/* Subroutine */ int slaqr0_(logical *wantt, logical *wantz, integer *n,
integer *ilo, integer *ihi, real *h__, integer *ldh, real *wr, real *
wi, integer *iloz, integer *ihiz, real *z__, integer *ldz, real *work,
integer *lwork, integer *info)
{
/* System generated locals */
integer h_dim1, h_offset, z_dim1, z_offset, i__1, i__2, i__3, i__4, i__5;
real r__1, r__2, r__3, r__4;
/* Local variables */
integer i__, k;
real aa, bb, cc, dd;
integer ld;
real cs;
integer nh, it, ks, kt;
real sn;
integer ku, kv, ls, ns;
real ss;
integer nw, inf, kdu, nho, nve, kwh, nsr, nwr, kwv, ndec, ndfl, kbot,
nmin;
real swap;
integer ktop;
real zdum[1] /* was [1][1] */;
integer kacc22, itmax, nsmax, nwmax, kwtop;
extern /* Subroutine */ int slanv2_(real *, real *, real *, real *, real *
, real *, real *, real *, real *, real *), slaqr3_(logical *,
logical *, integer *, integer *, integer *, integer *, real *,
integer *, integer *, integer *, real *, integer *, integer *,
integer *, real *, real *, real *, integer *, integer *, real *,
integer *, integer *, real *, integer *, real *, integer *),
slaqr4_(logical *, logical *, integer *, integer *, integer *,
real *, integer *, real *, real *, integer *, integer *, real *,
integer *, real *, integer *, integer *), slaqr5_(logical *,
logical *, integer *, integer *, integer *, integer *, integer *,
real *, real *, real *, integer *, integer *, integer *, real *,
integer *, real *, integer *, real *, integer *, integer *, real *
, integer *, integer *, real *, integer *);
integer nibble;
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *);
char jbcmpz[2];
extern /* Subroutine */ int slahqr_(logical *, logical *, integer *,
integer *, integer *, real *, integer *, real *, real *, integer *
, integer *, real *, integer *, integer *), slacpy_(char *,
integer *, integer *, real *, integer *, real *, integer *);
integer nwupbd;
logical sorted;
integer lwkopt;
/* -- LAPACK auxiliary routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SLAQR0 computes the eigenvalues of a Hessenberg matrix H */
/* and, optionally, the matrices T and Z from the Schur decomposition */
/* H = Z T Z**T, where T is an upper quasi-triangular matrix (the */
/* Schur form), and Z is the orthogonal matrix of Schur vectors. */
/* Optionally Z may be postmultiplied into an input orthogonal */
/* matrix Q so that this routine can give the Schur factorization */
/* of a matrix A which has been reduced to the Hessenberg form H */
/* by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T. */
/* Arguments */
/* ========= */
/* WANTT (input) LOGICAL */
/* = .TRUE. : the full Schur form T is required; */
/* = .FALSE.: only eigenvalues are required. */
/* WANTZ (input) LOGICAL */
/* = .TRUE. : the matrix of Schur vectors Z is required; */
/* = .FALSE.: Schur vectors are not required. */
/* N (input) INTEGER */
/* The order of the matrix H. N .GE. 0. */
/* ILO (input) INTEGER */
/* IHI (input) INTEGER */
/* It is assumed that H is already upper triangular in rows */
/* and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1, */
/* H(ILO,ILO-1) is zero. ILO and IHI are normally set by a */
/* previous call to SGEBAL, and then passed to SGEHRD when the */
/* matrix output by SGEBAL is reduced to Hessenberg form. */
/* Otherwise, ILO and IHI should be set to 1 and N, */
/* respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N. */
/* If N = 0, then ILO = 1 and IHI = 0. */
/* H (input/output) REAL array, dimension (LDH,N) */
/* On entry, the upper Hessenberg matrix H. */
/* On exit, if INFO = 0 and WANTT is .TRUE., then H contains */
/* the upper quasi-triangular matrix T from the Schur */
/* decomposition (the Schur form); 2-by-2 diagonal blocks */
/* (corresponding to complex conjugate pairs of eigenvalues) */
/* are returned in standard form, with H(i,i) = H(i+1,i+1) */
/* and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is */
/* .FALSE., then the contents of H are unspecified on exit. */
/* (The output value of H when INFO.GT.0 is given under the */
/* description of INFO below.) */
/* This subroutine may explicitly set H(i,j) = 0 for i.GT.j and */
/* j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N. */
/* LDH (input) INTEGER */
/* The leading dimension of the array H. LDH .GE. max(1,N). */
/* WR (output) REAL array, dimension (IHI) */
/* WI (output) REAL array, dimension (IHI) */
