/* slaqr3.f -- translated by f2c (version 20061008). You must link the resulting object file with libf2c: on Microsoft Windows system, link with libf2c.lib; on Linux or Unix systems, link with .../path/to/libf2c.a -lm or, if you install libf2c.a in a standard place, with -lf2c -lm -- in that order, at the end of the command line, as in cc *.o -lf2c -lm Source for libf2c is in /netlib/f2c/libf2c.zip, e.g., http://www.netlib.org/f2c/libf2c.zip */ #include "f2c.h" #include "blaswrap.h" /* Table of constant values */ static integer c__1 = 1; static integer c_n1 = -1; static logical c_true = TRUE_; static real c_b17 = 0.f; static real c_b18 = 1.f; static integer c__12 = 12; /* Subroutine */ int slaqr3_(logical *wantt, logical *wantz, integer *n, integer *ktop, integer *kbot, integer *nw, real *h__, integer *ldh, integer *iloz, integer *ihiz, real *z__, integer *ldz, integer *ns, integer *nd, real *sr, real *si, real *v, integer *ldv, integer *nh, real *t, integer *ldt, integer *nv, real *wv, integer *ldwv, real * work, integer *lwork) { /* System generated locals */ integer 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; real r__1, r__2, r__3, r__4, r__5, r__6; /* Builtin functions */ double sqrt(doublereal); /* Local variables */ integer i__, j, k; real s, aa, bb, cc, dd, cs, sn; integer jw; real evi, evk, foo; integer kln; real tau, ulp; integer lwk1, lwk2, lwk3; real beta; integer kend, kcol, info, nmin, ifst, ilst, ltop, krow; logical bulge; extern /* Subroutine */ int slarf_(char *, integer *, integer *, real *, integer *, real *, real *, integer *, real *), sgemm_( char *, char *, integer *, integer *, integer *, real *, real *, integer *, real *, integer *, real *, real *, integer *); integer infqr; extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *, integer *); integer kwtop; extern /* Subroutine */ int slanv2_(real *, real *, real *, real *, real * , real *, real *, real *, real *, real *), slaqr4_(logical *, logical *, integer *, integer *, integer *, real *, integer *, real *, real *, integer *, integer *, real *, integer *, real *, integer *, integer *), slabad_(real *, real *); extern doublereal slamch_(char *); extern /* Subroutine */ int sgehrd_(integer *, integer *, integer *, real *, integer *, real *, real *, integer *, integer *); real safmin; extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *); real safmax; extern /* Subroutine */ int slarfg_(integer *, real *, real *, integer *, real *), slahqr_(logical *, logical *, integer *, integer *, integer *, real *, integer *, real *, real *, integer *, integer * , real *, integer *, integer *), slacpy_(char *, integer *, integer *, real *, integer *, real *, integer *), slaset_( char *, integer *, integer *, real *, real *, real *, integer *); logical sorted; extern /* Subroutine */ int strexc_(char *, integer *, real *, integer *, real *, integer *, integer *, integer *, real *, integer *), sormhr_(char *, char *, integer *, integer *, integer *, integer *, real *, integer *, real *, real *, integer *, real *, integer *, integer *); real smlnum; integer 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 orthogonal 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 orthogonal 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 quasi-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 orthogonal matrix Z is updated so */ /* so that the orthogonal 