/* Subroutine */ int dlasq1_(integer *n, doublereal *d__, doublereal *e,
doublereal *work, integer *info)
{
/* System generated locals */
integer i__1, i__2;
doublereal d__1, d__2, d__3;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
integer i__;
doublereal eps;
extern /* Subroutine */ int dlas2_(doublereal *, doublereal *, doublereal
*, doublereal *, doublereal *);
doublereal scale;
integer iinfo;
doublereal sigmn;
extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *,
doublereal *, integer *);
doublereal sigmx;
extern /* Subroutine */ int dlasq2_(integer *, doublereal *, integer *);
extern doublereal dlamch_(char *);
extern /* Subroutine */ int dlascl_(char *, integer *, integer *,
doublereal *, doublereal *, integer *, integer *, doublereal *,
integer *, integer *);
doublereal safmin;
extern /* Subroutine */ int xerbla_(char *, integer *), dlasrt_(
char *, integer *, doublereal *, integer *);
/* -- LAPACK routine (version 3.1) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* DLASQ1 computes the singular values of a real N-by-N bidiagonal */
/* matrix with diagonal D and off-diagonal E. The singular values */
/* are computed to high relative accuracy, in the absence of */
/* denormalization, underflow and overflow. The algorithm was first */
/* presented in */
/* "Accurate singular values and differential qd algorithms" by K. V. */
/* Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230, */
/* 1994, */
/* and the present implementation is described in "An implementation of */
/* the dqds Algorithm (Positive Case)", LAPACK Working Note. */
/* Arguments */
/* ========= */
/* N (input) INTEGER */
/* The number of rows and columns in the matrix. N >= 0. */
/* D (input/output) DOUBLE PRECISION array, dimension (N) */
/* On entry, D contains the diagonal elements of the */
/* bidiagonal matrix whose SVD is desired. On normal exit, */
/* D contains the singular values in decreasing order. */
/* E (input/output) DOUBLE PRECISION array, dimension (N) */
/* On entry, elements E(1:N-1) contain the off-diagonal elements */
/* of the bidiagonal matrix whose SVD is desired. */
/* On exit, E is overwritten. */
/* WORK (workspace) DOUBLE PRECISION array, dimension (4*N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* > 0: the algorithm failed */
/* = 1, a split was marked by a positive value in E */
/* = 2, current block of Z not diagonalized after 30*N */
/* iterations (in inner while loop) */
/* = 3, termination criterion of outer while loop not met */
/* (program created more than N unreduced blocks) */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Parameter adjustments */
--work;
--e;
--d__;
/* Function Body */
*info = 0;
if (*n < 0) {
*info = -2;
i__1 = -(*info);
xerbla_("DLASQ1", &i__1);
return 0;
} else if (*n == 0) {
return 0;
} else if (*n == 1) {
d__[1] = abs(d__[1]);
return 0;
} else if (*n == 2) {
dlas2_(&d__[1], &e[1], &d__[2], &sigmn, &sigmx);
d__[1] = sigmx;
d__[2] = sigmn;
return 0;
}
/* Estimate the largest singular value. */
sigmx = 0.;
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
d__[i__] = (d__1 = d__[i__], abs(d__1));
/* Computing MAX */
d__2 = sigmx, d__3 = (d__1 = e[i__], abs(d__1));
sigmx = max(d__2,d__3);
/* L10: */
}
d__[*n] = (d__1 = d__[*n], abs(d__1));
/* Early return if SIGMX is zero (matrix is already diagonal). */
if (sigmx == 0.) {
dlasrt_("D", n, &d__[1], &iinfo);
return 0;
}
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Computing MAX */
d__1 = sigmx, d__2 = d__[i__];
sigmx = max(d__1,d__2);
/* L20: */
}
/* Copy D and E into WORK (in the Z format) and scale (squaring the */
/* input data makes scaling by a power of the radix pointless). */
eps = dlamch_("Precision");
safmin = dlamch_("Safe minimum");
scale = sqrt(eps / safmin);
dcopy_(n, &d__[1], &c__1, &work[1], &c__2);
i__1 = *n - 1;
dcopy_(&i__1, &e[1], &c__1, &work[2], &c__2);
i__1 = (*n << 1) - 1;
i__2 = (*n << 1) - 1;
dlascl_("G", &c__0, &c__0, &sigmx, &scale, &i__1, &c__1, &work[1], &i__2,
&iinfo);
/* Compute the q's and e's. */
i__1 = (*n << 1) - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Computing 2nd power */
d__1 = work[i__];
work[i__] = d__1 * d__1;
/* L30: */
}
work[*n * 2] = 0.;
dlasq2_(n, &work[1], info);
if (*info == 0) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
d__[i__] = sqrt(work[i__]);
/* L40: */
}
dlascl_("G", &c__0, &c__0, &scale, &sigmx, n, &c__1, &d__[1], n, &
iinfo);
}
return 0;
/* End of DLASQ1 */
} /* dlasq1_ */
/* 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 dlarre_(integer *n, doublereal *d__, doublereal *e,
doublereal *tol, integer *nsplit, integer *isplit, integer *m,
doublereal *w, doublereal *woff, doublereal *gersch, doublereal *work,
integer *info)
{
/* -- LAPACK auxiliary routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
June 30, 1999
Purpose
=======
Given the tridiagonal matrix T, DLARRE sets "small" off-diagonal
elements to zero, and for each unreduced block T_i, it finds
(i) the numbers sigma_i
(ii) the base T_i - sigma_i I = L_i D_i L_i^T representations and
(iii) eigenvalues of each L_i D_i L_i^T.