/* The real and imaginary parts, respectively, of the computed */
/* eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI) */
/* and WI(ILO:IHI). If two eigenvalues are computed as a */
/* complex conjugate pair, they are stored in consecutive */
/* elements of WR and WI, say the i-th and (i+1)th, with */
/* WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then */
/* the eigenvalues are stored in the same order as on the */
/* diagonal of the Schur form returned in H, with */
/* WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal */
/* block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and */
/* WI(i+1) = -WI(i). */
/* ILOZ (input) INTEGER */
/* IHIZ (input) INTEGER */
/* Specify the rows of Z to which transformations must be */
/* applied if WANTZ is .TRUE.. */
/* 1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N. */
/* Z (input/output) REAL array, dimension (LDZ,IHI) */
/* If WANTZ is .FALSE., then Z is not referenced. */
/* If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is */
/* replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the */
/* orthogonal Schur factor of H(ILO:IHI,ILO:IHI). */
/* (The output value of Z when INFO.GT.0 is given under */
/* the description of INFO below.) */
/* LDZ (input) INTEGER */
/* The leading dimension of the array Z. if WANTZ is .TRUE. */
/* then LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1. */
/* WORK (workspace/output) REAL array, dimension LWORK */
/* On exit, if LWORK = -1, WORK(1) returns an estimate of */
/* the optimal value for LWORK. */
/* LWORK (input) INTEGER */
/* The dimension of the array WORK. LWORK .GE. max(1,N) */
/* is sufficient, but LWORK typically as large as 6*N may */
/* be required for optimal performance. A workspace query */
/* to determine the optimal workspace size is recommended. */
/* If LWORK = -1, then SLAQR0 does a workspace query. */
/* In this case, SLAQR0 checks the input parameters and */
/* estimates the optimal workspace size for the given */
/* values of N, ILO and IHI. The estimate is returned */
/* in WORK(1). No error message related to LWORK is */
/* issued by XERBLA. Neither H nor Z are accessed. */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* .GT. 0: if INFO = i, SLAQR0 failed to compute all of */
/* the eigenvalues. Elements 1:ilo-1 and i+1:n of WR */
/* and WI contain those eigenvalues which have been */
/* successfully computed. (Failures are rare.) */
/* If INFO .GT. 0 and WANT is .FALSE., then on exit, */
/* the remaining unconverged eigenvalues are the eigen- */
/* values of the upper Hessenberg matrix rows and */
/* columns ILO through INFO of the final, output */
/* value of H. */
/* If INFO .GT. 0 and WANTT is .TRUE., then on exit */
/* (*) (initial value of H)*U = U*(final value of H) */
/* where U is an orthogonal matrix. The final */
/* value of H is upper Hessenberg and quasi-triangular */
/* in rows and columns INFO+1 through IHI. */
/* If INFO .GT. 0 and WANTZ is .TRUE., then on exit */
/* (final value of Z(ILO:IHI,ILOZ:IHIZ) */
/* = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U */
/* where U is the orthogonal matrix in (*) (regard- */
/* less of the value of WANTT.) */
/* If INFO .GT. 0 and WANTZ is .FALSE., then Z is not */
/* accessed. */
/* ================================================================ */
/* Based on contributions by */
/* Karen Braman and Ralph Byers, Department of Mathematics, */
/* University of Kansas, USA */
/* ================================================================ */
/* References: */
/* K. Braman, R. Byers and R. Mathias, The Multi-Shift QR */
/* Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 */
/* Performance, SIAM Journal of Matrix Analysis, volume 23, pages */
/* 929--947, 2002. */
/* K. Braman, R. Byers and R. Mathias, The Multi-Shift QR */
/* Algorithm Part II: Aggressive Early Deflation, SIAM Journal */
/* of Matrix Analysis, volume 23, pages 948--973, 2002. */
/* ================================================================ */
/* .. Parameters .. */
/* ==== Matrices of order NTINY or smaller must be processed by */
/* . SLAHQR because of insufficient subdiagonal scratch space. */
/* . (This is a hard limit.) ==== */
/* ==== Exceptional deflation windows: try to cure rare */
/* . slow convergence by varying the size of the */
/* . deflation window after KEXNW iterations. ==== */
