/* Subroutine */ int slasq1_(integer *n, real *d__, real *e, real *work,
integer *info)
{
/* System generated locals */
integer i__1, i__2;
real r__1, r__2, r__3;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
integer i__;
real eps;
extern /* Subroutine */ int slas2_(real *, real *, real *, real *, real *)
;
real scale;
integer iinfo;
real sigmn, sigmx;
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *), slasq2_(integer *, real *, integer *);
extern doublereal slamch_(char *);
real safmin;
extern /* Subroutine */ int xerbla_(char *, integer *), slascl_(
char *, integer *, integer *, real *, real *, integer *, integer *
, real *, integer *, integer *), slasrt_(char *, integer *
, real *, integer *);
/* -- LAPACK routine (version 3.2) -- */
/* -- Contributed by Osni Marques of the Lawrence Berkeley National -- */
/* -- Laboratory and Beresford Parlett of the Univ. of California at -- */
/* -- Berkeley -- */
/* -- November 2008 -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SLASQ1 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) REAL 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) REAL 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) REAL 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_("SLASQ1", &i__1);
return 0;
} else if (*n == 0) {
return 0;
} else if (*n == 1) {
d__[1] = dabs(d__[1]);
return 0;
} else if (*n == 2) {
slas2_(&d__[1], &e[1], &d__[2], &sigmn, &sigmx);
d__[1] = sigmx;
d__[2] = sigmn;
return 0;
}
/* Estimate the largest singular value. */
sigmx = 0.f;
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
d__[i__] = (r__1 = d__[i__], dabs(r__1));
/* Computing MAX */
r__2 = sigmx, r__3 = (r__1 = e[i__], dabs(r__1));
sigmx = dmax(r__2,r__3);
/* L10: */
}
d__[*n] = (r__1 = d__[*n], dabs(r__1));
/* Early return if SIGMX is zero (matrix is already diagonal). */
if (sigmx == 0.f) {
slasrt_("D", n, &d__[1], &iinfo);
return 0;
}
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Computing MAX */
r__1 = sigmx, r__2 = d__[i__];
sigmx = dmax(r__1,r__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 = slamch_("Precision");
safmin = slamch_("Safe minimum");
scale = sqrt(eps / safmin);
scopy_(n, &d__[1], &c__1, &work[1], &c__2);
i__1 = *n - 1;
scopy_(&i__1, &e[1], &c__1, &work[2], &c__2);
i__1 = (*n << 1) - 1;
i__2 = (*n << 1) - 1;
slascl_("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 */
r__1 = work[i__];
work[i__] = r__1 * r__1;
/* L30: */
}
work[*n * 2] = 0.f;
slasq2_(n, &work[1], info);
if (*info == 0) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
d__[i__] = sqrt(work[i__]);
/* L40: */
}
slascl_("G", &c__0, &c__0, &scale, &sigmx, n, &c__1, &d__[1], n, &
iinfo);
}
return 0;
/* End of SLASQ1 */
} /* slasq1_ */
/* Subroutine */ int slarre_(char* range, integer* n, real* vl, real* vu,
integer* il, integer* iu, real* d__, real* e, real* e2, real* rtol1,
real* rtol2, real* spltol, integer* nsplit, integer* isplit, integer *
m, real* w, real* werr, real* wgap, integer* iblock, integer* indexw,
real* gers, real* pivmin, real* work, integer* iwork, integer* info) {
/* System generated locals */
integer i__1, i__2;
real r__1, r__2, r__3;
/* Builtin functions */
double sqrt(doublereal), log(doublereal);
/* Local variables */
integer i__, j;
real s1, s2;
integer mb;
real gl;
integer in, mm;
real gu;
integer cnt;
real eps, tau, tmp, rtl;
integer cnt1, cnt2;
real tmp1, eabs;
integer iend, jblk;
real eold;
integer indl;
real dmax__, emax;
integer wend, idum, indu;
real rtol;
integer iseed[4];
real avgap, sigma;
extern logical lsame_(char*, char*);
integer iinfo;
logical norep;
extern /* Subroutine */ int scopy_(integer*, real*, integer*, real*,
integer*), slasq2_(integer*, real*, integer*);
integer ibegin;
logical forceb;
integer irange;
real sgndef;
extern doublereal slamch_(char*);
integer wbegin;
real safmin, spdiam;
extern /* Subroutine */ int slarra_(integer*, real*, real*, real*,
real*, real*, integer*, integer*, integer*);
