/* Subroutine */ int claqr0_(logical *wantt, logical *wantz, integer *n,
integer *ilo, integer *ihi, complex *h__, integer *ldh, complex *w,
integer *iloz, integer *ihiz, complex *z__, integer *ldz, complex *
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, r__5, r__6, r__7, r__8;
complex q__1, q__2, q__3, q__4, q__5;
/* Local variables */
integer i__, k;
real s;
complex aa, bb, cc, dd;
integer ld, nh, it, ks, kt, ku, kv, ls, ns, nw;
complex tr2, det;
integer inf, kdu, nho, nve, kwh, nsr, nwr, kwv, ndec, ndfl, kbot, nmin;
complex swap;
integer ktop;
complex zdum[1] /* was [1][1] */;
integer kacc22, itmax, nsmax, nwmax, kwtop;
integer nibble;
char jbcmpz[1];
complex rtdisc;
integer nwupbd;
logical sorted;
integer lwkopt;
/* -- LAPACK auxiliary routine (version 3.2) -- */
/* November 2006 */
/* Purpose */
/* ======= */
/* CLAQR0 computes the eigenvalues of a Hessenberg matrix H */
/* and, optionally, the matrices T and Z from the Schur decomposition */
/* H = Z T Z**H, where T is an upper triangular matrix (the */
/* Schur form), and Z is the unitary matrix of Schur vectors. */
/* Optionally Z may be postmultiplied into an input unitary */
/* 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 unitary matrix Q: A = Q*H*Q**H = (QZ)*H*(QZ)**H. */
/* 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 CGEBAL, and then passed to CGEHRD when the */
/* matrix output by CGEBAL 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) COMPLEX 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 triangular matrix T from the Schur */
/* decomposition (the Schur form). If INFO = 0 and WANT 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 */
/* LDH (input) INTEGER */
/* The leading dimension of the array H. LDH .GE. max(1,N). */
/* W (output) COMPLEX array, dimension (N) */
/* The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored */
/* in W(ILO:IHI). 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 W(i) = H(i,i). */
/* Z (input/output) COMPLEX 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) COMPLEX 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 CLAQR0 does a workspace query. */
/* In this case, CLAQR0 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, CLAQR0 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 a unitary matrix. The final */
/* value of H is upper Hessenberg and 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 unitary 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. */
/* ================================================================ */
/* ==== Matrices of order NTINY or smaller must be processed by */
/* . CLAHQR 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 constant WILK1 is used to form the exceptional */
/* . shifts. ==== */
/* Parameter adjustments */
h_dim1 = *ldh;
h_offset = 1 + h_dim1;
h__ -= h_offset;
--w;
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].r = 1.f, work[1].i = 0.f;
return 0;
}
if (*n <= 11) {
/* ==== Tiny matrices must use CLAHQR. ==== */
lwkopt = 1;
if (*lwork != -1) {
clahqr_(wantt, wantz, n, ilo, ihi, &h__[h_offset], ldh, &w[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, "CLAQR0", 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, "CLAQR0", 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 CLAQR3 ==== */