/* ==== Exceptional shifts: try to cure rare slow convergence */
/* . with ad-hoc exceptional shifts every KEXSH iterations. */
/* . ==== */
/* ==== The constants WILK1 and WILK2 are used to form the */
/* . exceptional shifts. ==== */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Parameter adjustments */
h_dim1 = *ldh;
h_offset = 1 + h_dim1;
h__ -= h_offset;
--wr;
--wi;
z_dim1 = *ldz;
z_offset = 1 + z_dim1;
z__ -= z_offset;
--work;
/* Function Body */
*info = 0;
/* ==== Quick return for N = 0: nothing to do. ==== */
if (*n == 0) {
work[1] = 1.f;
return 0;
}
if (*n <= 11) {
/* ==== Tiny matrices must use SLAHQR. ==== */
lwkopt = 1;
if (*lwork != -1) {
slahqr_(wantt, wantz, n, ilo, ihi, &h__[h_offset], ldh, &wr[1], &
wi[1], iloz, ihiz, &z__[z_offset], ldz, info);
}
} else {
/* ==== Use small bulge multi-shift QR with aggressive early */
/* . deflation on larger-than-tiny matrices. ==== */
/* ==== Hope for the best. ==== */
*info = 0;
/* ==== Set up job flags for ILAENV. ==== */
if (*wantt) {
*(unsigned char *)jbcmpz = 'S';
} else {
*(unsigned char *)jbcmpz = 'E';
}
if (*wantz) {
*(unsigned char *)&jbcmpz[1] = 'V';
} else {
*(unsigned char *)&jbcmpz[1] = 'N';
}
/* ==== NWR = recommended deflation window size. At this */
/* . point, N .GT. NTINY = 11, so there is enough */
/* . subdiagonal workspace for NWR.GE.2 as required. */
/* . (In fact, there is enough subdiagonal space for */
/* . NWR.GE.3.) ==== */
nwr = ilaenv_(&c__13, "SLAQR0", jbcmpz, n, ilo, ihi, lwork);
nwr = max(2,nwr);
/* Computing MIN */
i__1 = *ihi - *ilo + 1, i__2 = (*n - 1) / 3, i__1 = min(i__1,i__2);
nwr = min(i__1,nwr);
/* ==== NSR = recommended number of simultaneous shifts. */
/* . At this point N .GT. NTINY = 11, so there is at */
/* . enough subdiagonal workspace for NSR to be even */
/* . and greater than or equal to two as required. ==== */
nsr = ilaenv_(&c__15, "SLAQR0", jbcmpz, n, ilo, ihi, lwork);
/* Computing MIN */
i__1 = nsr, i__2 = (*n + 6) / 9, i__1 = min(i__1,i__2), i__2 = *ihi -
*ilo;
nsr = min(i__1,i__2);
/* Computing MAX */
i__1 = 2, i__2 = nsr - nsr % 2;
nsr = max(i__1,i__2);
/* ==== Estimate optimal workspace ==== */
/* ==== Workspace query call to SLAQR3 ==== */
i__1 = nwr + 1;
slaqr3_(wantt, wantz, n, ilo, ihi, &i__1, &h__[h_offset], ldh, iloz,
ihiz, &z__[z_offset], ldz, &ls, &ld, &wr[1], &wi[1], &h__[
h_offset], ldh, n, &h__[h_offset], ldh, n, &h__[h_offset],
ldh, &work[1], &c_n1);
/* ==== Optimal workspace = MAX(SLAQR5, SLAQR3) ==== */
/* Computing MAX */
i__1 = nsr * 3 / 2, i__2 = (integer) work[1];
lwkopt = max(i__1,i__2);
/* ==== Quick return in case of workspace query. ==== */
if (*lwork == -1) {
work[1] = (real) lwkopt;
return 0;
}
/* ==== SLAHQR/SLAQR0 crossover point ==== */
nmin = ilaenv_(&c__12, "SLAQR0", jbcmpz, n, ilo, ihi, lwork);
nmin = max(11,nmin);
/* ==== Nibble crossover point ==== */
nibble = ilaenv_(&c__14, "SLAQR0", jbcmpz, n, ilo, ihi, lwork);
nibble = max(0,nibble);
/* ==== Accumulate reflections during ttswp? Use block */
/* . 2-by-2 structure during matrix-matrix multiply? ==== */
kacc22 = ilaenv_(&c__16, "SLAQR0", jbcmpz, n, ilo, ihi, lwork);
kacc22 = max(0,kacc22);
kacc22 = min(2,kacc22);
/* ==== NWMAX = the largest possible deflation window for */
/* . which there is sufficient workspace. ==== */
/* Computing MIN */
i__1 = (*n - 1) / 3, i__2 = *lwork / 2;
nwmax = min(i__1,i__2);
nw = nwmax;
/* ==== NSMAX = the Largest number of simultaneous shifts */
/* . for which there is sufficient workspace. ==== */
/* Computing MIN */
i__1 = (*n + 6) / 9, i__2 = (*lwork << 1) / 3;
nsmax = min(i__1,i__2);
nsmax -= nsmax % 2;
/* ==== NDFL: an iteration count restarted at deflation. ==== */
ndfl = 1;
/* ==== ITMAX = iteration limit ==== */
/* Computing MAX */
i__1 = 10, i__2 = *ihi - *ilo + 1;
itmax = max(i__1,i__2) * 30;
/* ==== Last row and column in the active block ==== */
kbot = *ihi;
/* ==== Main Loop ==== */
i__1 = itmax;
for (it = 1; it <= i__1; ++it) {
/* ==== Done when KBOT falls below ILO ==== */
if (kbot < *ilo) {
goto L90;
}
/* ==== Locate active block ==== */
i__2 = *ilo + 1;
for (k = kbot; k >= i__2; --k) {
if (h__[k + (k - 1) * h_dim1] == 0.f) {
goto L20;
}
/* L10: */
}
k = *ilo;
L20:
ktop = k;
/* ==== Select deflation window size: */
/* . Typical Case: */
/* . If possible and advisable, nibble the entire */
/* . active block. If not, use size MIN(NWR,NWMAX) */
/* . or MIN(NWR+1,NWMAX) depending upon which has */