logical usedqd;
real clwdth, isleft;
extern /* Subroutine */ int slarrb_(integer*, real*, real*, integer*,
integer*, real*, real*, integer*, real*, real*, real*,
real*, integer*, real*, real*, integer*, integer*), slarrc_(
char*, integer*, real*, real*, real*, real*, real*,
integer*, integer*, integer*, integer*), slarrd_(char
*, char*, integer*, real*, real*, integer*, integer*, real *
, real*, real*, real*, real*, real*, integer*, integer*,
integer*, real*, real*, real*, real*, integer*, integer*,
real*, integer*, integer*), slarrk_(integer*,
integer*, real*, real*, real*, real*, real*, real*, real*,
real*, integer*);
real isrght, bsrtol, dpivot;
extern /* Subroutine */ int slarnv_(integer*, integer*, integer*, real
*);
/* -- LAPACK auxiliary routine (version 3.1) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* To find the desired eigenvalues of a given real symmetric */
/* tridiagonal matrix T, SLARRE 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 */
/* SSTEMR 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 SLASQ2) to */
/* conpute all and then discard any unwanted one. */
/* As an added benefit, SLARRE 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) REAL */
/* VU (input/output) REAL */
/* 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', SLARRE 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) REAL 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) REAL 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) REAL 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) REAL */
/* RTOL2 (input) REAL */
/* Parameters for bisection. */
/* An interval [LEFT,RIGHT] has converged if */
/* RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) ) */
/* SPLTOL (input) REAL */
/* 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) REAL 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 ( SLARRE may use the */
/* remaining N-M elements as workspace). */
/* WERR (output) REAL array, dimension (N) */
/* The error bound on the corresponding eigenvalue in W. */
/* WGAP (output) REAL 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) REAL 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) REAL array, dimension (6*N) */
/* Workspace. */
/* IWORK (workspace) INTEGER array, dimension (5*N) */
/* Workspace. */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* > 0: A problem occured in SLARRE. */
/* < 0: One of the called subroutines signaled an internal problem. */
/* Needs inspection of the corresponding parameter IINFO */
/* for further information. */
/* =-1: Problem in SLARRD. */
/* = 2: No base representation could be found in MAXTRY iterations. */
/* Increasing MAXTRY and recompilation might be a remedy. */
/* =-3: Problem in SLARRB when computing the refined root */
/* representation for SLASQ2. */
/* =-4: Problem in SLARRB when preforming bisection on the */
/* desired part of the spectrum. */
/* =-5: Problem in SLASQ2. */
/* =-6: Problem in SLASQ2. */
/* 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 */
/* ===================================================================== */
/* .. 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 */
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 = slamch_("S");
eps = slamch_("P");
/* Set parameters */
rtl = eps * 100.f;
/* If one were ever to ask for less initial precision in BSRTOL, */
/* one should keep in mind that for the subset case, the extremal */
/* eigenvalues must be at least as accurate as the current setting */
/* (eigenvalues in the middle need not as much accuracy) */
bsrtol = sqrt(eps) * 5e-4f;
/* 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.f;
wgap[1] = 0.f;
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.f;
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.f;
emax = 0.f;
e[*n] = 0.f;
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
werr[i__] = 0.f;
wgap[i__] = 0.f;
eabs = (r__1 = e[i__], dabs(r__1));
if (eabs >= emax) {
emax = eabs;
