#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int slahqr_(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, integer * info) { /* -- LAPACK auxiliary routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 Purpose ======= SLAHQR is an auxiliary routine called by SHSEQR to update the eigenvalues and Schur decomposition already computed by SHSEQR, by dealing with the Hessenberg submatrix in rows and columns ILO to IHI. 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 >= 0. ILO (input) INTEGER IHI (input) INTEGER It is assumed that H is already upper quasi-triangular in rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless ILO = 1). SLAHQR works primarily with the Hessenberg submatrix in rows and columns ILO to IHI, but applies transformations to all of H if WANTT is .TRUE.. 1 <= ILO <= max(1,IHI); IHI <= N. H (input/output) REAL array, dimension (LDH,N) On entry, the upper Hessenberg matrix H. On exit, if INFO is zero and if WANTT is .TRUE., H is upper quasi-triangular in rows and columns ILO:IHI, with any 2-by-2 diagonal blocks in standard form. If INFO is zero and WANTT is .FALSE., the contents of H are unspecified on exit. The output state of H if INFO is nonzero is given below under the description of INFO. LDH (input) INTEGER The leading dimension of the array H. LDH >= max(1,N). WR (output) REAL array, dimension (N) WI (output) REAL array, dimension (N) The real and imaginary parts, respectively, of the computed eigenvalues ILO to IHI are stored in the corresponding elements of WR and WI. 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) > 0 and WI(i+1) < 0. If WANTT is .TRUE., 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 <= ILOZ <= ILO; IHI <= IHIZ <= N. Z (input/output) REAL array, dimension (LDZ,N) If WANTZ is .TRUE., on entry Z must contain the current matrix Z of transformations accumulated by SHSEQR, and on exit Z has been updated; transformations are applied only to the submatrix Z(ILOZ:IHIZ,ILO:IHI). If WANTZ is .FALSE., Z is not referenced. LDZ (input) INTEGER The leading dimension of the array Z. LDZ >= max(1,N). INFO (output) INTEGER = 0: successful exit .GT. 0: If INFO = i, SLAHQR failed to compute all the eigenvalues ILO to IHI in a total of 30 iterations per eigenvalue; elements i+1:ihi of WR and WI contain those eigenvalues which have been successfully computed. If INFO .GT. 0 and WANTT is .FALSE., then on exit, the remaining unconverged eigenvalues are the eigenvalues of the upper Hessenberg matrix rows and columns ILO thorugh 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 orthognal 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) = (initial value of Z)*U where U is the orthogonal matrix in (*) (regardless of the value of WANTT.) Further Details =============== 02-96 Based on modifications by David Day, Sandia National Laboratory, USA 12-04 Further modifications by Ralph Byers, University of Kansas, USA This is a modified version of SLAHQR from LAPACK version 3.0. It is (1) more robust against overflow and underflow and (2) adopts the more conservative Ahues & Tisseur stopping criterion (LAWN 122, 1997). ========================================================= Parameter adjustments */ /* Table of constant values */ static integer c__1 = 1; /* System generated locals */ integer h_dim1, h_offset, z_dim1, z_offset, i__1, i__2, i__3; real r__1, r__2, r__3, r__4; /* Builtin functions */ double sqrt(doublereal); /* Local variables */ static integer i__, j, k, l, m; static real s, v[3]; static integer i1, i2; static real t1, t2, t3, v2, v3, aa, ab, ba, bb, h11, h12, h21, h22, cs; static integer nh; static real sn; static integer nr; static real tr; static integer nz; static real det, h21s; static integer its; static real ulp, sum, tst, rt1i, rt2i, rt1r, rt2r; extern /* Subroutine */ int srot_(integer *, real *, integer *, real *, integer *, real *, real *), scopy_(integer *, real *, integer *, real *, integer *), slanv2_(real *, real *, real *, real *, real * , real *, real *, real *, real *, real *), slabad_(real *, real *) ; extern doublereal slamch_(char *); static real safmin; extern /* Subroutine */ int slarfg_(integer *, real *, real *, integer *, real *); static real safmax, rtdisc, smlnum; h_dim1 = *ldh; h_offset = 1 + h_dim1; h__ -= h_offset; --wr; --wi; z_dim1 = *ldz; z_offset = 1 + z_dim1; z__ -= z_offset; /* Function Body */ *info = 0; /* Quick return if possible */ if (*n == 0) { return 0; } if (*ilo == *ihi) { wr[*ilo] = h__[*ilo + *ilo * h_dim1]; wi[*ilo] = 0.f; return 0; } /* ==== clear out the trash ==== */ i__1 = *ihi - 3; for (j = *ilo; j <= i__1; ++j) { h__[j + 2 + j * h_dim1] = 0.f; h__[j + 3 + j * h_dim1] = 0.f; /* L10: */ } if (*ilo <= *ihi - 2) { h__[*ihi + (*ihi - 2) * h_dim1] = 0.f; } nh = *ihi - *ilo + 1; nz = *ihiz - *iloz + 1; /* Set machine-dependent constants for the stopping criterion. */ safmin = slamch_("SAFE MINIMUM"); safmax = 1.f / safmin; slabad_(&safmin, &safmax); ulp = slamch_("PRECISION"); smlnum = safmin * ((real) nh / ulp); /* I1 and I2 are the indices of the first row and last column of H to which transformations must be applied. If eigenvalues only are being computed, I1 and I2 are set inside the main loop. */ if (*wantt) { i1 = 1; i2 = *n; } /* The main loop begins here. I is the loop index and decreases from IHI to ILO in steps of 1 or 2. Each iteration of the loop works with the active submatrix in rows and columns L to I. Eigenvalues I+1 to IHI have already converged. Either L = ILO or H(L,L-1) is negligible so that the matrix splits. */ i__ = *ihi; L20: l = *ilo; if (i__ < *ilo) { goto L160; } /* Perform QR iterations on rows and columns ILO to I until a submatrix of order 1 or 2 splits off at the bottom because a subdiagonal element has become negligible. */ for (its = 0; its <= 30; ++its) { /* Look for a single small subdiagonal element. */ i__1 = l + 1; for (k = i__; k >= i__1; --k) { if ((r__1 = h__[k + (k - 1) * h_dim1], dabs(r__1)) <= smlnum) { goto L40; } tst = (r__1 = h__[k - 1 + (k - 1) * h_dim1], dabs(r__1)) + (r__2 = h__[k + k * h_dim1], dabs(r__2)); if (tst == 0.f) { if (k - 2 >= *ilo) { tst += (r__1 = h__[k - 1 + (k - 2) * h_dim1], dabs(r__1)); } if (k + 1 <= *ihi) { tst += (r__1 = h__[k + 1 + k * h_dim1], dabs(r__1)); } } /* ==== The following is a conservative small subdiagonal . deflation criterion due to Ahues & Tisseur (LAWN 122, . 1997). It has better mathematical foundation and . improves accuracy in some cases. ==== */ if ((r__1 = h__[k + (k - 1) * h_dim1], dabs(r__1)) <= ulp * tst) { /* Computing MAX */ r__3 = (r__1 = h__[k + (k - 1) * h_dim1], dabs(r__1)), r__4 = (r__2 = h__[k - 1 + k * h_dim1], dabs(r__2)); ab = dmax(r__3,r__4); /* Computing MIN */ r__3 = (r__1 = h__[k + (k - 1) * h_dim1], dabs(r__1)), r__4 = (r__2 = h__[k - 1 + k * h_dim1], dabs(r__2)); ba = dmin(r__3,r__4); /* Computing MAX */ r__3 = (r__1 = h__[k + k * h_dim1], dabs(r__1)), r__4 = (r__2 = h__[k - 1 + (k - 1) * h_dim1] - h__[k + k * h_dim1], dabs(r__2)); aa = dmax(r__3,r__4); /* Computing MIN */ r__3 = (r__1 = h__[k + k * h_dim1], dabs(r__1)), r__4 = (r__2 = h__[k - 1 + (k - 1) * h_dim1] - h__[k + k * h_dim1], dabs(r__2)); bb = dmin(r__3,r__4); s = aa + ab; /* Computing MAX */ r__1 = smlnum, r__2 = ulp * (bb * (aa / s)); if (ba * (ab / s) <= dmax(r__1,r__2)) { goto L40; } } /* L30: */ } L40: l = k; if (l > *ilo) { /* H(L,L-1) is negligible */ h__[l + (l - 1) * h_dim1] = 0.f; } /* Exit from loop if a submatrix of order 1 or 2 has split off. */ if (l >= i__ - 1) { goto L150; } /* Now the active submatrix is in rows and columns L to I. If eigenvalues only are being computed, only the active submatrix need be transformed. */ if (! (*wantt)) { i1 = l; i2 = i__; } if (its == 10 || its == 20) { /* Exceptional shift. */ h11 = s * .75f + h__[i__ + i__ * h_dim1]; h12 = s * -.4375f; h21 = s; h22 = h11; } else { /* Prepare to use Francis' double shift (i.e. 2nd degree generalized Rayleigh quotient) */ h11 = h__[i__ - 1 + (i__ - 1) * h_dim1]; h21 = h__[i__ + (i__ - 1) * h_dim1]; h12 = h__[i__ - 1 + i__ * h_dim1]; h22 = h__[i__ + i__ * h_dim1]; } s = dabs(h11) + dabs(h12) + dabs(h21) + dabs(h22); if (s == 0.f) { rt1r = 0.f; rt1i = 0.f; rt2r = 0.f; rt2i = 0.f; } else { h11 /= s; h21 /= s; h12 /= s; h22 /= s; tr = (h11 + h22) / 2.f; det = (h11 - tr) * (h22 - tr) - h12 * h21; rtdisc = sqrt((dabs(det))); if (det >= 0.f) { /* ==== complex conjugate shifts ==== */ rt1r = tr * s; rt2r = rt1r; rt1i = rtdisc * s; rt2i = -rt1i; } else { /* ==== real shifts (use only one of them) ==== */ rt1r = tr + rtdisc; rt2r = tr - rtdisc; if ((r__1 = rt1r - h22, dabs(r__1)) <= (r__2 = rt2r - h22, dabs(r__2))) { rt1r *= s; rt2r = rt1r; } else { rt2r *= s; rt1r = rt2r; } rt1i = 0.f; rt2i = 0.f; } } /* Look for two consecutive small subdiagonal elements. */ i__1 = l; for (m = i__ - 2; m >= i__1; --m) { /* Determine the effect of starting the double-shift QR iteration at row M, and see if this would make H(M,M-1) negligible. (The following uses scaling to avoid overflows and most underflows.) */ h21s = h__[m + 1 + m * h_dim1]; s = (r__1 = h__[m + m * h_dim1] - rt2r, dabs(r__1)) + dabs(rt2i) + dabs(h21s); h21s = h__[m + 1 + m * h_dim1] / s; v[0] = h21s * h__[m + (m + 1) * h_dim1] + (h__[m + m * h_dim1] - rt1r) * ((h__[m + m * h_dim1] - rt2r) / s) - rt1i * (rt2i / s); v[1] = h21s * (h__[m + m * h_dim1] + h__[m + 1 + (m + 1) * h_dim1] - rt1r - rt2r); v[2] = h21s * h__[m + 2 + (m + 1) * h_dim1]; s = dabs(v[0]) + dabs(v[1]) + dabs(v[2]); v[0] /= s; v[1] /= s; v[2] /= s; if (m == l) { goto L60; } if ((r__1 = h__[m + (m - 1) * h_dim1], dabs(r__1)) * (dabs(v[1]) + dabs(v[2])) <= ulp * dabs(v[0]) * ((r__2 = h__[m - 1 + ( m - 1) * h_dim1], dabs(r__2)) + (r__3 = h__[m + m * h_dim1], dabs(r__3)) + (r__4 = h__[m + 1 + (m + 1) * h_dim1], dabs(r__4)))) { goto L60; } /* L50: */ } L60: /* Double-shift QR step */ i__1 = i__ - 1; for (k = m; k <= i__1; ++k) { /* The first iteration of this loop determines a reflection G from the vector V and applies it from left and right to H, thus creating a nonzero bulge below the subdiagonal. Each subsequent iteration determines a reflection G to restore the Hessenberg form in the (K-1)th column, and thus chases the bulge one step toward the bottom of the active submatrix. NR is the order of G. Computing MIN */ i__2 = 3, i__3 = i__ - k + 1; nr = min(i__2,i__3); if (k > m) { scopy_(&nr, &h__[k + (k - 