/* . the smaller corresponding subdiagonal entry */
/* . (a heuristic). */
/* . */
/* . Exceptional Case: */
/* . If there have been no deflations in KEXNW or */
/* . more iterations, then vary the deflation window */
/* . size. At first, because, larger windows are, */
/* . in general, more powerful than smaller ones, */
/* . rapidly increase the window to the maximum possible. */
/* . Then, gradually reduce the window size. ==== */
nh = kbot - ktop + 1;
nwupbd = min(nh,nwmax);
if (ndfl < 5) {
nw = min(nwupbd,nwr);
} else {
/* Computing MIN */
i__2 = nwupbd, i__3 = nw << 1;
nw = min(i__2,i__3);
}
if (nw < nwmax) {
if (nw >= nh - 1) {
nw = nh;
} else {
kwtop = kbot - nw + 1;
if ((r__1 = h__[kwtop + (kwtop - 1) * h_dim1], dabs(r__1))
> (r__2 = h__[kwtop - 1 + (kwtop - 2) * h_dim1],
dabs(r__2))) {
++nw;
}
}
}
if (ndfl < 5) {
ndec = -1;
} else if (ndec >= 0 || nw >= nwupbd) {
++ndec;
if (nw - ndec < 2) {
ndec = 0;
}
nw -= ndec;
}
/* ==== Aggressive early deflation: */
/* . split workspace under the subdiagonal into */
/* . - an nw-by-nw work array V in the lower */
/* . left-hand-corner, */
/* . - an NW-by-at-least-NW-but-more-is-better */
/* . (NW-by-NHO) horizontal work array along */
/* . the bottom edge, */
/* . - an at-least-NW-but-more-is-better (NHV-by-NW) */
/* . vertical work array along the left-hand-edge. */
/* . ==== */
kv = *n - nw + 1;
kt = nw + 1;
nho = *n - nw - 1 - kt + 1;
kwv = nw + 2;
nve = *n - nw - kwv + 1;
/* ==== Aggressive early deflation ==== */
slaqr3_(wantt, wantz, n, &ktop, &kbot, &nw, &h__[h_offset], ldh,
iloz, ihiz, &z__[z_offset], ldz, &ls, &ld, &wr[1], &wi[1],
&h__[kv + h_dim1], ldh, &nho, &h__[kv + kt * h_dim1],
ldh, &nve, &h__[kwv + h_dim1], ldh, &work[1], lwork);
/* ==== Adjust KBOT accounting for new deflations. ==== */
kbot -= ld;
/* ==== KS points to the shifts. ==== */
ks = kbot - ls + 1;
/* ==== Skip an expensive QR sweep if there is a (partly */
/* . heuristic) reason to expect that many eigenvalues */
/* . will deflate without it. Here, the QR sweep is */
/* . skipped if many eigenvalues have just been deflated */
/* . or if the remaining active block is small. */
if (ld == 0 || ld * 100 <= nw * nibble && kbot - ktop + 1 > min(
nmin,nwmax)) {
/* ==== NS = nominal number of simultaneous shifts. */
/* . This may be lowered (slightly) if SLAQR3 */
/* . did not provide that many shifts. ==== */
/* Computing MIN */
/* Computing MAX */
i__4 = 2, i__5 = kbot - ktop;
i__2 = min(nsmax,nsr), i__3 = max(i__4,i__5);
ns = min(i__2,i__3);
ns -= ns % 2;
/* ==== If there have been no deflations */
/* . in a multiple of KEXSH iterations, */
/* . then try exceptional shifts. */
/* . Otherwise use shifts provided by */
/* . SLAQR3 above or from the eigenvalues */
/* . of a trailing principal submatrix. ==== */
if (ndfl % 6 == 0) {
ks = kbot - ns + 1;
/* Computing MAX */
i__3 = ks + 1, i__4 = ktop + 2;
i__2 = max(i__3,i__4);
for (i__ = kbot; i__ >= i__2; i__ += -2) {
ss = (r__1 = h__[i__ + (i__ - 1) * h_dim1], dabs(r__1)
) + (r__2 = h__[i__ - 1 + (i__ - 2) * h_dim1],
dabs(r__2));
aa = ss * .75f + h__[i__ + i__ * h_dim1];
bb = ss;
cc = ss * -.4375f;
dd = aa;
slanv2_(&aa, &bb, &cc, &dd, &wr[i__ - 1], &wi[i__ - 1]
, &wr[i__], &wi[i__], &cs, &sn);
/* L30: */
}
if (ks == ktop) {
wr[ks + 1] = h__[ks + 1 + (ks + 1) * h_dim1];
wi[ks + 1] = 0.f;
wr[ks] = wr[ks + 1];
wi[ks] = wi[ks + 1];
}
} else {
/* ==== Got NS/2 or fewer shifts? Use SLAQR4 or */
/* . SLAHQR on a trailing principal submatrix to */
/* . get more. (Since NS.LE.NSMAX.LE.(N+6)/9, */
/* . there is enough space below the subdiagonal */
/* . to fit an NS-by-NS scratch array.) ==== */
if (kbot - ks + 1 <= ns / 2) {
ks = kbot - ns + 1;
kt = *n - ns + 1;
slacpy_("A", &ns, &ns, &h__[ks + ks * h_dim1], ldh, &
h__[kt + h_dim1], ldh);
if (ns > nmin) {
slaqr4_(&c_false, &c_false, &ns, &c__1, &ns, &h__[
kt + h_dim1], ldh, &wr[ks], &wi[ks], &
c__1, &c__1, zdum, &c__1, &work[1], lwork,
&inf);
} else {
slahqr_(&c_false, &c_false, &ns, &c__1, &ns, &h__[
kt + h_dim1], ldh, &wr[ks], &wi[ks], &
c__1, &c__1, zdum, &c__1, &inf);
}
ks += inf;
/* ==== In case of a rare QR failure use */
/* . eigenvalues of the trailing 2-by-2 */
/* . principal submatrix. ==== */
if (ks >= kbot) {
aa = h__[kbot - 1 + (kbot - 1) * h_dim1];
cc = h__[kbot + (kbot - 1) * h_dim1];
bb = h__[kbot - 1 + kbot * h_dim1];
dd = h__[kbot + kbot * h_dim1];