The representations and eigenvalues found are then used by
DSTEGR to compute the eigenvectors of a symmetric tridiagonal
matrix. Currently, the base representations are limited to being
positive or negative definite, and the eigenvalues of the definite
matrices are found by the dqds algorithm (subroutine DLASQ2). As
an added benefit, DLARRE also outputs the n Gerschgorin
intervals for each L_i D_i L_i^T.
Arguments
=========
N (input) INTEGER
The order of the matrix.
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 (n-1) subdiagonal elements of the tridiagonal
matrix T; E(N) need not be set.
On exit, the subdiagonal elements of the unit bidiagonal
matrices L_i.
TOL (input) DOUBLE PRECISION
The threshold for splitting. If on input |E(i)| < TOL, then
the matrix T is split into smaller blocks.
NSPLIT (input) INTEGER
The number of blocks T splits into. 1 <= NSPLIT <= N.
ISPLIT (output) INTEGER array, dimension (2*N)
The splitting points, at which T breaks up into submatrices.
The first submatrix 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 the 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.
WOFF (output) DOUBLE PRECISION array, dimension (N)
The NSPLIT base points sigma_i.
GERSCH (output) DOUBLE PRECISION array, dimension (2*N)
The n Gerschgorin intervals.
WORK (input) DOUBLE PRECISION array, dimension (4*N???)
Workspace.
INFO (output) INTEGER
Output error code from DLASQ2
Further Details
===============
Based on contributions by
Inderjit Dhillon, IBM Almaden, USA
Osni Marques, LBNL/NERSC, USA
=====================================================================
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
/* System generated locals */
integer i__1, i__2;
doublereal d__1, d__2;
/* Local variables */
static doublereal offd;
static integer iend, jblk, i__, j;
static doublereal s, delta, sigma;
extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *,
doublereal *, integer *);
static doublereal width;
extern /* Subroutine */ int dlasq2_(integer *, doublereal *, integer *);
static doublereal gl;
static integer in;
extern doublereal dlamch_(char *);
static doublereal gu;
static integer ibegin;
static doublereal sgndef;
static integer maxcnt, cnt;
static doublereal eps, tau, nrm, tmp1;
--work;
--gersch;
--woff;
--w;
--isplit;
--e;
--d__;
/* Function Body */
*info = 0;
eps = dlamch_("Precision");
/* Compute Splitting Points */
*nsplit = 1;
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
if ((d__1 = e[i__], abs(d__1)) <= *tol) {
isplit[*nsplit] = i__;
++(*nsplit);
}
/* L10: */
}
isplit[*nsplit] = *n;
ibegin = 1;
i__1 = *nsplit;
for (jblk = 1; jblk <= i__1; ++jblk) {
iend = isplit[jblk];
if (ibegin == iend) {
w[ibegin] = d__[ibegin];
woff[jblk] = 0.;
ibegin = iend + 1;
goto L170;
}
in = iend - ibegin + 1;
/* Form the n Gerschgorin intervals */
gl = d__[ibegin] - (d__1 = e[ibegin], abs(d__1));
gu = d__[ibegin] + (d__1 = e[ibegin], abs(d__1));
gersch[(ibegin << 1) - 1] = gl;
gersch[ibegin * 2] = gu;
gersch[(iend << 1) - 1] = d__[iend] - (d__1 = e[iend - 1], abs(d__1));
gersch[iend * 2] = d__[iend] + (d__1 = e[iend - 1], abs(d__1));
/* Computing MIN */
d__1 = gersch[(iend << 1) - 1];
gl = min(d__1,gl);
/* Computing MAX */
d__1 = gersch[iend * 2];
gu = max(d__1,gu);
i__2 = iend - 1;
for (i__ = ibegin + 1; i__ <= i__2; ++i__) {
offd = (d__1 = e[i__ - 1], abs(d__1)) + (d__2 = e[i__], abs(d__2))
;
gersch[(i__ << 1) - 1] = d__[i__] - offd;