i__1 = nwr + 1;
claqr3_(wantt, wantz, n, ilo, ihi, &i__1, &h__[h_offset], ldh, iloz,
ihiz, &z__[z_offset], ldz, &ls, &ld, &w[1], &h__[h_offset],
ldh, n, &h__[h_offset], ldh, n, &h__[h_offset], ldh, &work[1],
&c_n1);
/* ==== Optimal workspace = MAX(CLAQR5, CLAQR3) ==== */
/* Computing MAX */
i__1 = nsr * 3 / 2, i__2 = (integer) work[1].r;
lwkopt = max(i__1,i__2);
/* ==== Quick return in case of workspace query. ==== */
if (*lwork == -1) {
r__1 = (real) lwkopt;
q__1.r = r__1, q__1.i = 0.f;
work[1].r = q__1.r, work[1].i = q__1.i;
return 0;
}
/* ==== CLAHQR/CLAQR0 crossover point ==== */
nmin = ilaenv_(&c__12, "CLAQR0", jbcmpz, n, ilo, ihi, lwork);
nmin = max(11,nmin);
/* ==== Nibble crossover point ==== */
nibble = ilaenv_(&c__14, "CLAQR0", 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, "CLAQR0", 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 L80;
}
/* ==== Locate active block ==== */
i__2 = *ilo + 1;
for (k = kbot; k >= i__2; --k) {
i__3 = k + (k - 1) * h_dim1;
if (h__[i__3].r == 0.f && h__[i__3].i == 0.f) {
goto L20;
}
}
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;
i__2 = kwtop + (kwtop - 1) * h_dim1;
i__3 = kwtop - 1 + (kwtop - 2) * h_dim1;
if ((r__1 = h__[i__2].r, dabs(r__1)) + (r__2 = r_imag(&
h__[kwtop + (kwtop - 1) * h_dim1]), dabs(r__2)) >
(r__3 = h__[i__3].r, dabs(r__3)) + (r__4 = r_imag(
&h__[kwtop - 1 + (kwtop - 2) * h_dim1]), dabs(
r__4))) {
++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 ==== */
claqr3_(wantt, wantz, n, &ktop, &kbot, &nw, &h__[h_offset], ldh,
iloz, ihiz, &z__[z_offset], ldz, &ls, &ld, &w[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 CLAQR3 */
/* . 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 */
/* . CLAQR3 above or from the eigenvalues */
/* . of a trailing principal submatrix. ==== */
if (ndfl % 6 == 0) {
ks = kbot - ns + 1;
i__2 = ks + 1;
for (i__ = kbot; i__ >= i__2; i__ += -2) {
i__3 = i__;
i__4 = i__ + i__ * h_dim1;
i__5 = i__ + (i__ - 1) * h_dim1;
r__3 = ((r__1 = h__[i__5].r, dabs(r__1)) + (r__2 =
r_imag(&h__[i__ + (i__ - 1) * h_dim1]), dabs(
r__2))) * .75f;
q__1.r = h__[i__4].r + r__3, q__1.i = h__[i__4].i;
w[i__3].r = q__1.r, w[i__3].i = q__1.i;
i__3 = i__ - 1;
i__4 = i__;
w[i__3].r = w[i__4].r, w[i__3].i = w[i__4].i;
}
} else {
/* ==== Got NS/2 or fewer shifts? Use CLAQR4 or */
/* . CLAHQR 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;
clacpy_("A", &ns, &ns, &h__[ks + ks * h_dim1], ldh, &
h__[kt + h_dim1], ldh);
if (ns > nmin) {
claqr4_(&c_false, &c_false, &ns, &c__1, &ns, &h__[
kt + h_dim1], ldh, &w[ks], &c__1, &c__1,
zdum, &c__1, &work[1], lwork, &inf);
} else {
clahqr_(&c_false, &c_false, &ns, &c__1, &ns, &h__[
kt + h_dim1], ldh, &w[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. Scale to avoid */
/* . overflows, underflows and subnormals. */
/* . (The scale factor S can not be zero, */
/* . because H(KBOT,KBOT-1) is nonzero.) ==== */
if (ks >= kbot) {
i__2 = kbot - 1 + (kbot - 1) * h_dim1;
i__3 = kbot + (kbot - 1) * h_dim1;
i__4 = kbot - 1 + kbot * h_dim1;
i__5 = kbot + kbot * h_dim1;
s = (r__1 = h__[i__2].r, dabs(r__1)) + (r__2 =
r_imag(&h__[kbot - 1 + (kbot - 1) *
h_dim1]), dabs(r__2)) + ((r__3 = h__[i__3]
.r, dabs(r__3)) + (r__4 = r_imag(&h__[
