerschgorin bound as shift to get pos def matrix */
/* for dqds */
sigma = isleft;
}
else
{
/* use approximation of the first desired eigenvalue of the */
/* block as shift */
sigma = max(isleft,*vl);
}
sgndef = 1.;
}
else
{
if (irange == 1 && ! forceb)
{
sigma = min(isrght,gu);
}
else if (usedqd)
{
/* use Gerschgorin bound as shift to get neg def matrix */
/* for dqds */
sigma = isrght;
}
else
{
/* use approximation of the first desired eigenvalue of the */
/* block as shift */
sigma = min(isrght,*vu);
}
sgndef = -1.;
}
/* An initial SIGMA has been chosen that will be used for computing */
/* T - SIGMA I = L D L^T */
/* Define the increment TAU of the shift in case the initial shift */
/* needs to be refined to obtain a factorization with not too much */
/* element growth. */
if (usedqd)
{
/* The initial SIGMA was to the outer end of the spectrum */
/* the matrix is definite and we need not retreat. */
tau = spdiam * eps * *n + *pivmin * 2.;
/* Computing MAX */
d__1 = tau;
d__2 = eps * 2. * abs(sigma); // , expr subst
tau = max(d__1,d__2);
}
else
{
if (mb > 1)
{
clwdth = w[wend] + werr[wend] - w[wbegin] - werr[wbegin];
avgap = (d__1 = clwdth / (doublereal) (wend - wbegin), abs( d__1));
if (sgndef == 1.)
{
/* Computing MAX */
d__1 = wgap[wbegin];
tau = max(d__1,avgap) * .5;
/* Computing MAX */
d__1 = tau;
d__2 = werr[wbegin]; // , expr subst
tau = max(d__1,d__2);
}
else
{
/* Computing MAX */
d__1 = wgap[wend - 1];
tau = max(d__1,avgap) * .5;
/* Computing MAX */
d__1 = tau;
d__2 = werr[wend]; // , expr subst
tau = max(d__1,d__2);
}
}
else
{
tau = werr[wbegin];
}
}
for (idum = 1;
idum <= 6;
++idum)
{
/* Compute L D L^T factorization of tridiagonal matrix T - sigma I. */
/* Store D in WORK(1:IN), L in WORK(IN+1:2*IN), and reciprocals of */
/* pivots in WORK(2*IN+1:3*IN) */
dpivot = d__[ibegin] - sigma;
work[1] = dpivot;
dmax__ = abs(work[1]);
j = ibegin;
i__2 = in - 1;
for (i__ = 1;
i__ <= i__2;
++i__)
{
work[(in << 1) + i__] = 1. / work[i__];
tmp = e[j] * work[(in << 1) + i__];
work[in + i__] = tmp;
dpivot = d__[j + 1] - sigma - tmp * e[j];
work[i__ + 1] = dpivot;
/* Computing MAX */
d__1 = dmax__;
d__2 = abs(dpivot); // , expr subst
dmax__ = max(d__1,d__2);
++j;
/* L70: */
}
/* check for element growth */
if (dmax__ > spdiam * 64.)
{
norep = TRUE_;
}
else
{
norep = FALSE_;
}
if (usedqd && ! norep)
{
/* Ensure the definiteness of the representation */
/* All entries of D (of L D L^T) must have the same sign */
i__2 = in;
for (i__ = 1;
i__ <= i__2;
++i__)
{
tmp = sgndef * work[i__];
if (tmp < 0.)
{
norep = TRUE_;
}
/* L71: */
}
}
if (norep)
{
/* Note that in the case of IRANGE=ALLRNG, we use the Gerschgorin */
/* shift which makes the matrix definite. So we should end up */
/* here really only in the case of IRANGE = VALRNG or INDRNG. */
if (idum == 5)
{
if (sgndef == 1.)