1) * h_dim1], &c__1, v, &c__1); } slarfg_(&nr, v, &v[1], &c__1, &t1); if (k > m) { h__[k + (k - 1) * h_dim1] = v[0]; h__[k + 1 + (k - 1) * h_dim1] = 0.f; if (k < i__ - 1) { h__[k + 2 + (k - 1) * h_dim1] = 0.f; } } else if (m > l) { h__[k + (k - 1) * h_dim1] = -h__[k + (k - 1) * h_dim1]; } v2 = v[1]; t2 = t1 * v2; if (nr == 3) { v3 = v[2]; t3 = t1 * v3; /* Apply G from the left to transform the rows of the matrix in columns K to I2. */ i__2 = i2; for (j = k; j <= i__2; ++j) { sum = h__[k + j * h_dim1] + v2 * h__[k + 1 + j * h_dim1] + v3 * h__[k + 2 + j * h_dim1]; h__[k + j * h_dim1] -= sum * t1; h__[k + 1 + j * h_dim1] -= sum * t2; h__[k + 2 + j * h_dim1] -= sum * t3; /* L70: */ } /* Apply G from the right to transform the columns of the matrix in rows I1 to min(K+3,I). Computing MIN */ i__3 = k + 3; i__2 = min(i__3,i__); for (j = i1; j <= i__2; ++j) { sum = h__[j + k * h_dim1] + v2 * h__[j + (k + 1) * h_dim1] + v3 * h__[j + (k + 2) * h_dim1]; h__[j + k * h_dim1] -= sum * t1; h__[j + (k + 1) * h_dim1] -= sum * t2; h__[j + (k + 2) * h_dim1] -= sum * t3; /* L80: */ } if (*wantz) { /* Accumulate transformations in the matrix Z */ i__2 = *ihiz; for (j = *iloz; j <= i__2; ++j) { sum = z__[j + k * z_dim1] + v2 * z__[j + (k + 1) * z_dim1] + v3 * z__[j + (k + 2) * z_dim1]; z__[j + k * z_dim1] -= sum * t1; z__[j + (k + 1) * z_dim1] -= sum * t2; z__[j + (k + 2) * z_dim1] -= sum * t3; /* L90: */ } } } else if (nr == 2) { /* Apply G from the left to transform the rows of the matrix in columns K to I2. */ i__2 = i2; for (j = k; j <= i__2; ++j) { sum = h__[k + j * h_dim1] + v2 * h__[k + 1 + j * h_dim1]; h__[k + j * h_dim1] -= sum * t1; h__[k + 1 + j * h_dim1] -= sum * t2; /* L100: */ } /* Apply G from the right to transform the columns of the matrix in rows I1 to min(K+3,I). */ i__2 = i__; for (j = i1; j <= i__2; ++j) { sum = h__[j + k * h_dim1] + v2 * h__[j + (k + 1) * h_dim1] ; h__[j + k * h_dim1] -= sum * t1; h__[j + (k + 1) * h_dim1] -= sum * t2; /* L110: */ } if (*wantz) { /* Accumulate transformations in the matrix Z */ i__2 = *ihiz; for (j = *iloz; j <= i__2; ++j) { sum = z__[j + k * z_dim1] + v2 * z__[j + (k + 1) * z_dim1]; z__[j + k * z_dim1] -= sum * t1; z__[j + (k + 1) * z_dim1] -= sum * t2; /* L120: */ } } } /* L130: */ } /* L140: */ } /* Failure to converge in remaining number of iterations */ *info = i__; return 0; L150: if (l == i__) { /* H(I,I-1) is negligible: one eigenvalue has converged. */ wr[i__] = h__[i__ + i__ * h_dim1]; wi[i__] = 0.f; } else if (l == i__ - 1) { /* H(I-1,I-2) is negligible: a pair of eigenvalues have converged. Transform the 2-by-2 submatrix to standard Schur form, and compute and store the eigenvalues. */ slanv2_(&h__[i__ - 1 + (i__ - 1) * h_dim1], &h__[i__ - 1 + i__ * h_dim1], &h__[i__ + (i__ - 1) * h_dim1], &h__[i__ + i__ * h_dim1], &wr[i__ - 1], &wi[i__ - 1], &wr[i__], &wi[i__], &cs, &sn); if (*wantt) { /* Apply the transformation to the rest of H. */ if (i2 > i__) { i__1 = i2 - i__; srot_(&i__1, &h__[i__ - 1 + (i__ + 1) * h_dim1], ldh, &h__[ i__ + (i__ + 1) * h_dim1], ldh, &cs, &sn); } i__1 = i__ - i1 - 1; srot_(&i__1, &h__[i1 + (i__ - 1) * h_dim1], &c__1, &h__[i1 + i__ * h_dim1], &c__1, &cs, &sn); } if (*wantz) { /* Apply the transformation to Z. */ srot_(&nz, &z__[*iloz + (i__ - 1) * z_dim1], &c__1, &z__[*iloz + i__ * z_dim1], &c__1, &cs, &sn); } } /* return to start of the main loop with new value of I. */ i__ = l - 1; goto L20; L160: return 0; /* End of SLAHQR */ } /* slahqr_ */