kbot + (kbot - 1) * h_dim1]), dabs(r__4)))
+ ((r__5 = h__[i__4].r, dabs(r__5)) + (
r__6 = r_imag(&h__[kbot - 1 + kbot *
h_dim1]), dabs(r__6))) + ((r__7 = h__[
i__5].r, dabs(r__7)) + (r__8 = r_imag(&
h__[kbot + kbot * h_dim1]), dabs(r__8)));
i__2 = kbot - 1 + (kbot - 1) * h_dim1;
q__1.r = h__[i__2].r / s, q__1.i = h__[i__2].i /
s;
aa.r = q__1.r, aa.i = q__1.i;
i__2 = kbot + (kbot - 1) * h_dim1;
q__1.r = h__[i__2].r / s, q__1.i = h__[i__2].i /
s;
cc.r = q__1.r, cc.i = q__1.i;
i__2 = kbot - 1 + kbot * h_dim1;
q__1.r = h__[i__2].r / s, q__1.i = h__[i__2].i /
s;
bb.r = q__1.r, bb.i = q__1.i;
i__2 = kbot + kbot * h_dim1;
q__1.r = h__[i__2].r / s, q__1.i = h__[i__2].i /
s;
dd.r = q__1.r, dd.i = q__1.i;
q__2.r = aa.r + dd.r, q__2.i = aa.i + dd.i;
q__1.r = q__2.r / 2.f, q__1.i = q__2.i / 2.f;
tr2.r = q__1.r, tr2.i = q__1.i;
q__3.r = aa.r - tr2.r, q__3.i = aa.i - tr2.i;
q__4.r = dd.r - tr2.r, q__4.i = dd.i - tr2.i;
q__2.r = q__3.r * q__4.r - q__3.i * q__4.i,
q__2.i = q__3.r * q__4.i + q__3.i *
q__4.r;
q__5.r = bb.r * cc.r - bb.i * cc.i, q__5.i = bb.r
* cc.i + bb.i * cc.r;
q__1.r = q__2.r - q__5.r, q__1.i = q__2.i -
q__5.i;
det.r = q__1.r, det.i = q__1.i;
q__2.r = -det.r, q__2.i = -det.i;
c_sqrt(&q__1, &q__2);
rtdisc.r = q__1.r, rtdisc.i = q__1.i;
i__2 = kbot - 1;
q__2.r = tr2.r + rtdisc.r, q__2.i = tr2.i +
rtdisc.i;
q__1.r = s * q__2.r, q__1.i = s * q__2.i;
w[i__2].r = q__1.r, w[i__2].i = q__1.i;
i__2 = kbot;
q__2.r = tr2.r - rtdisc.r, q__2.i = tr2.i -
rtdisc.i;
q__1.r = s * q__2.r, q__1.i = s * q__2.i;
w[i__2].r = q__1.r, w[i__2].i = q__1.i;
ks = kbot - 1;
}
}
if (kbot - ks + 1 > ns) {
/* ==== Sort the shifts (Helps a little) ==== */
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__) {
i__4 = i__;
i__5 = i__ + 1;
if ((r__1 = w[i__4].r, dabs(r__1)) + (r__2 =
r_imag(&w[i__]), dabs(r__2)) < (r__3 =
w[i__5].r, dabs(r__3)) + (r__4 =
r_imag(&w[i__ + 1]), dabs(r__4))) {
sorted = FALSE_;
i__4 = i__;
swap.r = w[i__4].r, swap.i = w[i__4].i;
i__4 = i__;
i__5 = i__ + 1;
w[i__4].r = w[i__5].r, w[i__4].i = w[i__5]
.i;
i__4 = i__ + 1;
w[i__4].r = swap.r, w[i__4].i = swap.i;
}
}
}
L60:
;
}
}
/* ==== If there are only two shifts, then use */
/* . only one. ==== */
if (kbot - ks + 1 == 2) {
i__2 = kbot;
i__3 = kbot + kbot * h_dim1;
q__2.r = w[i__2].r - h__[i__3].r, q__2.i = w[i__2].i -
h__[i__3].i;
q__1.r = q__2.r, q__1.i = q__2.i;
i__4 = kbot - 1;
i__5 = kbot + kbot * h_dim1;
q__4.r = w[i__4].r - h__[i__5].r, q__4.i = w[i__4].i -
h__[i__5].i;
q__3.r = q__4.r, q__3.i = q__4.i;
if ((r__1 = q__1.r, dabs(r__1)) + (r__2 = r_imag(&q__1),
dabs(r__2)) < (r__3 = q__3.r, dabs(r__3)) + (r__4
= r_imag(&q__3), dabs(r__4))) {
i__2 = kbot - 1;
i__3 = kbot;
w[i__2].r = w[i__3].r, w[i__2].i = w[i__3].i;
} else {
i__2 = kbot;
i__3 = kbot - 1;
w[i__2].r = w[i__3].r, w[i__2].i = w[i__3].i;
}
}
/* ==== 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 ==== */
claqr5_(wantt, wantz, &kacc22, n, &ktop, &kbot, &ns, &w[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 ==== */
}
/* ==== Iteration limit exceeded. Set INFO to show where */
/* . the problem occurred and exit. ==== */
*info = kbot;
L80:
;
}
/* ==== Return the optimal value of LWORK. ==== */
r__1 = (real) lwkopt;
q__1.r = r__1, q__1.i = 0.f;
