#include "blaswrap.h" /* -- translated by f2c (version 19990503). You must link the resulting object file with the libraries: -lf2c -lm (in that order) */ #include "f2c.h" /* Common Block Declarations */ struct { real ops, itcnt; } latime_; #define latime_1 latime_ /* Table of constant values */ static integer c__2 = 2; static integer c__1 = 1; /* Subroutine */ int clahqr_(logical *wantt, logical *wantz, integer *n, integer *ilo, integer *ihi, complex *h__, integer *ldh, complex *w, integer *iloz, integer *ihiz, complex *z__, integer *ldz, 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; complex q__1, q__2, q__3, q__4; /* Builtin functions */ double r_imag(complex *); void c_sqrt(complex *, complex *), r_cnjg(complex *, complex *); double c_abs(complex *); /* Local variables */ static complex temp; static real opst; static integer i__, j, k, l, m; static real s; static complex t, u, v[2], x, y; extern /* Subroutine */ int cscal_(integer *, complex *, complex *, integer *), ccopy_(integer *, complex *, integer *, complex *, integer *); static real rtemp; static integer i1, i2; static real rwork[1]; static complex t1; static real t2; static complex v2; static real h10; static complex h11; static real h21; static complex h22; static integer nh; extern /* Subroutine */ int clarfg_(integer *, complex *, complex *, integer *, complex *); extern /* Complex */ VOID cladiv_(complex *, complex *, complex *); extern doublereal slamch_(char *); static integer nz; extern doublereal clanhs_(char *, integer *, complex *, integer *, real *); static real smlnum; static complex h11s; static integer itn, its; static real ulp; static complex sum; static real tst1; #define h___subscr(a_1,a_2) (a_2)*h_dim1 + a_1 #define h___ref(a_1,a_2) h__[h___subscr(a_1,a_2)] #define z___subscr(a_1,a_2) (a_2)*z_dim1 + a_1 #define z___ref(a_1,a_2) z__[z___subscr(a_1,a_2)] /* -- LAPACK auxiliary routine (instrumented to count operations) -- Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., Courant Institute, Argonne National Lab, and Rice University June 30, 1999 Common block to return operation count. Purpose ======= CLAHQR is an auxiliary routine called by CHSEQR to update the eigenvalues and Schur decomposition already computed by CHSEQR, 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 triangular in rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless ILO = 1). CLAHQR 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) COMPLEX array, dimension (LDH,N) On entry, the upper Hessenberg matrix H. On exit, if WANTT is .TRUE., H is upper triangular in rows and columns ILO:IHI, with any 2-by-2 diagonal blocks in standard form. If WANTT is .FALSE., the contents of H are unspecified on exit. LDH (input) INTEGER The leading dimension of the array H. LDH >= max(1,N). W (output) COMPLEX array, dimension (N) The computed eigenvalues ILO to IHI are stored in the corresponding elements of W. If WANTT is .TRUE., 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). 