/* Computing MIN */
d__1 = gersch[(i__ << 1) - 1];
gl = min(d__1,gl);
gersch[i__ * 2] = d__[i__] + offd;
/* Computing MAX */
d__1 = gersch[i__ * 2];
gu = max(d__1,gu);
/* L20: */
}
/* Computing MAX */
d__1 = abs(gl), d__2 = abs(gu);
nrm = max(d__1,d__2);
/* Find the number SIGMA where the base representation
T - sigma I = L D L^T is to be formed. */
width = gu - gl;
i__2 = iend - 1;
for (i__ = ibegin; i__ <= i__2; ++i__) {
work[i__] = e[i__] * e[i__];
/* L30: */
}
for (j = 1; j <= 2; ++j) {
if (j == 1) {
tau = gl + width * .25;
} else {
tau = gu - width * .25;
}
tmp1 = d__[ibegin] - tau;
if (tmp1 < 0.) {
cnt = 1;
} else {
cnt = 0;
}
i__2 = iend;
for (i__ = ibegin + 1; i__ <= i__2; ++i__) {
tmp1 = d__[i__] - tau - work[i__ - 1] / tmp1;
if (tmp1 < 0.) {
++cnt;
}
/* L40: */
}
if (cnt == 0) {
gl = tau;
} else if (cnt == in) {
gu = tau;
}
if (j == 1) {
maxcnt = cnt;
sigma = gl;
sgndef = 1.;
} else {
if (in - cnt > maxcnt) {
sigma = gu;
sgndef = -1.;
}
}
/* L50: */
}
/* Find the base L D L^T representation */
work[in * 3] = 1.;
delta = eps;
tau = sgndef * nrm;
L60:
sigma -= delta * tau;
work[1] = d__[ibegin] - sigma;
j = ibegin;
i__2 = in - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
work[(in << 1) + i__] = 1. / work[(i__ << 1) - 1];
tmp1 = e[j] * work[(in << 1) + i__];
work[(i__ << 1) + 1] = d__[j + 1] - sigma - tmp1 * e[j];
work[i__ * 2] = tmp1;
++j;
/* L70: */
}
for (i__ = in; i__ >= 1; --i__) {
tmp1 = sgndef * work[(i__ << 1) - 1];
if (tmp1 < 0. || work[(in << 1) + i__] == 0. || ! (tmp1 > 0. ||
tmp1 < 1.)) {
delta *= 2.;
goto L60;
}
--j;
/* L80: */
}
j = ibegin;
d__[ibegin] = work[1];
work[1] = abs(work[1]);
i__2 = in - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
tmp1 = e[j];
e[j] = work[i__ * 2];
work[i__ * 2] = (d__1 = tmp1 * work[i__ * 2], abs(d__1));
++j;
d__[j] = work[(i__ << 1) + 1];
work[(i__ << 1) + 1] = (d__1 = work[(i__ << 1) + 1], abs(d__1));
/* L90: */
}
dlasq2_(&in, &work[1], info);
tau = sgndef * work[in];
work[in * 3] = 1.;
delta = eps * 2.;
L100:
tau *= 1. - delta;
s = -tau;
j = ibegin;
i__2 = in - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
work[i__] = d__[j] + s;
work[(in << 1) + i__] = 1. / work[i__];
/* WORK( N+I ) = ( E( I ) * D( I ) ) / WORK( I ) */
work[in + i__] = e[j] * d__[j] * work[(in << 1) + i__];
s = s * work[in + i__] * e[j] - tau;
++j;
/* L110: */
}
work[in] = d__[iend] + s;
/* Checking to see if all the diagonal elements of the new
L D L^T representation have the same sign */
for (i__ = in; i__ >= 1; --i__) {
tmp1 = sgndef * work[i__];
if (tmp1 < 0. || work[(in << 1) + i__] == 0. || ! (tmp1 > 0. ||
tmp1 < 1.)) {
delta *= 2.;
goto L100;
}
/* L120: */
}
sigma += tau;
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);
woff[jblk] = sigma;
/* Update the n Gerschgorin intervals */
i__2 = iend;
for (i__ = ibegin; i__ <= i__2; ++i__) {
gersch[(i__ << 1) - 1] -= sigma;
gersch[i__ * 2] -= sigma;
/* L130: */
}
/* Compute the eigenvalues of L D L^T. */
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));
dlasq2_(&in, &work[1], info);
j = ibegin;
if (sgndef > 0.) {
i__2 = in;
for (i__ = 1; i__ <= i__2; ++i__) {
w[j] = work[in - i__ + 1];
++j;
/* L150: */
}
} else {
i__2 = in;
for (i__ = 1; i__ <= i__2; ++i__) {
w[j] = -work[i__];
++j;
/* L160: */
}
}
ibegin = iend + 1;
L170:
;
}
*m = *n;
return 0;
/* End of DLARRE */
} /* dlarre_ */
/* 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 */
}