work[1].r = q__1.r, work[1].i = q__1.i;
/* ==== End of CLAQR0 ==== */
return 0;
} /* claqr0_ */
int claqr3_(int *wantt, int *wantz, int *n,
int *ktop, int *kbot, int *nw, complex *h__, int *ldh,
int *iloz, int *ihiz, complex *z__, int *ldz, int *
ns, int *nd, complex *sh, complex *v, int *ldv, int *nh,
complex *t, int *ldt, int *nv, complex *wv, int *ldwv,
complex *work, int *lwork)
{
/* System generated locals */
int h_dim1, h_offset, t_dim1, t_offset, v_dim1, v_offset, wv_dim1,
wv_offset, z_dim1, z_offset, i__1, i__2, i__3, i__4;
float r__1, r__2, r__3, r__4, r__5, r__6;
complex q__1, q__2;
/* Builtin functions */
double r_imag(complex *);
void r_cnjg(complex *, complex *);
/* Local variables */
int i__, j;
complex s;
int jw;
float foo;
int kln;
complex tau;
int knt;
float ulp;
int lwk1, lwk2, lwk3;
complex beta;
int kcol, info, nmin, ifst, ilst, ltop, krow;
extern int clarf_(char *, int *, int *, complex *
, int *, complex *, complex *, int *, complex *),
cgemm_(char *, char *, int *, int *, int *, complex *,
complex *, int *, complex *, int *, complex *, complex *,
int *), ccopy_(int *, complex *, int
*, complex *, int *);
int infqr, kwtop;
extern int claqr4_(int *, int *, int *,
int *, int *, complex *, int *, complex *, int *,
int *, complex *, int *, complex *, int *, int *),
slabad_(float *, float *), cgehrd_(int *, int *, int *,
complex *, int *, complex *, complex *, int *, int *)
, clarfg_(int *, complex *, complex *, int *, complex *);
extern double slamch_(char *);
extern int clahqr_(int *, int *, int *,
int *, int *, complex *, int *, complex *, int *,
int *, complex *, int *, int *), clacpy_(char *,
int *, int *, complex *, int *, complex *, int *), claset_(char *, int *, int *, complex *, complex
*, complex *, int *);
float safmin;
extern int ilaenv_(int *, char *, char *, int *, int *,
int *, int *);
float safmax;
extern int ctrexc_(char *, int *, complex *, int
*, complex *, int *, int *, int *, int *),
cunmhr_(char *, char *, int *, int *, int *, int
*, complex *, int *, complex *, complex *, int *, complex
*, int *, int *);
float smlnum;
int lwkopt;
/* -- LAPACK auxiliary routine (version 3.2.1) -- */
/* Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd.. */
/* -- April 2009 -- */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* ****************************************************************** */
/* Aggressive early deflation: */
/* This subroutine accepts as input an upper Hessenberg matrix */
/* H and performs an unitary similarity transformation */
/* designed to detect and deflate fully converged eigenvalues from */
/* a trailing principal submatrix. On output H has been over- */
/* written by a new Hessenberg matrix that is a perturbation of */
/* an unitary similarity transformation of H. It is to be */
/* hoped that the final version of H has many zero subdiagonal */
/* entries. */
/* ****************************************************************** */
/* WANTT (input) LOGICAL */
/* If .TRUE., then the Hessenberg matrix H is fully updated */
/* so that the triangular Schur factor may be */
/* computed (in cooperation with the calling subroutine). */
/* If .FALSE., then only enough of H is updated to preserve */
/* the eigenvalues. */
/* WANTZ (input) LOGICAL */