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) COMPLEX array, dimension (LDZ,N) If WANTZ is .TRUE., on entry Z must contain the current matrix Z of transformations accumulated by CHSEQR, 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 > 0: if INFO = i, CLAHQR failed to compute all the eigenvalues ILO to IHI in a total of 30*(IHI-ILO+1) iterations; elements i+1:ihi of W contain those eigenvalues which have been successfully computed. ===================================================================== Parameter adjustments */ h_dim1 = *ldh; h_offset = 1 + h_dim1 * 1; h__ -= h_offset; --w; z_dim1 = *ldz; z_offset = 1 + z_dim1 * 1; z__ -= z_offset; /* Function Body */ *info = 0; /* ** Initialize */ opst = 0.f; /* ** Quick return if possible */ if (*n == 0) { return 0; } if (*ilo == *ihi) { i__1 = *ilo; i__2 = h___subscr(*ilo, *ilo); w[i__1].r = h__[i__2].r, w[i__1].i = h__[i__2].i; return 0; } nh = *ihi - *ilo + 1; nz = *ihiz - *iloz + 1; /* Set machine-dependent constants for the stopping criterion. If norm(H) <= sqrt(OVFL), overflow should not occur. */ ulp = slamch_("Precision"); smlnum = slamch_("Safe minimum") / 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; } /* ITN is the total number of QR iterations allowed. */ itn = nh * 30; /* The main loop begins here. I is the loop index and decreases from IHI to ILO in steps of 1. 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; L10: if (i__ < *ilo) { goto L130; } /* Perform QR iterations on rows and columns ILO to I until a submatrix of order 1 splits off at the bottom because a subdiagonal element has become negligible. */ l = *ilo; i__1 = itn; for (its = 0; its <= i__1; ++its) { /* Look for a single small subdiagonal element. */ i__2 = l + 1; for (k = i__; k >= i__2; --k) { i__3 = h___subscr(k - 1, k - 1); i__4 = h___subscr(k, k); tst1 = (r__1 = h__[i__3].r, dabs(r__1)) + (r__2 = r_imag(&h___ref( k - 1, k - 1)), dabs(r__2)) + ((r__3 = h__[i__4].r, dabs( r__3)) + (r__4 = r_imag(&h___ref(k, k)), dabs(r__4))); if (tst1 == 0.f) { i__3 = i__ - l + 1; tst1 = clanhs_("1", &i__3, &h___ref(l, l), ldh, rwork); /* ** Increment op count */ latime_1.ops += (i__ - l + 1) * 5 * (i__ - l) / 2; /* ** */ } i__3 = h___subscr(k, k - 1); /* Computing MAX */ r__2 = ulp * tst1; if ((r__1 = h__[i__3].r, dabs(r__1)) <= dmax(r__2,smlnum)) { goto L30; } /* L20: */ } L30: l = k; /* ** Increment op count */ opst += (i__ - l + 1) * 5; /* ** */ if (l > *ilo) { /* H(L,L-1) is negligible */ i__2 = h___subscr(l, l - 1); h__[i__2].r = 0.f, h__[i__2].i = 0.f; } /* Exit from loop if a submatrix of order 1 has split off. */ if (l >= i__) { goto L120; } /* 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. */ i__2 = h___subscr(i__, i__ - 1); s = (r__1 = h__[i__2].r, dabs(r__1)) * .75f; i__2 = h___subscr(i__, i__); q__1.r = s + h__[i__2].r, q__1.i = h__[i__2].i; t.r = q__1.r, t.i = q__1.i; /* ** Increment op count */ opst += 1; /* ** */ } else { /* Wilkinson's shift. */ i__2 = h___subscr(i__, i__); t.r = h__[i__2].r, t.i = h__[i__2].i; i__2 = h___subscr(i__ - 1, i__); i__3 = h___subscr(i__, i__ - 1); r__1 = h__[i__3].r; q__1.r = r__1 * h__[i__2].r, q__1.i = r__1 * h__[i__2].i; u.r = q__1.r, u.i = q__1.i; /* ** Increment op count */ opst += 2; /* ** */ if (u.r != 0.f || u.i != 0.f) { i__2 = h___subscr(i__ - 1, i__ - 1); q__2.r = h__[i__2].r - t.r, q__2.i = h__[i__2].i - t.i; q__1.r = q__2.r * .5f, q__1.i = q__2.i * .5f; x.r = q__1.r, x.i = q__1.i; q__3.r = x.r * x.r - x.i * x.i, q__3.i = x.r * x.i + x.i * x.r; q__2.r = q__3.r + u.r, q__2.i = q__3.i + u.i; c_sqrt(&q__1, &q__2); y.r = q__1.r, y.i = q__1.i; if (x.r * y.r + r_imag(&x) * r_imag(&y) < 0.f) { q__1.r = -y.r, q__1.i = -y.i; y.r = q__1.r, y.i = q__1.i; } q__3.r = x.r + y.r, q__3.i = x.i + y.i; cladiv_(&q__2, &u, &q__3); q__1.r = t.r - q__2.r, q__1.i = t.i - q__2.i; t.r = q__1.r, t.i = q__1.i; /* ** Increment op count */ opst += 20; /* ** */ } } /* Look for two consecutive small subdiagonal elements. */ i__2 = l + 1; for (m = i__ - 1; m >= i__2; --m) { /* Determine the effect of starting the single-shift QR iteration at row M, and see if this would make H(M,M-1) negligible. */ i__3 = h___subscr(m, m); h11.r = h__[i__3].r, h11.i = h__[i__3].i; i__3 = h___subscr(m + 1, m + 1); h22.r = h__[i__3].r, h22.i = h__[i__3].i; q__1.r = h11.r - t.r, q__1.i = h11.i - t.i; h11s.r = q__1.r, h11s.i = q__1.i; i__3 = h___subscr(m + 1, m); h21 = h__[i__3].r; s = (r__1 = h11s.r, dabs(r__1)) + (r__2 = r_imag(&h11s), dabs( r__2)) + dabs(h21); q__1.r = h11s.r / s, q__1.i = h11s.i / s; h11s.r = q__1.r, h11s.i = q__1.i; h21 /= s; v[0].r = h11s.r, v[0].i = h11s.i; v[1].r = h21, v[1].i = 0.f; i__3 = h___subscr(m, m - 1); h10 = h__[i__3].r; tst1 = ((r__1 = h11s.r, dabs(r__1)) + (r__2 = r_imag(&h11s), dabs( r__2))) * ((r__3 = h11.r, dabs(r__3)) + (r__4 = r_imag(& h11), dabs(r__4)) + ((r__5 = h22.r, dabs(r__5)) + (r__6 = r_imag(&h22), dabs(r__6)))); if ((r__1 = h10 * h21, dabs(r__1)) <= ulp * tst1) { goto L50; } /* L40: */ } i__2 = h___subscr(l, l); h11.r = h__[i__2].r, h11.i = h__[i__2].i; i__2 = h___subscr(l + 1, l + 1); h22.r = h__[i__2].r, h22.i = h__[i__2].i; q__1.r = h11.r - t.r, q__1.i = h11.i - t.i; h11s.r = q__1.r, h11s.i = q__1.i; i__2 = h___subscr(l + 1, l); h21 = h__[i__2].r; s = (r__1 = h11s.r, dabs(r__1)) + (r__2 = r_imag(&h11s), dabs(r__2)) + dabs(h21); q__1.r = h11s.r / s, q__1.i = h11s.i / s; h11s.r = q__1.r, h11s.i = q__1.i; h21 /= s; v[0].r = h11s.r, v[0].i = h11s.i; v[1].r = h21, v[1].i = 0.f; L50: /* ** Increment op count */ opst += (i__ - m) * 14; /* ** Single-shift QR step */ i__2 = i__ - 1; for (k = m; k <= i__2; ++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. V(2) is always real before the call to CLARFG, and hence after