/* -- translated by f2c (version 20100827).
You must link the resulting object file with libf2c:
on Microsoft Windows system, link with libf2c.lib;
on Linux or Unix systems, link with .../path/to/libf2c.a -lm
or, if you install libf2c.a in a standard place, with -lf2c -lm
-- in that order, at the end of the command line, as in
cc *.o -lf2c -lm
Source for libf2c is in /netlib/f2c/libf2c.zip, e.g.,
http://www.netlib.org/f2c/libf2c.zip
*/
#include "f2c.h"
/* Table of constant values */
static integer c__1 = 1;
static integer c__2 = 2;
static integer c__10 = 10;
static integer c__3 = 3;
static integer c__4 = 4;
static integer c__11 = 11;
/* > \brief \b DLASQ2 computes all the eigenvalues of the symmetric positive definite tridiagonal matrix assoc
iated with the qd Array Z to high relative accuracy. Used by sbdsqr and sstegr.
=========== DOCUMENTATION ===========
Online html documentation available at
http://www.netlib.org/lapack/explore-html/
> \htmlonly
> Download DLASQ2 + dependencies
> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasq2.
f">
> [TGZ]</a>
> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasq2.
f">
> [ZIP]</a>
> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasq2.
f">
> [TXT]</a>
> \endhtmlonly
Definition:
===========
SUBROUTINE DLASQ2( N, Z, INFO )
INTEGER INFO, N
DOUBLE PRECISION Z( * )
> \par Purpose:
=============
>
> \verbatim
>
> DLASQ2 computes all the eigenvalues of the symmetric positive
> definite tridiagonal matrix associated with the qd array Z to high
> relative accuracy are computed to high relative accuracy, in the
> absence of denormalization, underflow and overflow.
>
> To see the relation of Z to the tridiagonal matrix, let L be a
> unit lower bidiagonal matrix with subdiagonals Z(2,4,6,,..) and
> let U be an upper bidiagonal matrix with 1's above and diagonal
> Z(1,3,5,,..). The tridiagonal is L*U or, if you prefer, the
> symmetric tridiagonal to which it is similar.
>
> Note : DLASQ2 defines a logical variable, IEEE, which is true
> on machines which follow ieee-754 floating-point standard in their
> handling of infinities and NaNs, and false otherwise. This variable
> is passed to DLASQ3.
> \endverbatim
Arguments:
==========
> \param[in] N
> \verbatim
> N is INTEGER
> The number of rows and columns in the matrix. N >= 0.
> \endverbatim
>
> \param[in,out] Z
> \verbatim
> Z is DOUBLE PRECISION array, dimension ( 4*N )
> On entry Z holds the qd array. On exit, entries 1 to N hold
> the eigenvalues in decreasing order, Z( 2*N+1 ) holds the
> trace, and Z( 2*N+2 ) holds the sum of the eigenvalues. If
> N > 2, then Z( 2*N+3 ) holds the iteration count, Z( 2*N+4 )
> holds NDIVS/NIN^2, and Z( 2*N+5 ) holds the percentage of
> shifts that failed.
> \endverbatim
>
> \param[out] INFO
> \verbatim
> INFO is INTEGER
> = 0: successful exit
> < 0: if the i-th argument is a scalar and had an illegal
> value, then INFO = -i, if the i-th argument is an
> array and the j-entry had an illegal value, then
> INFO = -(i*100+j)
> > 0: the algorithm failed
> = 1, a split was marked by a positive value in E
> = 2, current block of Z not diagonalized after 100*N
> iterations (in inner while loop). On exit Z holds
> a qd array with the same eigenvalues as the given Z.
> = 3, termination criterion of outer while loop not met
> (program created more than N unreduced blocks)
> \endverbatim
Authors:
========
> \author Univ. of Tennessee
> \author Univ. of California Berkeley
> \author Univ. of Colorado Denver
> \author NAG Ltd.
> \date September 2012
> \ingroup auxOTHERcomputational
> \par Further Details:
=====================
>
> \verbatim
>
> Local Variables: I0:N0 defines a current unreduced segment of Z.
> The shifts are accumulated in SIGMA. Iteration count is in ITER.
> Ping-pong is controlled by PP (alternates between 0 and 1).