/* If .TRUE., then the unitary matrix Z is updated so */
/* so that the unitary Schur factor may be computed */
/* (in cooperation with the calling subroutine). */
/* If .FALSE., then Z is not referenced. */
/* N (input) INTEGER */
/* The order of the matrix H and (if WANTZ is .TRUE.) the */
/* order of the unitary matrix Z. */
/* KTOP (input) INTEGER */
/* It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0. */
/* KBOT and KTOP together determine an isolated block */
/* along the diagonal of the Hessenberg matrix. */
/* KBOT (input) INTEGER */
/* It is assumed without a check that either */
/* KBOT = N or H(KBOT+1,KBOT)=0. KBOT and KTOP together */
/* determine an isolated block along the diagonal of the */
/* Hessenberg matrix. */
/* NW (input) INTEGER */
/* Deflation window size. 1 .LE. NW .LE. (KBOT-KTOP+1). */
/* H (input/output) COMPLEX array, dimension (LDH,N) */
/* On input the initial N-by-N section of H stores the */
/* Hessenberg matrix undergoing aggressive early deflation. */
/* On output H has been transformed by a unitary */
/* similarity transformation, perturbed, and the returned */
/* to Hessenberg form that (it is to be hoped) has some */
/* zero subdiagonal entries. */
/* LDH (input) int */
/* Leading dimension of H just as declared in the calling */
/* subroutine. N .LE. LDH */
/* 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. IHIZ .LE. N. */
/* Z (input/output) COMPLEX array, dimension (LDZ,N) */
/* IF WANTZ is .TRUE., then on output, the unitary */
/* similarity transformation mentioned above has been */
/* accumulated into Z(ILOZ:IHIZ,ILO:IHI) from the right. */
/* If WANTZ is .FALSE., then Z is unreferenced. */
/* LDZ (input) int */
/* The leading dimension of Z just as declared in the */
/* calling subroutine. 1 .LE. LDZ. */
/* NS (output) int */
/* The number of unconverged (ie approximate) eigenvalues */
/* returned in SR and SI that may be used as shifts by the */
/* calling subroutine. */
/* ND (output) int */
/* The number of converged eigenvalues uncovered by this */
/* subroutine. */
/* SH (output) COMPLEX array, dimension KBOT */
/* On output, approximate eigenvalues that may */
/* be used for shifts are stored in SH(KBOT-ND-NS+1) */
/* through SR(KBOT-ND). Converged eigenvalues are */
/* stored in SH(KBOT-ND+1) through SH(KBOT). */
/* V (workspace) COMPLEX array, dimension (LDV,NW) */
/* An NW-by-NW work array. */
/* LDV (input) int scalar */
/* The leading dimension of V just as declared in the */
/* calling subroutine. NW .LE. LDV */
/* NH (input) int scalar */
/* The number of columns of T. NH.GE.NW. */
/* T (workspace) COMPLEX array, dimension (LDT,NW) */
/* LDT (input) int */
/* The leading dimension of T just as declared in the */
/* calling subroutine. NW .LE. LDT */
/* NV (input) int */
/* The number of rows of work array WV available for */
/* workspace. NV.GE.NW. */
/* WV (workspace) COMPLEX array, dimension (LDWV,NW) */
/* LDWV (input) int */
/* The leading dimension of W just as declared in the */
/* calling subroutine. NW .LE. LDV */
/* WORK (workspace) COMPLEX array, dimension LWORK. */
/* On exit, WORK(1) is set to an estimate of the optimal value */
/* of LWORK for the given values of N, NW, KTOP and KBOT. */