slanv2_(&aa, &bb, &cc, &dd, &wr[kbot - 1], &wi[
kbot - 1], &wr[kbot], &wi[kbot], &cs, &sn)
;
ks = kbot - 1;
}
}
if (kbot - ks + 1 > ns) {
/* ==== Sort the shifts (Helps a little) */
/* . Bubble sort keeps complex conjugate */
/* . pairs together. ==== */
sorted = FALSE_;
i__2 = ks + 1;
for (k = kbot; k >= i__2; --k) {
if (sorted) {
goto L60;
}
sorted = TRUE_;
i__3 = k - 1;
for (i__ = ks; i__ <= i__3; ++i__) {
if ((r__1 = wr[i__], dabs(r__1)) + (r__2 = wi[
i__], dabs(r__2)) < (r__3 = wr[i__ +
1], dabs(r__3)) + (r__4 = wi[i__ + 1],
dabs(r__4))) {
sorted = FALSE_;
swap = wr[i__];
wr[i__] = wr[i__ + 1];
wr[i__ + 1] = swap;
swap = wi[i__];
wi[i__] = wi[i__ + 1];
wi[i__ + 1] = swap;
}
/* L40: */
}
/* L50: */
}
L60:
;
}
/* ==== Shuffle shifts into pairs of real shifts */
/* . and pairs of complex conjugate shifts */
/* . assuming complex conjugate shifts are */
/* . already adjacent to one another. (Yes, */
/* . they are.) ==== */
i__2 = ks + 2;
for (i__ = kbot; i__ >= i__2; i__ += -2) {
if (wi[i__] != -wi[i__ - 1]) {
swap = wr[i__];
wr[i__] = wr[i__ - 1];
wr[i__ - 1] = wr[i__ - 2];
wr[i__ - 2] = swap;
swap = wi[i__];
wi[i__] = wi[i__ - 1];
wi[i__ - 1] = wi[i__ - 2];
wi[i__ - 2] = swap;
}
/* L70: */
}
}
/* ==== If there are only two shifts and both are */
/* . real, then use only one. ==== */
if (kbot - ks + 1 == 2) {
if (wi[kbot] == 0.f) {
if ((r__1 = wr[kbot] - h__[kbot + kbot * h_dim1],
dabs(r__1)) < (r__2 = wr[kbot - 1] - h__[kbot
+ kbot * h_dim1], dabs(r__2))) {
wr[kbot - 1] = wr[kbot];
} else {
wr[kbot] = wr[kbot - 1];
}
}
}
/* ==== Use up to NS of the the smallest magnatiude */
/* . shifts. If there aren't NS shifts available, */
/* . then use them all, possibly dropping one to */
/* . make the number of shifts even. ==== */
/* Computing MIN */
i__2 = ns, i__3 = kbot - ks + 1;
ns = min(i__2,i__3);
ns -= ns % 2;
ks = kbot - ns + 1;
/* ==== Small-bulge multi-shift QR sweep: */
/* . split workspace under the subdiagonal into */
/* . - a KDU-by-KDU work array U in the lower */
/* . left-hand-corner, */
/* . - a KDU-by-at-least-KDU-but-more-is-better */
/* . (KDU-by-NHo) horizontal work array WH along */
/* . the bottom edge, */
/* . - and an at-least-KDU-but-more-is-better-by-KDU */
/* . (NVE-by-KDU) vertical work WV arrow along */
/* . the left-hand-edge. ==== */
kdu = ns * 3 - 3;
ku = *n - kdu + 1;
kwh = kdu + 1;
nho = *n - kdu - 3 - (kdu + 1) + 1;
kwv = kdu + 4;
nve = *n - kdu - kwv + 1;
/* ==== Small-bulge multi-shift QR sweep ==== */
slaqr5_(wantt, wantz, &kacc22, n, &ktop, &kbot, &ns, &wr[ks],
&wi[ks], &h__[h_offset], ldh, iloz, ihiz, &z__[
z_offset], ldz, &work[1], &c__3, &h__[ku + h_dim1],
ldh, &nve, &h__[kwv + h_dim1], ldh, &nho, &h__[ku +
kwh * h_dim1], ldh);
}
/* ==== Note progress (or the lack of it). ==== */
if (ld > 0) {
ndfl = 1;
} else {
++ndfl;
}
/* ==== End of main loop ==== */
/* L80: */
}
/* ==== Iteration limit exceeded. Set INFO to show where */
/* . the problem occurred and exit. ==== */
*info = kbot;
L90:
;
}
/* ==== Return the optimal value of LWORK. ==== */
work[1] = (real) lwkopt;
/* ==== End of SLAQR0 ==== */
return 0;
} /* slaqr0_ */
/* Subroutine */
int slaqr0_(logical *wantt, logical *wantz, integer *n, integer *ilo, integer *ihi, real *h__, integer *ldh, real *wr, real * wi, integer *iloz, integer *ihiz, real *z__, integer *ldz, real *work, integer *lwork, integer *info)
{
/* System generated locals */
integer h_dim1, h_offset, z_dim1, z_offset, i__1, i__2, i__3, i__4, i__5;
real r__1, r__2, r__3, r__4;
/* Local variables */
integer i__, k;
real aa, bb, cc, dd;
integer ld;
real cs;
integer nh, it, ks, kt;
real sn;
integer ku, kv, ls, ns;
real ss;
integer nw, inf, kdu, nho, nve, kwh, nsr, nwr, kwv, ndec, ndfl, kbot, nmin;
real swap;
integer ktop;
real zdum[1] /* was [1][1] */
;
integer kacc22, itmax, nsmax, nwmax, kwtop;
extern /* Subroutine */
int slanv2_(real *, real *, real *, real *, real * , real *, real *, real *, real *, real *), slaqr3_(logical *, logical *, integer *, integer *, integer *, integer *, real *, integer *, integer *, integer *, real *, integer *, integer *, integer *, real *, real *, real *, integer *, integer *, real *, integer *, integer *, real *, integer *, real *, integer *), slaqr4_(logical *, logical *, integer *, integer *, integer *, real *, integer *, real *, real *, integer *, integer *, real *, integer *, real *, integer *, integer *), slaqr5_(logical *, logical *, integer *, integer *, integer *, integer *, integer *, real *, real *, real *, integer *, integer *, integer *, real *, integer *, real *, integer *, real *, integer *, integer *, real * , integer *, integer *, real *, integer *);