> \endverbatim
>
=====================================================================
Subroutine */ int igraphdlasq2_(integer *n, doublereal *z__, integer *info)
{
/* System generated locals */
integer i__1, i__2, i__3;
doublereal d__1, d__2;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
doublereal d__, e, g;
integer k;
doublereal s, t;
integer i0, i1, i4, n0, n1;
doublereal dn;
integer pp;
doublereal dn1, dn2, dee, eps, tau, tol;
integer ipn4;
doublereal tol2;
logical ieee;
integer nbig;
doublereal dmin__, emin, emax;
integer kmin, ndiv, iter;
doublereal qmin, temp, qmax, zmax;
integer splt;
doublereal dmin1, dmin2;
integer nfail;
doublereal desig, trace, sigma;
integer iinfo;
doublereal tempe, tempq;
integer ttype;
extern /* Subroutine */ int igraphdlasq3_(integer *, integer *, doublereal *,
integer *, doublereal *, doublereal *, doublereal *, doublereal *,
integer *, integer *, integer *, logical *, integer *,
doublereal *, doublereal *, doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *);
extern doublereal igraphdlamch_(char *);
doublereal deemin;
integer iwhila, iwhilb;
doublereal oldemn, safmin;
extern /* Subroutine */ int igraphxerbla_(char *, integer *, ftnlen);
extern integer igraphilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *, ftnlen, ftnlen);
extern /* Subroutine */ int igraphdlasrt_(char *, integer *, doublereal *,
integer *);
/* -- LAPACK computational 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
=====================================================================
Test the input arguments.
(in case DLASQ2 is not called by DLASQ1)
Parameter adjustments */
--z__;
/* Function Body */
*info = 0;
eps = igraphdlamch_("Precision");
safmin = igraphdlamch_("Safe minimum");
tol = eps * 100.;
/* Computing 2nd power */
d__1 = tol;
tol2 = d__1 * d__1;
if (*n < 0) {
*info = -1;
igraphxerbla_("DLASQ2", &c__1, (ftnlen)6);
return 0;
} else if (*n == 0) {
return 0;
} else if (*n == 1) {
/* 1-by-1 case. */
if (z__[1] < 0.) {
*info = -201;
igraphxerbla_("DLASQ2", &c__2, (ftnlen)6);
}
return 0;
} else if (*n == 2) {
/* 2-by-2 case. */
if (z__[2] < 0. || z__[3] < 0.) {
*info = -2;
igraphxerbla_("DLASQ2", &c__2, (ftnlen)6);
return 0;
} else if (z__[3] > z__[1]) {
d__ = z__[3];
z__[3] = z__[1];
z__[1] = d__;
}
z__[5] = z__[1] + z__[2] + z__[3];
if (z__[2] > z__[3] * tol2) {
t = (z__[1] - z__[3] + z__[2]) * .5;
s = z__[3] * (z__[2] / t);
if (s <= t) {
s = z__[3] * (z__[2] / (t * (sqrt(s / t + 1.) + 1.)));
} else {
s = z__[3] * (z__[2] / (t + sqrt(t) * sqrt(t + s)));
}
t = z__[1] + (s + z__[2]);
z__[3] *= z__[1] / t;
z__[1] = t;
}
z__[2] = z__[3];
z__[6] = z__[2] + z__[1];
return 0;
}
/* Check for negative data and compute sums of q's and e's. */
z__[*n * 2] = 0.;
emin = z__[2];
qmax = 0.;
zmax = 0.;
d__ = 0.;
e = 0.;