/* LWORK (input) int */
/* The dimension of the work array WORK. LWORK = 2*NW */
/* suffices, but greater efficiency may result from larger */
/* values of LWORK. */
/* If LWORK = -1, then a workspace query is assumed; CLAQR3 */
/* only estimates the optimal workspace size for the given */
/* values of N, NW, KTOP and KBOT. The estimate is returned */
/* in WORK(1). No error message related to LWORK is issued */
/* by XERBLA. Neither H nor Z are accessed. */
/* ================================================================ */
/* Based on contributions by */
/* Karen Braman and Ralph Byers, Department of Mathematics, */
/* University of Kansas, USA */
/* ================================================================ */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. Executable Statements .. */
/* ==== Estimate optimal workspace. ==== */
/* Parameter adjustments */
h_dim1 = *ldh;
h_offset = 1 + h_dim1;
h__ -= h_offset;
z_dim1 = *ldz;
z_offset = 1 + z_dim1;
z__ -= z_offset;
--sh;
v_dim1 = *ldv;
v_offset = 1 + v_dim1;
v -= v_offset;
t_dim1 = *ldt;
t_offset = 1 + t_dim1;
t -= t_offset;
wv_dim1 = *ldwv;
wv_offset = 1 + wv_dim1;
wv -= wv_offset;
--work;
/* Function Body */
/* Computing MIN */
i__1 = *nw, i__2 = *kbot - *ktop + 1;
jw = MIN(i__1,i__2);
if (jw <= 2) {
lwkopt = 1;
} else {
/* ==== Workspace query call to CGEHRD ==== */
i__1 = jw - 1;
cgehrd_(&jw, &c__1, &i__1, &t[t_offset], ldt, &work[1], &work[1], &
c_n1, &info);
lwk1 = (int) work[1].r;
/* ==== Workspace query call to CUNMHR ==== */
i__1 = jw - 1;
cunmhr_("R", "N", &jw, &jw, &c__1, &i__1, &t[t_offset], ldt, &work[1],
&v[v_offset], ldv, &work[1], &c_n1, &info);
lwk2 = (int) work[1].r;
/* ==== Workspace query call to CLAQR4 ==== */
claqr4_(&c_true, &c_true, &jw, &c__1, &jw, &t[t_offset], ldt, &sh[1],
&c__1, &jw, &v[v_offset], ldv, &work[1], &c_n1, &infqr);
lwk3 = (int) work[1].r;
/* ==== Optimal workspace ==== */
/* Computing MAX */
i__1 = jw + MAX(lwk1,lwk2);
lwkopt = MAX(i__1,lwk3);
}
/* ==== Quick return in case of workspace query. ==== */
if (*lwork == -1) {
r__1 = (float) lwkopt;
q__1.r = r__1, q__1.i = 0.f;
work[1].r = q__1.r, work[1].i = q__1.i;
return 0;
}
/* ==== Nothing to do ... */
/* ... for an empty active block ... ==== */
*ns = 0;
*nd = 0;
work[1].r = 1.f, work[1].i = 0.f;
if (*ktop > *kbot) {
return 0;
}
/* ... nor for an empty deflation window. ==== */
if (*nw < 1) {
return 0;
}
/* ==== Machine constants ==== */
safmin = slamch_("SAFE MINIMUM");
safmax = 1.f / safmin;
slabad_(&safmin, &safmax);
ulp = slamch_("PRECISION");
smlnum = safmin * ((float) (*n) / ulp);
/* ==== Setup deflation window ==== */
/* Computing MIN */
i__1 = *nw, i__2 = *kbot - *ktop + 1;
jw = MIN(i__1,i__2);
kwtop = *kbot - jw + 1;
if (kwtop == *ktop) {
s.r = 0.f, s.i = 0.f;
} else {
i__1 = kwtop + (kwtop - 1) * h_dim1;
s.r = h__[i__1].r, s.i = h__[i__1].i;
}
if (*kbot == kwtop) {
/* ==== 1-by-1 deflation window: not much to do ==== */
i__1 = kwtop;
i__2 = kwtop + kwtop * h_dim1;
sh[i__1].r = h__[i__2].r, sh[i__1].i = h__[i__2].i;
*ns = 1;
*nd = 0;
/* Computing MAX */
i__1 = kwtop + kwtop * h_dim1;
r__5 = smlnum, r__6 = ulp * ((r__1 = h__[i__1].r, ABS(r__1)) + (r__2
= r_imag(&h__[kwtop + kwtop * h_dim1]), ABS(r__2)));