integer nibble;
extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *);
char jbcmpz[2];
extern /* Subroutine */
int slahqr_(logical *, logical *, integer *, integer *, integer *, real *, integer *, real *, real *, integer * , integer *, real *, integer *, integer *), slacpy_(char *, integer *, integer *, real *, integer *, real *, integer *);
integer nwupbd;
logical sorted;
integer lwkopt;
/* -- 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 .. */
/* ==== Matrices of order NTINY or smaller must be processed by */
/* . SLAHQR because of insufficient subdiagonal scratch space. */
/* . (This is a hard limit.) ==== */
/* ==== Exceptional deflation windows: try to cure rare */
/* . slow convergence by varying the size of the */
/* . deflation window after KEXNW iterations. ==== */
/* ==== Exceptional shifts: try to cure rare slow convergence */
/* . with ad-hoc exceptional shifts every KEXSH iterations. */
/* . ==== */
/* ==== The constants WILK1 and WILK2 are used to form the */
/* . exceptional shifts. ==== */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Parameter adjustments */
h_dim1 = *ldh;
h_offset = 1 + h_dim1;
h__ -= h_offset;
--wr;
--wi;
z_dim1 = *ldz;
z_offset = 1 + z_dim1;
z__ -= z_offset;
--work;
/* Function Body */
*info = 0;
/* ==== Quick return for N = 0: nothing to do. ==== */
if (*n == 0)
{
work[1] = 1.f;
return 0;
}
if (*n <= 11)
{
/* ==== Tiny matrices must use SLAHQR. ==== */
lwkopt = 1;
if (*lwork != -1)
{
slahqr_(wantt, wantz, n, ilo, ihi, &h__[h_offset], ldh, &wr[1], & wi[1], iloz, ihiz, &z__[z_offset], ldz, info);
}
}
else
{
/* ==== Use small bulge multi-shift QR with aggressive early */
/* . deflation on larger-than-tiny matrices. ==== */
/* ==== Hope for the best. ==== */
*info = 0;
/* ==== Set up job flags for ILAENV. ==== */
if (*wantt)
{
*(unsigned char *)jbcmpz = 'S';
}
else
{
*(unsigned char *)jbcmpz = 'E';
}
if (*wantz)
{
*(unsigned char *)&jbcmpz[1] = 'V';
}
else
{
*(unsigned char *)&jbcmpz[1] = 'N';
}
/* ==== NWR = recommended deflation window size. At this */
/* . point, N .GT. NTINY = 11, so there is enough */
/* . subdiagonal workspace for NWR.GE.2 as required. */
/* . (In fact, there is enough subdiagonal space for */
/* . NWR.GE.3.) ==== */
nwr = ilaenv_(&c__13, "SLAQR0", jbcmpz, n, ilo, ihi, lwork);
nwr = max(2,nwr);
/* Computing MIN */
i__1 = *ihi - *ilo + 1;
i__2 = (*n - 1) / 3;
i__1 = min(i__1,i__2); // ; expr subst
nwr = min(i__1,nwr);
/* ==== NSR = recommended number of simultaneous shifts. */
/* . At this point N .GT. NTINY = 11, so there is at */
/* . enough subdiagonal workspace for NSR to be even */
/* . and greater than or equal to two as required. ==== */
nsr = ilaenv_(&c__15, "SLAQR0", jbcmpz, n, ilo, ihi, lwork);
/* Computing MIN */
i__1 = nsr, i__2 = (*n + 6) / 9;
i__1 = min(i__1,i__2);
i__2 = *ihi - *ilo; // ; expr subst
nsr = min(i__1,i__2);
/* Computing MAX */
i__1 = 2;
i__2 = nsr - nsr % 2; // , expr subst
nsr = max(i__1,i__2);
/* ==== Estimate optimal workspace ==== */
/* ==== Workspace query call to SLAQR3 ==== */
i__1 = nwr + 1;
slaqr3_(wantt, wantz, n, ilo, ihi, &i__1, &h__[h_offset], ldh, iloz, ihiz, &z__[z_offset], ldz, &ls, &ld, &wr[1], &wi[1], &h__[ h_offset], ldh, n, &h__[h_offset], ldh, n, &h__[h_offset], ldh, &work[1], &c_n1);
/* ==== Optimal workspace = MAX(SLAQR5, SLAQR3) ==== */
/* Computing MAX */
i__1 = nsr * 3 / 2;
i__2 = (integer) work[1]; // , expr subst
lwkopt = max(i__1,i__2);
/* ==== Quick return in case of workspace query. ==== */
if (*lwork == -1)
{
work[1] = (real) lwkopt;
return 0;
}
/* ==== SLAHQR/SLAQR0 crossover point ==== */
nmin = ilaenv_(&c__12, "SLAQR0", jbcmpz, n, ilo, ihi, lwork);
nmin = max(11,nmin);
/* ==== Nibble crossover point ==== */
nibble = ilaenv_(&c__14, "SLAQR0", jbcmpz, n, ilo, ihi, lwork);
nibble = max(0,nibble);
/* ==== Accumulate reflections during ttswp? Use block */
/* . 2-by-2 structure during matrix-matrix multiply? ==== */
kacc22 = ilaenv_(&c__16, "SLAQR0", jbcmpz, n, ilo, ihi, lwork);
kacc22 = max(0,kacc22);
kacc22 = min(2,kacc22);
/* ==== NWMAX = the largest possible deflation window for */
/* . which there is sufficient workspace. ==== */