{
/* The fudged Gerschgorin shift should succeed */
sigma = gl - spdiam * 2. * eps * *n - *pivmin * 4.;
}
else
{
sigma = gu + spdiam * 2. * eps * *n + *pivmin * 4.;
}
}
else
{
sigma -= sgndef * tau;
tau *= 2.;
}
}
else
{
/* an initial RRR is found */
goto L83;
}
/* L80: */
}
/* if the program reaches this point, no base representation could be */
/* found in MAXTRY iterations. */
*info = 2;
return 0;
L83: /* At this point, we have found an initial base representation */
/* T - SIGMA I = L D L^T with not too much element growth. */
/* Store the shift. */
e[iend] = sigma;
/* Store D and L. */
dcopy_(&in, &work[1], &c__1, &d__[ibegin], &c__1);
i__2 = in - 1;
dcopy_(&i__2, &work[in + 1], &c__1, &e[ibegin], &c__1);
if (mb > 1)
{
/* Perturb each entry of the base representation by a small */
/* (but random) relative amount to overcome difficulties with */
/* glued matrices. */
for (i__ = 1;
i__ <= 4;
++i__)
{
iseed[i__ - 1] = 1;
/* L122: */
}
i__2 = (in << 1) - 1;
dlarnv_(&c__2, iseed, &i__2, &work[1]);
i__2 = in - 1;
for (i__ = 1;
i__ <= i__2;
++i__)
{
d__[ibegin + i__ - 1] *= eps * 8. * work[i__] + 1.;
e[ibegin + i__ - 1] *= eps * 8. * work[in + i__] + 1.;
/* L125: */
}
d__[iend] *= eps * 4. * work[in] + 1.;
}
/* Don't update the Gerschgorin intervals because keeping track */
/* of the updates would be too much work in DLARRV. */
/* We update W instead and use it to locate the proper Gerschgorin */
/* intervals. */
/* Compute the required eigenvalues of L D L' by bisection or dqds */
if (! usedqd)
{
/* If DLARRD has been used, shift the eigenvalue approximations */
/* according to their representation. This is necessary for */
/* a uniform DLARRV since dqds computes eigenvalues of the */
/* shifted representation. In DLARRV, W will always hold the */
/* UNshifted eigenvalue approximation. */
i__2 = wend;
for (j = wbegin;
j <= i__2;
++j)
{
w[j] -= sigma;
werr[j] += (d__1 = w[j], abs(d__1)) * eps;
/* L134: */
}
/* call DLARRB to reduce eigenvalue error of the approximations */
/* from DLARRD */
i__2 = iend - 1;
for (i__ = ibegin;
i__ <= i__2;
++i__)
{
/* Computing 2nd power */
d__1 = e[i__];
work[i__] = d__[i__] * (d__1 * d__1);
/* L135: */
}
/* use bisection to find EV from INDL to INDU */
i__2 = indl - 1;
dlarrb_(&in, &d__[ibegin], &work[ibegin], &indl, &indu, rtol1, rtol2, &i__2, &w[wbegin], &wgap[wbegin], &werr[wbegin], & work[(*n << 1) + 1], &iwork[1], pivmin, &spdiam, &in, & iinfo);
if (iinfo != 0)
{
*info = -4;
return 0;
}
/* DLARRB computes all gaps correctly except for the last one */
/* Record distance to VU/GU */
/* Computing MAX */
d__1 = 0.;
d__2 = *vu - sigma - (w[wend] + werr[wend]); // , expr subst
wgap[wend] = max(d__1,d__2);
i__2 = indu;
for (i__ = indl;
i__ <= i__2;
++i__)
{
++(*m);
iblock[*m] = jblk;
indexw[*m] = i__;
/* L138: */
}
}
else
{
/* Call dqds to get all eigs (and then possibly delete unwanted */
/* eigenvalues). */
/* Note that dqds finds the eigenvalues of the L D L^T representation */
/* of T to high relative accuracy. High relative accuracy */
/* might be lost when the shift of the RRR is subtracted to obtain */
/* the eigenvalues of T. However, T is not guaranteed to define its */
/* eigenvalues to high relative accuracy anyway. */
/* Set RTOL to the order of the tolerance used in DLASQ2 */
/* This is an ESTIMATED error, the worst case bound is 4*N*EPS */
/* which is usually too large and requires unnecessary work to be */
/* done by bisection when computing the eigenvectors */
rtol = log((doublereal) in) * 4. * eps;
j = ibegin;
i__2 = in - 1;
for (i__ = 1;
i__ <= i__2;
++i__)
{
work[(i__ << 1) - 1] = (d__1 = d__[j], abs(d__1));
work[i__ * 2] = e[j] * e[j] * work[(i__ << 1) - 1];
++j;
/* L140: */
}
work[(in << 1) - 1] = (d__1 = d__[iend], abs(d__1));
work[in * 2] = 0.;
dlasq2_(&in, &work[1], &iinfo);
if (iinfo != 0)
{
/* If IINFO = -5 then an index is part of a tight cluster */
/* and should be changed. The index is in IWORK(1) and the */
/* gap is in WORK(N+1) */
*info = -5;
return 0;
}
else
{
/* Test that all eigenvalues are positive as expected */
i__2 = in;
for (i__ = 1;
i__ <= i__2;
++i__)
{
if (work[i__] < 0.)
{
*info = -6;
return 0;
}
/* L149: */
}
}
if (sgndef > 0.)
{
i__2 = indu;
for (i__ = indl;
i__ <= i__2;
++i__)
{
++(*m);
w[*m] = work[in - i__ + 1];
iblock[*m] = jblk;
indexw[*m] = i__;
/* L150: */
}
}
else
{
i__2 = indu;
for (i__ = indl;
i__ <= i__2;
++i__)
{
++(*m);
w[*m] = -work[i__];
iblock[*m] = jblk;
indexw[*m] = i__;
/* L160: */
}
}
i__2 = *m;
for (i__ = *m - mb + 1;
i__ <= i__2;
++i__)
{
/* the value of RTOL below should be the tolerance in DLASQ2 */
werr[i__] = rtol * (d__1 = w[i__], abs(d__1));
/* L165: */
}
i__2 = *m - 1;
for (i__ = *m - mb + 1;
i__ <= i__2;
++i__)
{
/* compute the right gap between the intervals */
/* Computing MAX */
d__1 = 0.;
d__2 = w[i__ + 1] - werr[i__ + 1] - (w[i__] + werr[ i__]); // , expr subst
wgap[i__] = max(d__1,d__2);
/* L166: */
}
/* Computing MAX */
d__1 = 0.;
d__2 = *vu - sigma - (w[*m] + werr[*m]); // , expr subst
wgap[*m] = max(d__1,d__2);
}
/* proceed with next block */
ibegin = iend + 1;
wbegin = wend + 1;
L170:
;
}
return 0;
/* end of DLARRE */
}