if ((r__3 = s.r, ABS(r__3)) + (r__4 = r_imag(&s), ABS(r__4)) <=
MAX(r__5,r__6)) {
*ns = 0;
*nd = 1;
if (kwtop > *ktop) {
i__1 = kwtop + (kwtop - 1) * h_dim1;
h__[i__1].r = 0.f, h__[i__1].i = 0.f;
}
}
work[1].r = 1.f, work[1].i = 0.f;
return 0;
}
/* ==== Convert to spike-triangular form. (In case of a */
/* . rare QR failure, this routine continues to do */
/* . aggressive early deflation using that part of */
/* . the deflation window that converged using INFQR */
/* . here and there to keep track.) ==== */
clacpy_("U", &jw, &jw, &h__[kwtop + kwtop * h_dim1], ldh, &t[t_offset],
ldt);
i__1 = jw - 1;
i__2 = *ldh + 1;
i__3 = *ldt + 1;
ccopy_(&i__1, &h__[kwtop + 1 + kwtop * h_dim1], &i__2, &t[t_dim1 + 2], &
i__3);
claset_("A", &jw, &jw, &c_b1, &c_b2, &v[v_offset], ldv);
nmin = ilaenv_(&c__12, "CLAQR3", "SV", &jw, &c__1, &jw, lwork);
if (jw > nmin) {
claqr4_(&c_true, &c_true, &jw, &c__1, &jw, &t[t_offset], ldt, &sh[
kwtop], &c__1, &jw, &v[v_offset], ldv, &work[1], lwork, &
infqr);
} else {
clahqr_(&c_true, &c_true, &jw, &c__1, &jw, &t[t_offset], ldt, &sh[
kwtop], &c__1, &jw, &v[v_offset], ldv, &infqr);
}
/* ==== Deflation detection loop ==== */
*ns = jw;
ilst = infqr + 1;
i__1 = jw;
for (knt = infqr + 1; knt <= i__1; ++knt) {
/* ==== Small spike tip deflation test ==== */
i__2 = *ns + *ns * t_dim1;
foo = (r__1 = t[i__2].r, ABS(r__1)) + (r__2 = r_imag(&t[*ns + *ns *
t_dim1]), ABS(r__2));
if (foo == 0.f) {
foo = (r__1 = s.r, ABS(r__1)) + (r__2 = r_imag(&s), ABS(r__2));
}
i__2 = *ns * v_dim1 + 1;
/* Computing MAX */
r__5 = smlnum, r__6 = ulp * foo;
if (((r__1 = s.r, ABS(r__1)) + (r__2 = r_imag(&s), ABS(r__2))) * ((
r__3 = v[i__2].r, ABS(r__3)) + (r__4 = r_imag(&v[*ns *
v_dim1 + 1]), ABS(r__4))) <= MAX(r__5,r__6)) {
/* ==== One more converged eigenvalue ==== */
--(*ns);
} else {
/* ==== One undeflatable eigenvalue. Move it up out of the */
/* . way. (CTREXC can not fail in this case.) ==== */
ifst = *ns;
ctrexc_("V", &jw, &t[t_offset], ldt, &v[v_offset], ldv, &ifst, &
ilst, &info);
++ilst;
}
/* L10: */
}
/* ==== Return to Hessenberg form ==== */
if (*ns == 0) {
s.r = 0.f, s.i = 0.f;
}
if (*ns < jw) {
/* ==== sorting the diagonal of T improves accuracy for */
/* . graded matrices. ==== */
i__1 = *ns;
for (i__ = infqr + 1; i__ <= i__1; ++i__) {
ifst = i__;
i__2 = *ns;
for (j = i__ + 1; j <= i__2; ++j) {
i__3 = j + j * t_dim1;
i__4 = ifst + ifst * t_dim1;
if ((r__1 = t[i__3].r, ABS(r__1)) + (r__2 = r_imag(&t[j + j *
t_dim1]), ABS(r__2)) > (r__3 = t[i__4].r, ABS(r__3)
) + (r__4 = r_imag(&t[ifst + ifst * t_dim1]), ABS(
r__4))) {
ifst = j;
}
/* L20: */
}
ilst = i__;
if (ifst != ilst) {
ctrexc_("V", &jw, &t[t_offset], ldt, &v[v_offset], ldv, &ifst,
&ilst, &info);
}
/* L30: */
}
}
/* ==== Restore shift/eigenvalue array from T ==== */
i__1 = jw;
for (i__ = infqr + 1; i__ <= i__1; ++i__) {
i__2 = kwtop + i__ - 1;
i__3 = i__ + i__ * t_dim1;
sh[i__2].r = t[i__3].r, sh[i__2].i = t[i__3].i;
/* L40: */
}
if (*ns < jw || s.r == 0.f && s.i == 0.f) {
if (*ns > 1 && (s.r != 0.f || s.i != 0.f)) {
/* ==== Reflect spike back into lower triangle ==== */
ccopy_(ns, &v[v_offset], ldv, &work[1], &c__1);
i__1 = *ns;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = i__;
r_cnjg(&q__1, &work[i__]);
work[i__2].r = q__1.r, work[i__2].i = q__1.i;