i__1 = *n - 1 << 1;
for (k = 1; k <= i__1; k += 2) {
if (z__[k] < 0.) {
*info = -(k + 200);
igraphxerbla_("DLASQ2", &c__2, (ftnlen)6);
return 0;
} else if (z__[k + 1] < 0.) {
*info = -(k + 201);
igraphxerbla_("DLASQ2", &c__2, (ftnlen)6);
return 0;
}
d__ += z__[k];
e += z__[k + 1];
/* Computing MAX */
d__1 = qmax, d__2 = z__[k];
qmax = max(d__1,d__2);
/* Computing MIN */
d__1 = emin, d__2 = z__[k + 1];
emin = min(d__1,d__2);
/* Computing MAX */
d__1 = max(qmax,zmax), d__2 = z__[k + 1];
zmax = max(d__1,d__2);
/* L10: */
}
if (z__[(*n << 1) - 1] < 0.) {
*info = -((*n << 1) + 199);
igraphxerbla_("DLASQ2", &c__2, (ftnlen)6);
return 0;
}
d__ += z__[(*n << 1) - 1];
/* Computing MAX */
d__1 = qmax, d__2 = z__[(*n << 1) - 1];
qmax = max(d__1,d__2);
zmax = max(qmax,zmax);
/* Check for diagonality. */
if (e == 0.) {
i__1 = *n;
for (k = 2; k <= i__1; ++k) {
z__[k] = z__[(k << 1) - 1];
/* L20: */
}
igraphdlasrt_("D", n, &z__[1], &iinfo);
z__[(*n << 1) - 1] = d__;
return 0;
}
trace = d__ + e;
/* Check for zero data. */
if (trace == 0.) {
z__[(*n << 1) - 1] = 0.;
return 0;
}
/* Check whether the machine is IEEE conformable. */
ieee = igraphilaenv_(&c__10, "DLASQ2", "N", &c__1, &c__2, &c__3, &c__4, (ftnlen)
6, (ftnlen)1) == 1 && igraphilaenv_(&c__11, "DLASQ2", "N", &c__1, &c__2,
&c__3, &c__4, (ftnlen)6, (ftnlen)1) == 1;
/* Rearrange data for locality: Z=(q1,qq1,e1,ee1,q2,qq2,e2,ee2,...). */
for (k = *n << 1; k >= 2; k += -2) {
z__[k * 2] = 0.;
z__[(k << 1) - 1] = z__[k];
z__[(k << 1) - 2] = 0.;
z__[(k << 1) - 3] = z__[k - 1];
/* L30: */
}
i0 = 1;
n0 = *n;
/* Reverse the qd-array, if warranted. */
if (z__[(i0 << 2) - 3] * 1.5 < z__[(n0 << 2) - 3]) {
ipn4 = i0 + n0 << 2;
i__1 = i0 + n0 - 1 << 1;
for (i4 = i0 << 2; i4 <= i__1; i4 += 4) {
temp = z__[i4 - 3];
z__[i4 - 3] = z__[ipn4 - i4 - 3];
z__[ipn4 - i4 - 3] = temp;
temp = z__[i4 - 1];
z__[i4 - 1] = z__[ipn4 - i4 - 5];
z__[ipn4 - i4 - 5] = temp;
/* L40: */
}
}
/* Initial split checking via dqd and Li's test. */
pp = 0;
for (k = 1; k <= 2; ++k) {
d__ = z__[(n0 << 2) + pp - 3];
i__1 = (i0 << 2) + pp;
for (i4 = (n0 - 1 << 2) + pp; i4 >= i__1; i4 += -4) {
if (z__[i4 - 1] <= tol2 * d__) {
z__[i4 - 1] = -0.;
d__ = z__[i4 - 3];
} else {
d__ = z__[i4 - 3] * (d__ / (d__ + z__[i4 - 1]));
}
/* L50: */
}
/* dqd maps Z to ZZ plus Li's test. */
emin = z__[(i0 << 2) + pp + 1];
d__ = z__[(i0 << 2) + pp - 3];
i__1 = (n0 - 1 << 2) + pp;
for (i4 = (i0 << 2) + pp; i4 <= i__1; i4 += 4) {
z__[i4 - (pp << 1) - 2] = d__ + z__[i4 - 1];
if (z__[i4 - 1] <= tol2 * d__) {
z__[i4 - 1] = -0.;
z__[i4 - (pp << 1) - 2] = d__;
z__[i4 - (pp << 1)] = 0.;
d__ = z__[i4 + 1];
} else if (safmin * z__[i4 + 1] < z__[i4 - (pp << 1) - 2] &&
safmin * z__[i4 - (pp << 1) - 2] < z__[i4 + 1]) {
temp = z__[i4 + 1] / z__[i4 - (pp << 1) - 2];
z__[i4 - (pp << 1)] = z__[i4 - 1] * temp;