the call T2 ( = T1*V(2) ) is also real. */ if (k > m) { ccopy_(&c__2, &h___ref(k, k - 1), &c__1, v, &c__1); } clarfg_(&c__2, v, &v[1], &c__1, &t1); /* ** Increment op count */ opst += 38; /* ** */ if (k > m) { i__3 = h___subscr(k, k - 1); h__[i__3].r = v[0].r, h__[i__3].i = v[0].i; i__3 = h___subscr(k + 1, k - 1); h__[i__3].r = 0.f, h__[i__3].i = 0.f; } v2.r = v[1].r, v2.i = v[1].i; q__1.r = t1.r * v2.r - t1.i * v2.i, q__1.i = t1.r * v2.i + t1.i * v2.r; t2 = q__1.r; /* Apply G from the left to transform the rows of the matrix in columns K to I2. */ i__3 = i2; for (j = k; j <= i__3; ++j) { r_cnjg(&q__3, &t1); i__4 = h___subscr(k, j); q__2.r = q__3.r * h__[i__4].r - q__3.i * h__[i__4].i, q__2.i = q__3.r * h__[i__4].i + q__3.i * h__[i__4].r; i__5 = h___subscr(k + 1, j); q__4.r = t2 * h__[i__5].r, q__4.i = t2 * h__[i__5].i; q__1.r = q__2.r + q__4.r, q__1.i = q__2.i + q__4.i; sum.r = q__1.r, sum.i = q__1.i; i__4 = h___subscr(k, j); i__5 = h___subscr(k, j); q__1.r = h__[i__5].r - sum.r, q__1.i = h__[i__5].i - sum.i; h__[i__4].r = q__1.r, h__[i__4].i = q__1.i; i__4 = h___subscr(k + 1, j); i__5 = h___subscr(k + 1, j); q__2.r = sum.r * v2.r - sum.i * v2.i, q__2.i = sum.r * v2.i + sum.i * v2.r; q__1.r = h__[i__5].r - q__2.r, q__1.i = h__[i__5].i - q__2.i; h__[i__4].r = q__1.r, h__[i__4].i = q__1.i; /* L60: */ } /* Apply G from the right to transform the columns of the matrix in rows I1 to min(K+2,I). Computing MIN */ i__4 = k + 2; i__3 = min(i__4,i__); for (j = i1; j <= i__3; ++j) { i__4 = h___subscr(j, k); q__2.r = t1.r * h__[i__4].r - t1.i * h__[i__4].i, q__2.i = t1.r * h__[i__4].i + t1.i * h__[i__4].r; i__5 = h___subscr(j, k + 1); q__3.r = t2 * h__[i__5].r, q__3.i = t2 * h__[i__5].i; q__1.r = q__2.r + q__3.r, q__1.i = q__2.i + q__3.i; sum.r = q__1.r, sum.i = q__1.i; i__4 = h___subscr(j, k); i__5 = h___subscr(j, k); q__1.r = h__[i__5].r - sum.r, q__1.i = h__[i__5].i - sum.i; h__[i__4].r = q__1.r, h__[i__4].i = q__1.i; i__4 = h___subscr(j, k + 1); i__5 = h___subscr(j, k + 1); r_cnjg(&q__3, &v2); q__2.r = sum.r * q__3.r - sum.i * q__3.i, q__2.i = sum.r * q__3.i + sum.i * q__3.r; q__1.r = h__[i__5].r - q__2.r, q__1.i = h__[i__5].i - q__2.i; h__[i__4].r = q__1.r, h__[i__4].i = q__1.i; /* L70: */ } /* ** Increment op count Computing MIN */ i__3 = 2, i__4 = i__ - k; latime_1.ops += (i2 - i1 + 2 + min(i__3,i__4)) * 20; /* ** */ if (*wantz) { /* Accumulate transformations in the matrix Z */ i__3 = *ihiz; for (j = *iloz; j <= i__3; ++j) { i__4 = z___subscr(j, k); q__2.r = t1.r * z__[i__4].r - t1.i * z__[i__4].i, q__2.i = t1.r * z__[i__4].i + t1.i * z__[i__4].r; i__5 = z___subscr(j, k + 1); q__3.r = t2 * z__[i__5].r, q__3.i = t2 * z__[i__5].i; q__1.r = q__2.r + q__3.r, q__1.i = q__2.i + q__3.i; sum.r = q__1.r, sum.i = q__1.i; i__4 = z___subscr(j, k); i__5 = z___subscr(j, k); q__1.r = z__[i__5].r - sum.r, q__1.i = z__[i__5].i - sum.i; z__[i__4].r = q__1.r, z__[i__4].i = q__1.i; i__4 = z___subscr(j, k + 1); i__5 = z___subscr(j, k + 1); r_cnjg(&q__3, &v2); q__2.r = sum.r * q__3.r - sum.i * q__3.i, q__2.i = sum.r * q__3.i + sum.i * q__3.r; q__1.r = z__[i__5].r - q__2.r, q__1.i = z__[i__5].i - q__2.i; z__[i__4].r = q__1.r, z__[i__4].i = q__1.i; /* L80: */ } /* ** Increment op count */ latime_1.ops += nz * 20; /* ** */ } if (k == m && m > l) { /* If the QR step was started at row M > L because two consecutive small subdiagonals were found, then extra scaling must be performed to ensure that H(M,M-1) remains real. */ q__1.r = 1.f - t1.r, q__1.i = 0.f - t1.i; temp.r = q__1.r, temp.i = q__1.i; r__1 = c_abs(&temp); q__1.r = temp.r / r__1, q__1.i = temp.i / r__1; temp.r = q__1.r, temp.i = q__1.i; i__3 = h___subscr(m + 1, m); i__4 = h___subscr(m + 1, m); r_cnjg(&q__2, &temp); q__1.r = h__[i__4].r * q__2.r - h__[i__4].i * q__2.i, q__1.i = h__[i__4].r * q__2.i + h__[i__4].i * q__2.r; h__[i__3].r = q__1.r, h__[i__3].i = q__1.i; if (m + 2 <= i__) { i__3 = h___subscr(m + 2, m + 1); i__4 = h___subscr(m + 2, m + 1); q__1.r = h__[i__4].r * temp.r - h__[i__4].i * temp.i, q__1.i = h__[i__4].r * temp.i + h__[i__4].i * temp.r; h__[i__3].r = q__1.r, h__[i__3].i = q__1.i; } i__3 = i__; for (j = m; j <= i__3; ++j) { if (j != m + 1) { if (i2 > j) { i__4 = i2 - j; cscal_(&i__4, &temp, &h___ref(j, j + 1), ldh); } i__4 = j - i1; r_cnjg(&q__1, &temp); cscal_(&i__4, &q__1, &h___ref(i1, j), &c__1); /* ** Increment op count */ opst += (i2 - i1 + 3) * 6; /* ** */ if (*wantz) { r_cnjg(&q__1, &temp); cscal_(&nz, &q__1, &z___ref(*iloz, j), &c__1); /* ** Increment op count */ opst += nz * 6; /* ** */ } } /* L90: */ } } /* L100: */ } /* Ensure that H(I,I-1) is real. */ i__2 = h___subscr(i__, i__ - 1); temp.r = h__[i__2].r, temp.i = h__[i__2].i; if (r_imag(&temp) != 0.f) { rtemp = c_abs(&temp); i__2 = h___subscr(i__, i__ - 1); h__[i__2].r = rtemp, h__[i__2].i = 0.f; q__1.r = temp.r / rtemp, q__1.i = temp.i / rtemp; temp.r = q__1.r, temp.i = q__1.i; if (i2 > i__) { i__2 = i2 - i__; r_cnjg(&q__1, &temp); cscal_(&i__2, &q__1, &h___ref(i__, i__ + 1), ldh); } i__2 = i__ - i1; cscal_(&i__2, &temp, &h___ref(i1, i__), &c__1); /* ** Increment op count */ opst += (i2 - i1 + 1) * 6; /* ** */ if (*wantz) { cscal_(&nz, &temp, &z___ref(*iloz, i__), &c__1); /* ** Increment op count */ opst += nz * 6; /* ** */ } } /* L110: */ } /* Failure to converge in remaining number of iterations */ *info = i__; return 0; L120: /* H(I,I-1) is negligible: one eigenvalue has converged. */ i__1 = i__; i__2 = h___subscr(i__, i__); w[i__1].r = h__[i__2].r, w[i__1].i = h__[i__2].i; /* Decrement number of remaining iterations, and return to start of the main loop with new value of I. */ itn -= its; i__ = l - 1; goto L10; L130: /* ** Compute final op count */ latime_1.ops += opst; /* ** */ return 0; /* End of CLAHQR */ } /* clahqr_ */ #undef z___ref #undef z___subscr #undef h___ref #undef h___subscr