}
tmp1 = eabs + eold;
gers[(i__ << 1) - 1] = d__[i__] - tmp1;
/* Computing MIN */
r__1 = gl, r__2 = gers[(i__ << 1) - 1];
gl = dmin(r__1, r__2);
gers[i__ * 2] = d__[i__] + tmp1;
/* Computing MAX */
r__1 = gu, r__2 = gers[i__ * 2];
gu = dmax(r__1, r__2);
eold = eabs;
/* L5: */
}
/* The minimum pivot allowed in the Sturm sequence for T */
/* Computing MAX */
/* Computing 2nd power */
r__3 = emax;
r__1 = 1.f, r__2 = r__3 * r__3;
*pivmin = safmin * dmax(r__1, r__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 */
slarra_(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 SLARRD 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)) */
/* SLARRD needs a WORK of size 4*N, IWORK of size 3*N */
slarrd_(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.f;
werr[i__] = 0.f;
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.f;
/* The gap for a single block doesn't matter for the later */
/* algorithm and is assigned an arbitrary large value */
wgap[*m] = 0.f;
iblock[*m] = jblk;
indexw[*m] = 1;
++wbegin;
}
/* E( IEND ) holds the shift for the initial RRR */
e[iend] = 0.f;
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.f;
/* 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 */
r__1 = gers[(i__ << 1) - 1];
gl = dmin(r__1, gl);
/* Computing MAX */
r__1 = gers[i__ * 2];
gu = dmax(r__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.f;
ibegin = iend + 1;
goto L170;
} else {
/* Decide whether dqds or bisection is more efficient */
usedqd = (real) mb > in * .5f && ! 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.f;
i__2 = wend - 1;
for (i__ = wbegin; i__ <= i__2; ++i__) {
/* Computing MAX */
r__1 = 0.f, r__2 = w[i__ + 1] - werr[i__ + 1] - (w[i__] +
werr[i__]);
wgap[i__] = dmax(r__1, r__2);
/* L30: */
}
/* Computing MAX */
r__1 = 0.f, r__2 = *vu - sigma - (w[wend] + werr[wend]);
wgap[wend] = dmax(r__1, r__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 */
slarrk_(&in, &c__1, &gl, &gu, &d__[ibegin], &e2[ibegin], pivmin, &
rtl, &tmp, &tmp1, &iinfo);
if (iinfo != 0) {
*info = -1;
return 0;
}
/* Computing MAX */
r__2 = gl, r__3 = tmp - tmp1 - eps * 100.f * (r__1 = tmp - tmp1,
dabs(r__1));
isleft = dmax(r__2, r__3);
slarrk_(&in, &in, &gl, &gu, &d__[ibegin], &e2[ibegin], pivmin, &
rtl, &tmp, &tmp1, &iinfo);
if (iinfo != 0) {
*info = -1;
return 0;
}
/* Computing MIN */
r__2 = gu, r__3 = tmp + tmp1 + eps * 100.f * (r__1 = tmp + tmp1,
dabs(r__1));
isrght = dmin(r__2, r__3);
/* Improve the estimate of the spectral diameter */
spdiam = isrght - isleft;
} else {
/* Case of bisection */
/* Find approximations to the wanted extremal eigenvalues */
/* Computing MAX */
r__2 = gl, r__3 = w[wbegin] - werr[wbegin] - eps * 100.f * (r__1 =
w[wbegin] - werr[wbegin], dabs(r__1));
isleft = dmax(r__2, r__3);
/* Computing MIN */
r__2 = gu, r__3 = w[wend] + werr[wend] + eps * 100.f * (r__1 = w[
wend] + werr[wend], dabs(r__1));
isrght = dmin(r__2, r__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 SLARRE 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 * .25f;
s2 = isrght - spdiam * .25f;
} else {
/* SLARRD has computed IBLOCK and INDEXW for each eigenvalue */
/* approximation. */
/* choose sigma */
if (usedqd) {
s1 = isleft + spdiam * .25f;
s2 = isrght - spdiam * .25f;
} else {
tmp = dmin(isrght, *vu) - dmax(isleft, *vl);
s1 = dmax(isleft, *vl) + tmp * .25f;
s2 = dmin(isrght, *vu) - tmp * .25f;
}
}
/* Compute the negcount at the 1/4 and 3/4 points */
if (mb > 1) {
slarrc_("T", &in, &s1, &s2, &d__[ibegin], &e[ibegin], pivmin, &
cnt, &cnt1, &cnt2, &iinfo);
}
if (mb == 1) {
sigma = gl;
sgndef = 1.f;
} else if (cnt1 - indl >= indu - cnt2) {
if (irange == 1 && ! forceb) {
sigma = dmax(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 = dmax(isleft, *vl);
}
sgndef = 1.f;
} else {
if (irange == 1 && ! forceb) {
sigma = dmin(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 = dmin(isrght, *vu);
}
sgndef = -1.f;
}
/* 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.f;