d__ *= temp;
} else {
z__[i4 - (pp << 1)] = z__[i4 + 1] * (z__[i4 - 1] / z__[i4 - (
pp << 1) - 2]);
d__ = z__[i4 + 1] * (d__ / z__[i4 - (pp << 1) - 2]);
}
/* Computing MIN */
d__1 = emin, d__2 = z__[i4 - (pp << 1)];
emin = min(d__1,d__2);
/* L60: */
}
z__[(n0 << 2) - pp - 2] = d__;
/* Now find qmax. */
qmax = z__[(i0 << 2) - pp - 2];
i__1 = (n0 << 2) - pp - 2;
for (i4 = (i0 << 2) - pp + 2; i4 <= i__1; i4 += 4) {
/* Computing MAX */
d__1 = qmax, d__2 = z__[i4];
qmax = max(d__1,d__2);
/* L70: */
}
/* Prepare for the next iteration on K. */
pp = 1 - pp;
/* L80: */
}
/* Initialise variables to pass to DLASQ3. */
ttype = 0;
dmin1 = 0.;
dmin2 = 0.;
dn = 0.;
dn1 = 0.;
dn2 = 0.;
g = 0.;
tau = 0.;
iter = 2;
nfail = 0;
ndiv = n0 - i0 << 1;
i__1 = *n + 1;
for (iwhila = 1; iwhila <= i__1; ++iwhila) {
if (n0 < 1) {
goto L170;
}
/* While array unfinished do
E(N0) holds the value of SIGMA when submatrix in I0:N0
splits from the rest of the array, but is negated. */
desig = 0.;
if (n0 == *n) {
sigma = 0.;
} else {
sigma = -z__[(n0 << 2) - 1];
}
if (sigma < 0.) {
*info = 1;
return 0;
}
/* Find last unreduced submatrix's top index I0, find QMAX and
EMIN. Find Gershgorin-type bound if Q's much greater than E's. */
emax = 0.;
if (n0 > i0) {
emin = (d__1 = z__[(n0 << 2) - 5], abs(d__1));
} else {
emin = 0.;
}
qmin = z__[(n0 << 2) - 3];
qmax = qmin;
for (i4 = n0 << 2; i4 >= 8; i4 += -4) {
if (z__[i4 - 5] <= 0.) {
goto L100;
}
if (qmin >= emax * 4.) {
/* Computing MIN */
d__1 = qmin, d__2 = z__[i4 - 3];
qmin = min(d__1,d__2);
/* Computing MAX */
d__1 = emax, d__2 = z__[i4 - 5];
emax = max(d__1,d__2);
}
/* Computing MAX */
d__1 = qmax, d__2 = z__[i4 - 7] + z__[i4 - 5];
qmax = max(d__1,d__2);
/* Computing MIN */
d__1 = emin, d__2 = z__[i4 - 5];
emin = min(d__1,d__2);
/* L90: */
}
i4 = 4;
L100:
i0 = i4 / 4;
pp = 0;
if (n0 - i0 > 1) {
dee = z__[(i0 << 2) - 3];
deemin = dee;
kmin = i0;
i__2 = (n0 << 2) - 3;
for (i4 = (i0 << 2) + 1; i4 <= i__2; i4 += 4) {
dee = z__[i4] * (dee / (dee + z__[i4 - 2]));
if (dee <= deemin) {
deemin = dee;
kmin = (i4 + 3) / 4;
}
/* L110: */
}
if (kmin - i0 << 1 < n0 - kmin && deemin <= z__[(n0 << 2) - 3] *
.5) {
ipn4 = i0 + n0 << 2;
pp = 2;
i__2 = i0 + n0 - 1 << 1;
for (i4 = i0 << 2; i4 <= i__2; i4 += 4) {
temp = z__[i4 - 3];
z__[i4 - 3] = z__[ipn4 - i4 - 3];
z__[ipn4 - i4 - 3] = temp;
temp = z__[i4 - 2];
z__[i4 - 2] = z__[ipn4 - i4 - 2];
z__[ipn4 - i4 - 2] = temp;
temp = z__[i4 - 1];
z__[i4 - 1] = z__[ipn4 - i4 - 5];
z__[ipn4 - i4 - 5] = temp;
temp = z__[i4];
z__[i4] = z__[ipn4 - i4 - 4];
z__[ipn4 - i4 - 4] = temp;
/* L120: */
}
}
}
/* Put -(initial shift) into DMIN.
Computing MAX */
d__1 = 0., d__2 = qmin - sqrt(qmin) * 2. * sqrt(emax);
dmin__ = -max(d__1,d__2);
/* Now I0:N0 is unreduced.
PP = 0 for ping, PP = 1 for pong.