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 orthogonal 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) REAL 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 an orthogonal */ /* similarity transformation, perturbed, and the returned */ /* to Hessenberg form that (it is to be hoped) has some */ /* zero subdiagonal entries. */ /* LDH (input) integer */ /* 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) REAL array, dimension (LDZ,N) */ /* IF WANTZ is .TRUE., then on output, the orthogonal */ /* 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) integer */ /* The leading dimension of Z just as declared in the */ /* calling subroutine. 1 .LE. LDZ. */ /* NS (output) integer */ /* The number of unconverged (ie approximate) eigenvalues */ /* returned in SR and SI that may be used as shifts by the */ /* calling subroutine. */ /* ND (output) integer */ /* The number of converged eigenvalues uncovered by this */ /* subroutine. */ /* SR (output) REAL array, dimension KBOT */ /* SI (output) REAL array, dimension KBOT */ /* On output, the real and imaginary parts of approximate */ /* eigenvalues that may be used for shifts are stored in */ /* SR(KBOT-ND-NS+1) through SR(KBOT-ND) and */ /* SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively. */ /* The real and imaginary parts of converged eigenvalues */ /* are stored in SR(KBOT-ND+1) through SR(KBOT) and */ /* SI(KBOT-ND+1) through SI(KBOT), respectively. */ /* V (workspace) REAL array, dimension (LDV,NW) */ /* An NW-by-NW work array. */ /* LDV (input) integer scalar */ /* The leading dimension of V just as declared in the */ /* calling subroutine. NW .LE. LDV */ /* NH (input) integer scalar */ /* The number of columns of T. NH.GE.NW. */ /* T (workspace) REAL array, dimension (LDT,NW) */ /* LDT (input) integer */ /* The leading dimension of T just as declared in the */ /* calling subroutine. NW .LE. LDT */ /* NV (input) integer */ /* The number of rows of work array WV available for */ /* workspace. NV.GE.NW. */ /* WV (workspace) REAL array, dimension (LDWV,NW) */ /* LDWV (input) integer */ /* The leading dimension of W just as declared in the */ /* calling subroutine. NW .LE. LDV */ /* WORK (workspace) REAL 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) integer */ /* 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; SLAQR3 */ /* 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 .. */ /* .. */ /* .. 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; --sr; --si; 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 SGEHRD ==== */ i__1 = jw - 1; sgehrd_(&jw, &c__1, &i__1, &t[t_offset], ldt, &work[1], &work[1], & c_n1, &info); lwk1 = (integer) work[1]; /* ==== Workspace query call to SORMHR ==== */ i__1 = jw - 1; sormhr_("R", "N", &jw, &jw, &c__1, &i__1, &t[t_offset], ldt, &work[1], &v[v_offset], ldv, &work[1], &c_n1, &info); lwk2 = (integer) work[1]; /* ==== Workspace query call to SLAQR4 ==== */ slaqr4_(&c_true, &c_true, &jw, &c__1, &jw, &t[t_offset], ldt, &sr[1], &si[1], &c__1, &jw, &v[v_offset], ldv, &work[1], &c_n1, & infqr); lwk3 = (integer) work[1]; /* ==== 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) { work[1] = (real) lwkopt; return 0; } /* ==== Nothing to do ... */ /* ... for an empty active block ... ==== */ *ns = 0; *nd = 0; work[1] = 1.