} else {
if (mb > 1) {
clwdth = w[wend] + werr[wend] - w[wbegin] - werr[wbegin];
avgap = (r__1 = clwdth / (real)(wend - wbegin), dabs(r__1));
if (sgndef == 1.f) {
/* Computing MAX */
r__1 = wgap[wbegin];
tau = dmax(r__1, avgap) * .5f;
/* Computing MAX */
r__1 = tau, r__2 = werr[wbegin];
tau = dmax(r__1, r__2);
} else {
/* Computing MAX */
r__1 = wgap[wend - 1];
tau = dmax(r__1, avgap) * .5f;
/* Computing MAX */
r__1 = tau, r__2 = werr[wend];
tau = dmax(r__1, r__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__ = dabs(work[1]);
j = ibegin;
i__2 = in - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
work[(in << 1) + i__] = 1.f / 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 */
r__1 = dmax__, r__2 = dabs(dpivot);
dmax__ = dmax(r__1, r__2);
++j;
/* L70: */
}
/* check for element growth */
if (dmax__ > spdiam * 64.f) {
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.f) {
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.f) {
/* The fudged Gerschgorin shift should succeed */
sigma = gl - spdiam * 2.f * eps * *n - *pivmin * 4.f;
} else {
sigma = gu + spdiam * 2.f * eps * *n + *pivmin * 4.f;
}
} else {
sigma -= sgndef * tau;
tau *= 2.f;
}
} 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. */
scopy_(&in, &work[1], &c__1, &d__[ibegin], &c__1);
i__2 = in - 1;
scopy_(&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;
slarnv_(&c__2, iseed, &i__2, &work[1]);
i__2 = in - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
d__[ibegin + i__ - 1] *= eps * 4.f * work[i__] + 1.f;
e[ibegin + i__ - 1] *= eps * 4.f * work[in + i__] + 1.f;
/* L125: */
}
d__[iend] *= eps * 4.f * work[in] + 1.f;
}
/* Don't update the Gerschgorin intervals because keeping track */
/* of the updates would be too much work in SLARRV. */
/* 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 SLARRD has been used, shift the eigenvalue approximations */
/* according to their representation. This is necessary for */
/* a uniform SLARRV since dqds computes eigenvalues of the */
/* shifted representation. In SLARRV, W will always hold the */
/* UNshifted eigenvalue approximation. */
i__2 = wend;
for (j = wbegin; j <= i__2; ++j) {
w[j] -= sigma;
werr[j] += (r__1 = w[j], dabs(r__1)) * eps;
/* L134: */
}
/* call SLARRB to reduce eigenvalue error of the approximations */
/* from SLARRD */
i__2 = iend - 1;
for (i__ = ibegin; i__ <= i__2; ++i__) {
/* Computing 2nd power */
r__1 = e[i__];
work[i__] = d__[i__] * (r__1 * r__1);
/* L135: */
}
/* use bisection to find EV from INDL to INDU */
i__2 = indl - 1;
slarrb_(&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;
}
/* SLARRB computes all gaps correctly except for the last one */
/* Record distance to VU/GU */
/* Computing MAX */
r__1 = 0.f, r__2 = *vu - sigma - (w[wend] + werr[wend]);
wgap[wend] = dmax(r__1, r__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 SLASQ2 */
/* 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((real) in) * 4.f * eps;
j = ibegin;
i__2 = in - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
work[(i__ << 1) - 1] = (r__1 = d__[j], dabs(r__1));
work[i__ * 2] = e[j] * e[j] * work[(i__ << 1) - 1];
++j;
/* L140: */
}
work[(in << 1) - 1] = (r__1 = d__[iend], dabs(r__1));
work[in * 2] = 0.f;
slasq2_(&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.f) {
*info = -6;
return 0;
}
/* L149: */
}
}
if (sgndef > 0.f) {
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 SLASQ2 */
werr[i__] = rtol * (r__1 = w[i__], dabs(r__1));
/* L165: */
}
i__2 = *m - 1;
for (i__ = *m - mb + 1; i__ <= i__2; ++i__) {
/* compute the right gap between the intervals */
/* Computing MAX */
r__1 = 0.f, r__2 = w[i__ + 1] - werr[i__ + 1] - (w[i__] +
werr[i__]);
wgap[i__] = dmax(r__1, r__2);
/* L166: */
}
/* Computing MAX */
r__1 = 0.f, r__2 = *vu - sigma - (w[*m] + werr[*m]);
wgap[*m] = dmax(r__1, r__2);
}
/* proceed with next block */
ibegin = iend + 1;
wbegin = wend + 1;
L170:
;
}
return 0;
/* end of SLARRE */
} /* slarre_ */