/* L50: */
}
beta.r = work[1].r, beta.i = work[1].i;
clarfg_(ns, &beta, &work[2], &c__1, &tau);
work[1].r = 1.f, work[1].i = 0.f;
i__1 = jw - 2;
i__2 = jw - 2;
claset_("L", &i__1, &i__2, &c_b1, &c_b1, &t[t_dim1 + 3], ldt);
r_cnjg(&q__1, &tau);
clarf_("L", ns, &jw, &work[1], &c__1, &q__1, &t[t_offset], ldt, &
work[jw + 1]);
clarf_("R", ns, ns, &work[1], &c__1, &tau, &t[t_offset], ldt, &
work[jw + 1]);
clarf_("R", &jw, ns, &work[1], &c__1, &tau, &v[v_offset], ldv, &
work[jw + 1]);
i__1 = *lwork - jw;
cgehrd_(&jw, &c__1, ns, &t[t_offset], ldt, &work[1], &work[jw + 1]
, &i__1, &info);
}
/* ==== Copy updated reduced window into place ==== */
if (kwtop > 1) {
i__1 = kwtop + (kwtop - 1) * h_dim1;
r_cnjg(&q__2, &v[v_dim1 + 1]);
q__1.r = s.r * q__2.r - s.i * q__2.i, q__1.i = s.r * q__2.i + s.i
* q__2.r;
h__[i__1].r = q__1.r, h__[i__1].i = q__1.i;
}
clacpy_("U", &jw, &jw, &t[t_offset], ldt, &h__[kwtop + kwtop * h_dim1]
, ldh);
i__1 = jw - 1;
i__2 = *ldt + 1;
i__3 = *ldh + 1;
ccopy_(&i__1, &t[t_dim1 + 2], &i__2, &h__[kwtop + 1 + kwtop * h_dim1],
&i__3);
/* ==== Accumulate orthogonal matrix in order update */
/* . H and Z, if requested. ==== */
if (*ns > 1 && (s.r != 0.f || s.i != 0.f)) {
i__1 = *lwork - jw;
cunmhr_("R", "N", &jw, ns, &c__1, ns, &t[t_offset], ldt, &work[1],
&v[v_offset], ldv, &work[jw + 1], &i__1, &info);
}
/* ==== Update vertical slab in H ==== */
if (*wantt) {
ltop = 1;
} else {
ltop = *ktop;
}
i__1 = kwtop - 1;
i__2 = *nv;
for (krow = ltop; i__2 < 0 ? krow >= i__1 : krow <= i__1; krow +=
i__2) {
/* Computing MIN */
i__3 = *nv, i__4 = kwtop - krow;
kln = MIN(i__3,i__4);
cgemm_("N", "N", &kln, &jw, &jw, &c_b2, &h__[krow + kwtop *
h_dim1], ldh, &v[v_offset], ldv, &c_b1, &wv[wv_offset],
ldwv);
clacpy_("A", &kln, &jw, &wv[wv_offset], ldwv, &h__[krow + kwtop *
h_dim1], ldh);
/* L60: */
}
/* ==== Update horizontal slab in H ==== */
if (*wantt) {
i__2 = *n;
i__1 = *nh;
for (kcol = *kbot + 1; i__1 < 0 ? kcol >= i__2 : kcol <= i__2;
kcol += i__1) {
/* Computing MIN */
i__3 = *nh, i__4 = *n - kcol + 1;
kln = MIN(i__3,i__4);
cgemm_("C", "N", &jw, &kln, &jw, &c_b2, &v[v_offset], ldv, &
h__[kwtop + kcol * h_dim1], ldh, &c_b1, &t[t_offset],
ldt);
clacpy_("A", &jw, &kln, &t[t_offset], ldt, &h__[kwtop + kcol *
h_dim1], ldh);
/* L70: */
}
}
/* ==== Update vertical slab in Z ==== */
if (*wantz) {
i__1 = *ihiz;
i__2 = *nv;
for (krow = *iloz; i__2 < 0 ? krow >= i__1 : krow <= i__1; krow +=
i__2) {
/* Computing MIN */
i__3 = *nv, i__4 = *ihiz - krow + 1;
kln = MIN(i__3,i__4);
cgemm_("N", "N", &kln, &jw, &jw, &c_b2, &z__[krow + kwtop *
z_dim1], ldz, &v[v_offset], ldv, &c_b1, &wv[wv_offset]
, ldwv);
clacpy_("A", &kln, &jw, &wv[wv_offset], ldwv, &z__[krow +
kwtop * z_dim1], ldz);
/* L80: */
}
}
}
/* ==== Return the number of deflations ... ==== */
*nd = jw - *ns;
/* ==== ... and the number of shifts. (Subtracting */
/* . INFQR from the spike length takes care */
/* . of the case of a rare QR failure while */
/* . calculating eigenvalues of the deflation */
/* . window.) ==== */
*ns -= infqr;
/* ==== Return optimal workspace. ==== */
r__1 = (float) lwkopt;
q__1.r = r__1, q__1.i = 0.f;
work[1].r = q__1.r, work[1].i = q__1.i;
/* ==== End of CLAQR3 ==== */
return 0;
} /* claqr3_ */