/* Computing MIN */
i__1 = (*n - 1) / 3;
i__2 = *lwork / 2; // , expr subst
nwmax = min(i__1,i__2);
nw = nwmax;
/* ==== NSMAX = the Largest number of simultaneous shifts */
/* . for which there is sufficient workspace. ==== */
/* Computing MIN */
i__1 = (*n + 6) / 9;
i__2 = (*lwork << 1) / 3; // , expr subst
nsmax = min(i__1,i__2);
nsmax -= nsmax % 2;
/* ==== NDFL: an iteration count restarted at deflation. ==== */
ndfl = 1;
/* ==== ITMAX = iteration limit ==== */
/* Computing MAX */
i__1 = 10;
i__2 = *ihi - *ilo + 1; // , expr subst
itmax = max(i__1,i__2) * 30;
/* ==== Last row and column in the active block ==== */
kbot = *ihi;
/* ==== Main Loop ==== */
i__1 = itmax;
for (it = 1;
it <= i__1;
++it)
{
/* ==== Done when KBOT falls below ILO ==== */
if (kbot < *ilo)
{
goto L90;
}
/* ==== Locate active block ==== */
i__2 = *ilo + 1;
for (k = kbot;
k >= i__2;
--k)
{
if (h__[k + (k - 1) * h_dim1] == 0.f)
{
goto L20;
}
/* L10: */
}
k = *ilo;
L20:
ktop = k;
/* ==== Select deflation window size: */
/* . Typical Case: */
/* . If possible and advisable, nibble the entire */
/* . active block. If not, use size MIN(NWR,NWMAX) */
/* . or MIN(NWR+1,NWMAX) depending upon which has */
/* . the smaller corresponding subdiagonal entry */
/* . (a heuristic). */
/* . */
/* . Exceptional Case: */
/* . If there have been no deflations in KEXNW or */
/* . more iterations, then vary the deflation window */
/* . size. At first, because, larger windows are, */
/* . in general, more powerful than smaller ones, */
/* . rapidly increase the window to the maximum possible. */
/* . Then, gradually reduce the window size. ==== */
nh = kbot - ktop + 1;
nwupbd = min(nh,nwmax);
if (ndfl < 5)
{
nw = min(nwupbd,nwr);
}
else
{
/* Computing MIN */
i__2 = nwupbd;
i__3 = nw << 1; // , expr subst
nw = min(i__2,i__3);
}
if (nw < nwmax)
{
if (nw >= nh - 1)
{
nw = nh;
}
else
{
kwtop = kbot - nw + 1;
if ((r__1 = h__[kwtop + (kwtop - 1) * h_dim1], f2c_abs(r__1)) > (r__2 = h__[kwtop - 1 + (kwtop - 2) * h_dim1], f2c_abs(r__2)))
{
++nw;
}
}
}
if (ndfl < 5)
{
ndec = -1;
}
else if (ndec >= 0 || nw >= nwupbd)
{
++ndec;
if (nw - ndec < 2)
{
ndec = 0;
}
nw -= ndec;
}
/* ==== Aggressive early deflation: */
/* . split workspace under the subdiagonal into */
/* . - an nw-by-nw work array V in the lower */
/* . left-hand-corner, */
/* . - an NW-by-at-least-NW-but-more-is-better */
/* . (NW-by-NHO) horizontal work array along */
/* . the bottom edge, */
/* . - an at-least-NW-but-more-is-better (NHV-by-NW) */
/* . vertical work array along the left-hand-edge. */
/* . ==== */
kv = *n - nw + 1;
kt = nw + 1;
nho = *n - nw - 1 - kt + 1;
kwv = nw + 2;
nve = *n - nw - kwv + 1;
/* ==== Aggressive early deflation ==== */
slaqr3_(wantt, wantz, n, &ktop, &kbot, &nw, &h__[h_offset], ldh, iloz, ihiz, &z__[z_offset], ldz, &ls, &ld, &wr[1], &wi[1], &h__[kv + h_dim1], ldh, &nho, &h__[kv + kt * h_dim1], ldh, &nve, &h__[kwv + h_dim1], ldh, &work[1], lwork);
/* ==== Adjust KBOT accounting for new deflations. ==== */
kbot -= ld;
/* ==== KS points to the shifts. ==== */
ks = kbot - ls + 1;
/* ==== Skip an expensive QR sweep if there is a (partly */
/* . heuristic) reason to expect that many eigenvalues */
/* . will deflate without it. Here, the QR sweep is */
/* . skipped if many eigenvalues have just been deflated */
/* . or if the remaining active block is small. */
if (ld == 0 || ld * 100 <= nw * nibble && kbot - ktop + 1 > min( nmin,nwmax))
{
/* ==== NS = nominal number of simultaneous shifts. */
/* . This may be lowered (slightly) if SLAQR3 */
/* . did not provide that many shifts. ==== */
/* Computing MIN */
/* Computing MAX */
i__4 = 2;
i__5 = kbot - ktop; // , expr subst
i__2 = min(nsmax,nsr);
i__3 = max(i__4,i__5); // , expr subst
ns = min(i__2,i__3);
ns -= ns % 2;
/* ==== If there have been no deflations */
/* . in a multiple of KEXSH iterations, */
/* . then try exceptional shifts. */
/* . Otherwise use shifts provided by */
/* . SLAQR3 above or from the eigenvalues */
/* . of a trailing principal submatrix. ==== */
if (ndfl % 6 == 0)
{
ks = kbot - ns + 1;
/* Computing MAX */
i__3 = ks + 1;
i__4 = ktop + 2; // , expr subst
i__2 = max(i__3,i__4);
for (i__ = kbot;
i__ >= i__2;
i__ += -2)
{
ss = (r__1 = h__[i__ + (i__ - 1) * h_dim1], f2c_abs(r__1)) + (r__2 = h__[i__ - 1 + (i__ - 2) * h_dim1], f2c_abs(r__2));
aa = ss * .75f + h__[i__ + i__ * h_dim1];
bb = ss;