PP = 2 indicates that flipping was applied to the Z array and
and that the tests for deflation upon entry in DLASQ3
should not be performed. */
nbig = (n0 - i0 + 1) * 100;
i__2 = nbig;
for (iwhilb = 1; iwhilb <= i__2; ++iwhilb) {
if (i0 > n0) {
goto L150;
}
/* While submatrix unfinished take a good dqds step. */
igraphdlasq3_(&i0, &n0, &z__[1], &pp, &dmin__, &sigma, &desig, &qmax, &
nfail, &iter, &ndiv, &ieee, &ttype, &dmin1, &dmin2, &dn, &
dn1, &dn2, &g, &tau);
pp = 1 - pp;
/* When EMIN is very small check for splits. */
if (pp == 0 && n0 - i0 >= 3) {
if (z__[n0 * 4] <= tol2 * qmax || z__[(n0 << 2) - 1] <= tol2 *
sigma) {
splt = i0 - 1;
qmax = z__[(i0 << 2) - 3];
emin = z__[(i0 << 2) - 1];
oldemn = z__[i0 * 4];
i__3 = n0 - 3 << 2;
for (i4 = i0 << 2; i4 <= i__3; i4 += 4) {
if (z__[i4] <= tol2 * z__[i4 - 3] || z__[i4 - 1] <=
tol2 * sigma) {
z__[i4 - 1] = -sigma;
splt = i4 / 4;
qmax = 0.;
emin = z__[i4 + 3];
oldemn = z__[i4 + 4];
} else {
/* Computing MAX */
d__1 = qmax, d__2 = z__[i4 + 1];
qmax = max(d__1,d__2);
/* Computing MIN */
d__1 = emin, d__2 = z__[i4 - 1];
emin = min(d__1,d__2);
/* Computing MIN */
d__1 = oldemn, d__2 = z__[i4];
oldemn = min(d__1,d__2);
}
/* L130: */
}
z__[(n0 << 2) - 1] = emin;
z__[n0 * 4] = oldemn;
i0 = splt + 1;
}
}
/* L140: */
}
*info = 2;
/* Maximum number of iterations exceeded, restore the shift
SIGMA and place the new d's and e's in a qd array.
This might need to be done for several blocks */
i1 = i0;
n1 = n0;
L145:
tempq = z__[(i0 << 2) - 3];
z__[(i0 << 2) - 3] += sigma;
i__2 = n0;
for (k = i0 + 1; k <= i__2; ++k) {
tempe = z__[(k << 2) - 5];
z__[(k << 2) - 5] *= tempq / z__[(k << 2) - 7];
tempq = z__[(k << 2) - 3];
z__[(k << 2) - 3] = z__[(k << 2) - 3] + sigma + tempe - z__[(k <<
2) - 5];
}
/* Prepare to do this on the previous block if there is one */
if (i1 > 1) {
n1 = i1 - 1;
while(i1 >= 2 && z__[(i1 << 2) - 5] >= 0.) {
--i1;
}
sigma = -z__[(n1 << 2) - 1];
goto L145;
}
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
z__[(k << 1) - 1] = z__[(k << 2) - 3];
/* Only the block 1..N0 is unfinished. The rest of the e's
must be essentially zero, although sometimes other data
has been stored in them. */
if (k < n0) {
z__[k * 2] = z__[(k << 2) - 1];
} else {
z__[k * 2] = 0.;
}
}
return 0;
/* end IWHILB */
L150:
/* L160: */
;
}
*info = 3;
return 0;
/* end IWHILA */
L170:
/* Move q's to the front. */
i__1 = *n;
for (k = 2; k <= i__1; ++k) {
z__[k] = z__[(k << 2) - 3];
/* L180: */
}
/* Sort and compute sum of eigenvalues. */
igraphdlasrt_("D", n, &z__[1], &iinfo);
e = 0.;
for (k = *n; k >= 1; --k) {
e += z__[k];
/* L190: */
}
/* Store trace, sum(eigenvalues) and information on performance. */
z__[(*n << 1) + 1] = trace;
z__[(*n << 1) + 2] = e;
z__[(*n << 1) + 3] = (doublereal) iter;
/* Computing 2nd power */
i__1 = *n;
z__[(*n << 1) + 4] = (doublereal) ndiv / (doublereal) (i__1 * i__1);
z__[(*n << 1) + 5] = nfail * 100. / (doublereal) iter;
return 0;
/* End of DLASQ2 */
} /* igraphdlasq2_ */