#include "blaswrap.h" #include "f2c.h" /* 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) { /* -- LAPACK auxiliary routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. 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 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). 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 increasing 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. ==== Parameter adjustments */ /* Table of constant values */ static integer c__13 = 13; static integer c__15 = 15; static integer c_n1 = -1; static integer c__12 = 12; static integer c__14 = 14; static integer c__16 = 16; static logical c_false = FALSE_; static integer c__1 = 1; static integer c__3 = 3; /* 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; /* Builtin functions */ double r_imag(complex *); void c_sqrt(complex *, complex *); /* Local variables */ static integer i__, k; static real s; static complex aa, bb, cc, dd; static integer ld, nh, it, ks, kt, ku, kv, ls, ns, nw; static complex tr2, det; static integer inf, kdu, nho, nve, kwh, nsr, nwr, kwv, ndfl, kbot, nmin; static complex swap; static integer ktop; static complex zdum[1] /* was [1][1] */; static integer kacc22; static logical nwinc; static integer itmax, nsmax, nwmax, kwtop; extern /* Subroutine */ int claqr3_(logical *, logical *, integer *, integer *, integer *, integer *, complex *, integer *, integer *, integer *, complex *, integer *, integer *, integer *, complex *, complex *, integer *, integer *, complex *, integer *, integer *, complex *, integer *, complex *, integer *), claqr4_(logical *, logical *, integer *, integer *, integer *, complex *, integer *, complex *, integer *, integer *, complex *, integer *, complex *, integer *, integer *), claqr5_(logical *, logical *, integer *, integer *, integer *, integer *, integer *, complex *, complex *, integer *, integer *, integer *, complex *, integer *, complex *, integer *, complex *, integer *, integer *, complex *, integer *, integer *, complex *, integer *); static integer nibble; extern /* Subroutine */ int clahqr_(logical *, logical *, integer *, integer *, integer *, complex *, integer *, complex *, integer *, integer *, complex *, integer *, integer *), clacpy_(char *, integer *, integer *, complex *, integer *, complex *, integer *); extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *, ftnlen, ftnlen); static char jbcmpz[2]; static complex rtdisc; static logical sorted; static integer lwkopt; 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; } /* ==== 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'; } /* ==== Tiny matrices must use CLAHQR. ==== */ if (*n <= 11) { /* ==== Estimate optimal workspace. ==== */ 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; /* ==== 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, ( ftnlen)6, (ftnlen)1); 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); nw = 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, ( ftnlen)6, (ftnlen)1); /* 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, ( ftnlen)6, (ftnlen)1); nmin = max(11,nmin); /* ==== Nibble crossover point ==== */ nibble = ilaenv_(&c__14, "CLAQR0", jbcmpz, n, ilo, ihi, lwork, ( ftnlen)6, (ftnlen)1); 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, ( ftnlen)6, (ftnlen)1); 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); /* ==== 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; } /* L10: */ } k = *ilo; L20: ktop = k; /* ==== Select deflation window size ==== */ nh = kbot - ktop + 1; if (ndfl < 5 || nh < nw) { /* ==== Typical deflation window. If possible and . advisable, nibble the entire active block. . If not, use size NWR or NWR+1 depending upon . which has the smaller corresponding subdiagonal . entry (a heuristic). ==== */ nwinc = TRUE_; if (nh <= min(nmin,nwmax)) { nw = nh; } else { /* Computing MIN */ i__2 = min(nwr,nh); nw = min(i__2,nwmax); 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; } } } } } else { /* ==== Exceptional deflation window. 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 up to the maximum reasonable and possible. . Then maybe try a slightly smaller window. ==== */ if (nwinc && nw < min(nwmax,nh)) { /* Computing MIN */ i__2 = min(nwmax,nh), i__3 = nw << 1; nw = min(i__2,i__3); } else { nwinc = FALSE_; if (nw == nh && nh > 2) { nw = nh - 1; } } } /* ==== 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; /* L30: */ } } 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; } /* L40: */ } /* L50: */ } 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 ==== L70: */ } /* ==== 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_ */