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 * ((real) (*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 = 0.f; } else { s = h__[kwtop + (kwtop - 1) * h_dim1]; } if (*kbot == kwtop) { /* ==== 1-by-1 deflation window: not much to do ==== */ sr[kwtop] = h__[kwtop + kwtop * h_dim1]; si[kwtop] = 0.f; *ns = 1; *nd = 0; /* Computing MAX */ r__2 = smlnum, r__3 = ulp * (r__1 = h__[kwtop + kwtop * h_dim1], dabs( r__1)); if (dabs(s) <= dmax(r__2,r__3)) { *ns = 0; *nd = 1; if (kwtop > *ktop) { h__[kwtop + (kwtop - 1) * h_dim1] = 0.f; } } work[1] = 1.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.) ==== */ slacpy_("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; scopy_(&i__1, &h__[kwtop + 1 + kwtop * h_dim1], &i__2, &t[t_dim1 + 2], & i__3); slaset_("A", &jw, &jw, &c_b17, &c_b18, &v[v_offset], ldv); nmin = ilaenv_(&c__12, "SLAQR3", "SV", &jw, &c__1, &jw, lwork); if (jw > nmin) { slaqr4_(&c_true, &c_true, &jw, &c__1, &jw, &t[t_offset], ldt, &sr[ kwtop], &si[kwtop], &c__1, &jw, &v[v_offset], ldv, &work[1], lwork, &infqr); } else { slahqr_(&c_true, &c_true, &jw, &c__1, &jw, &t[t_offset], ldt, &sr[ kwtop], &si[kwtop], &c__1, &jw, &v[v_offset], ldv, &infqr); } /* ==== STREXC needs a clean margin near the diagonal ==== */ i__1 = jw - 3; for (j = 1; j <= i__1; ++j) { t[j + 2 + j * t_dim1] = 0.f; t[j + 3 + j * t_dim1] = 0.f; /* L10: */ } if (jw > 2) { t[jw + (jw - 2) * t_dim1] = 0.f; } /* ==== Deflation detection loop ==== */ *ns = jw; ilst = infqr + 1; L20: if (ilst <= *ns) { if (*ns == 1) { bulge = FALSE_; } else { bulge = t[*ns + (*ns - 1) * t_dim1] != 0.f; } /* ==== Small spike tip test for deflation ==== */ if (! bulge) { /* ==== Real eigenvalue ==== */ foo = (r__1 = t[*ns + *ns * t_dim1], dabs(r__1)); if (foo == 0.f) { foo = dabs(s); } /* Computing MAX */ r__2 = smlnum, r__3 = ulp * foo; if ((r__1 = s * v[*ns * v_dim1 + 1], dabs(r__1)) <= dmax(r__2, r__3)) { /* ==== Deflatable ==== */ --(*ns); } else { /* ==== Undeflatable. Move it up out of the way. */ /* . (STREXC can not fail in this case.) ==== */ ifst = *ns; strexc_("V", &jw, &t[t_offset], ldt, &v[v_offset], ldv, &ifst, &ilst, &work[1], &info); ++ilst; } } else { /* ==== Complex conjugate pair ==== */ foo = (r__3 = t[*ns + *ns * t_dim1], dabs(r__3)) + sqrt((r__1 = t[ *ns + (*ns - 1) * t_dim1], dabs(r__1))) * sqrt((r__2 = t[* ns - 1 + *ns * t_dim1], dabs(r__2))); if (foo == 0.f) { foo = dabs(s); } /* Computing MAX */ r__3 = (r__1 = s * v[*ns * v_dim1 + 1], dabs(r__1)), r__4 = (r__2 = s * v[(*ns - 1) * v_dim1 + 1], dabs(r__2)); /* Computing MAX */ r__5 = smlnum, r__6 = ulp * foo; if (dmax(r__3,r__4) <= dmax(r__5,r__6)) { /* ==== Deflatable ==== */ *ns += -2; } else { /* ==== Undeflatable. Move them up out of the way. */ /* . Fortunately, STREXC does the right thing with */ /* . ILST in case of a rare exchange failure. ==== */ ifst = *ns; strexc_("V", &jw, &t[t_offset], ldt, &v[v_offset], ldv, &ifst, &ilst, &work[1], &info); ilst += 2; } } /* ==== End deflation detection loop ==== */ goto L20; } /* ==== Return to Hessenberg form ==== */ if (*ns == 0) { s = 0.f; } if (*ns < jw) { /* ==== sorting diagonal blocks of T improves accuracy for */ /* . graded matrices. Bubble sort deals well with */ /* . exchange failures. ==== */ sorted = FALSE_; i__ = *ns + 1; L30: if (sorted) { goto L50; } sorted = TRUE_; kend = i__ - 1; i__ = infqr + 1; if (i__ == *ns) { k = i__ + 1; } else if (t[i__ + 1 + i__ * t_dim1] == 0.f) { k = i__ + 1; } else { k = i__ + 2; } L40: if (k <= kend) { if (k == i__ + 1) { evi = (r__1 = t[i__ + i__ * t_dim1], dabs(r__1)); } else { evi = (r__3 = t[i__ + i__ * t_dim1], dabs(r__3)) + sqrt((r__1 = t[i__ + 1 + i__ * t_dim1], dabs(r__1))) * sqrt(( r__2 = t[i__ + (i__ + 1) * t_dim1], dabs(r__2))); } if (k == kend) { evk = (r__1 = t[k + k * t_dim1], dabs(r__1)); } else if (t[k + 1 + k * t_dim1] == 0.f) { evk = (r__1 = t[k + k * t_dim1], dabs(r__1)); } else { evk = (r__3 = t[k + k * t_dim1], dabs(r__3)) + sqrt((r__1 = t[ k + 1 + k * t_dim1], dabs(r__1))) * sqrt((r__2 = t[k + (k + 1) * t_dim1], dabs(r__2))); } if (evi >= evk) { i__ = k; } else { sorted = FALSE_; ifst = i__; ilst = k; strexc_("V", &jw, &t[t_offset], ldt, &v[v_offset], ldv, &ifst, &ilst, &work[1], &info); if (info == 0) { i__ = ilst; } else { i__ = k; } } if (i__ == kend) { k = i__ + 1; } else if (t[i__ + 1 + i__ * t_dim1] == 0.f) { k = i__ + 1; } else { k = i__ + 2; } goto L40; } goto L30; L50: ; } /* ==== Restore shift/eigenvalue array from T ==== */ i__ = jw; L60: if (i__ >= infqr + 1) { if (i__ == infqr + 1) { sr[kwtop + i__ - 1] = t[i__ + i__ * t_dim1]; si[kwtop + i__ - 1] = 0.f; --i__; } else if (t[i__ + (i__ - 1) * t_dim1] == 0.f) { sr[kwtop + i__ - 1] = t[i__ + i__ * t_dim1]; si[kwtop + i__ - 1] = 0.f; --i__; } else { aa = t[i__ - 1 + (i__ - 1) * t_dim1]; cc = t[i__ + (i__ - 1) * t_dim1]; bb = t[i__ - 1 + i__ * t_dim1]; dd = t[i__ + i__ * t_dim1]; slanv2_(&aa, &bb, &cc, &dd, &sr[kwtop + i__ - 2], &si[kwtop + i__ - 2], &sr[kwtop + i__ - 1], &si[kwtop + i__ - 1], &cs, & sn); i__ += -2; } goto L60; } if (*ns < jw || s == 0.f) { if (*ns > 1 && s != 0.f) { /* ==== Reflect spike back into lower triangle ==== */ scopy_(ns, &v[v_offset], ldv, &work[1], &c__1); beta = work[1]; slarfg_(ns, &beta, &work[2], &c__1, &tau); work[1] = 1.f; i__1 = jw - 2; i__2 = jw - 2; slaset_("L", &i__1, &i__2, &c_b17, &c_b17, &t[t_dim1 + 3], ldt); slarf_("L", ns, &jw, &work[1], &c__1, &tau, &t[t_offset], ldt, & work[jw + 1]); slarf_("R", ns, ns, &work[1], &c__1, &tau, &t[t_offset], ldt, & work[jw + 1]); slarf_("R", &jw, ns, &work[1], &c__1, &tau, &v[v_offset], ldv, & work[jw + 1]); i__1 = *lwork - jw; sgehrd_(&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) { h__[kwtop + (kwtop - 1) * h_dim1] = s * v[v_dim1 + 1]; } slacpy_("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; scopy_(&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 != 0.f) { i__1 = *lwork - jw; sormhr_("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); sgemm_("N", "N", &kln, &jw, &jw, &c_b18, &h__[krow + kwtop * h_dim1], ldh, &v[v_offset], ldv, &c_b17, &wv[wv_offset], ldwv); slacpy_("A", &kln, &jw, &wv[wv_offset], ldwv, &h__[krow + kwtop * h_dim1], ldh); /* L70: */ } /* ==== 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); sgemm_("C", "N", &jw, &kln, &jw, &c_b18, &v[v_offset], ldv, & h__[kwtop + kcol * h_dim1], ldh, &c_b17, &t[t_offset], ldt); slacpy_("A", &jw, &kln, &t[t_offset], ldt, &h__[kwtop + kcol * h_dim1], ldh); /* L80: */ } } /* ==== 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); sgemm_("N", "N", &kln, &jw, &jw, &c_b18, &z__[krow + kwtop * z_dim1], ldz, &v[v_offset], ldv, &c_b17, &wv[ wv_offset], ldwv); slacpy_("A", &kln, &jw, &wv[wv_offset], ldwv, &z__[krow + kwtop * z_dim1], ldz); /* L90: */ } } } /* ==== 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. ==== */ work[1] = (real) lwkopt; /* ==== End of SLAQR3 ==== */ return 0; } /* slaqr3_ */