cc = ss * -.4375f;
dd = aa;
slanv2_(&aa, &bb, &cc, &dd, &wr[i__ - 1], &wi[i__ - 1] , &wr[i__], &wi[i__], &cs, &sn);
/* L30: */
}
if (ks == ktop)
{
wr[ks + 1] = h__[ks + 1 + (ks + 1) * h_dim1];
wi[ks + 1] = 0.f;
wr[ks] = wr[ks + 1];
wi[ks] = wi[ks + 1];
}
}
else
{
/* ==== Got NS/2 or fewer shifts? Use SLAQR4 or */
/* . SLAHQR on a trailing principal submatrix to */
/* . get more. (Since NS.LE.NSMAX.LE.(N+6)/9, */
/* . there is enough space below the subdiagonal */
/* . to fit an NS-by-NS scratch array.) ==== */
if (kbot - ks + 1 <= ns / 2)
{
ks = kbot - ns + 1;
kt = *n - ns + 1;
slacpy_("A", &ns, &ns, &h__[ks + ks * h_dim1], ldh, & h__[kt + h_dim1], ldh);
if (ns > nmin)
{
slaqr4_(&c_false, &c_false, &ns, &c__1, &ns, &h__[ kt + h_dim1], ldh, &wr[ks], &wi[ks], & c__1, &c__1, zdum, &c__1, &work[1], lwork, &inf);
}
else
{
slahqr_(&c_false, &c_false, &ns, &c__1, &ns, &h__[ kt + h_dim1], ldh, &wr[ks], &wi[ks], & c__1, &c__1, zdum, &c__1, &inf);
}
ks += inf;
/* ==== In case of a rare QR failure use */
/* . eigenvalues of the trailing 2-by-2 */
/* . principal submatrix. ==== */
if (ks >= kbot)
{
aa = h__[kbot - 1 + (kbot - 1) * h_dim1];
cc = h__[kbot + (kbot - 1) * h_dim1];
bb = h__[kbot - 1 + kbot * h_dim1];
dd = h__[kbot + kbot * h_dim1];
slanv2_(&aa, &bb, &cc, &dd, &wr[kbot - 1], &wi[ kbot - 1], &wr[kbot], &wi[kbot], &cs, &sn) ;
ks = kbot - 1;
}
}
if (kbot - ks + 1 > ns)
{
/* ==== Sort the shifts (Helps a little) */
/* . Bubble sort keeps complex conjugate */
/* . pairs together. ==== */
sorted = FALSE_;
i__2 = ks + 1;
for (k = kbot;
k >= i__2;
--k)
{
if (sorted)
{
goto L60;
}
sorted = TRUE_;
i__3 = k - 1;
for (i__ = ks;
i__ <= i__3;
++i__)
{
if ((r__1 = wr[i__], f2c_abs(r__1)) + (r__2 = wi[ i__], f2c_abs(r__2)) < (r__3 = wr[i__ + 1] , f2c_abs(r__3)) + (r__4 = wi[i__ + 1], f2c_abs(r__4)))
{
sorted = FALSE_;
swap = wr[i__];
wr[i__] = wr[i__ + 1];
wr[i__ + 1] = swap;
swap = wi[i__];
wi[i__] = wi[i__ + 1];
wi[i__ + 1] = swap;
}
/* L40: */
}
/* L50: */
}
L60:
;
}
/* ==== Shuffle shifts into pairs of real shifts */
/* . and pairs of complex conjugate shifts */
/* . assuming complex conjugate shifts are */
/* . already adjacent to one another. (Yes, */
/* . they are.) ==== */
i__2 = ks + 2;
for (i__ = kbot;
i__ >= i__2;
i__ += -2)
{
if (wi[i__] != -wi[i__ - 1])
{
swap = wr[i__];
wr[i__] = wr[i__ - 1];
wr[i__ - 1] = wr[i__ - 2];
wr[i__ - 2] = swap;
swap = wi[i__];
wi[i__] = wi[i__ - 1];
wi[i__ - 1] = wi[i__ - 2];
wi[i__ - 2] = swap;
}
/* L70: */
}
}
/* ==== If there are only two shifts and both are */
/* . real, then use only one. ==== */
if (kbot - ks + 1 == 2)
{
if (wi[kbot] == 0.f)
{
if ((r__1 = wr[kbot] - h__[kbot + kbot * h_dim1], f2c_abs( r__1)) < (r__2 = wr[kbot - 1] - h__[kbot + kbot * h_dim1], f2c_abs(r__2)))
{
wr[kbot - 1] = wr[kbot];
}
else
{
wr[kbot] = wr[kbot - 1];
}
}
}
/* ==== Use up to NS of the the smallest magnatiude */
/* . shifts. If there aren't NS shifts available, */
/* . then use them all, possibly dropping one to */
/* . make the number of shifts even. ==== */
/* Computing MIN */
i__2 = ns;
i__3 = kbot - ks + 1; // , expr subst
ns = min(i__2,i__3);
ns -= ns % 2;
ks = kbot - ns + 1;
/* ==== Small-bulge multi-shift QR sweep: */
/* . split workspace under the subdiagonal into */
/* . - a KDU-by-KDU work array U in the lower */
/* . left-hand-corner, */
/* . - a KDU-by-at-least-KDU-but-more-is-better */
/* . (KDU-by-NHo) horizontal work array WH along */
/* . the bottom edge, */
/* . - and an at-least-KDU-but-more-is-better-by-KDU */
/* . (NVE-by-KDU) vertical work WV arrow along */
/* . the left-hand-edge. ==== */
kdu = ns * 3 - 3;
ku = *n - kdu + 1;
kwh = kdu + 1;
nho = *n - kdu - 3 - (kdu + 1) + 1;
kwv = kdu + 4;
nve = *n - kdu - kwv + 1;
/* ==== Small-bulge multi-shift QR sweep ==== */
slaqr5_(wantt, wantz, &kacc22, n, &ktop, &kbot, &ns, &wr[ks], &wi[ks], &h__[h_offset], ldh, iloz, ihiz, &z__[ z_offset], ldz, &work[1], &c__3, &h__[ku + h_dim1], ldh, &nve, &h__[kwv + h_dim1], ldh, &nho, &h__[ku + kwh * h_dim1], ldh);
}
/* ==== Note progress (or the lack of it). ==== */
if (ld > 0)
{
ndfl = 1;
}
else
{
++ndfl;
}
/* ==== End of main loop ==== */
/* L80: */
}
/* ==== Iteration limit exceeded. Set INFO to show where */
/* . the problem occurred and exit. ==== */
*info = kbot;
L90:
;
}
/* ==== Return the optimal value of LWORK. ==== */
work[1] = (real) lwkopt;
/* ==== End of SLAQR0 ==== */
return 0;
}
