#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__1 = 1; /* 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) { /* System generated locals */ integer h_dim1, h_offset, z_dim1, z_offset, i__1, i__2, i__3, i__4; real r__1, r__2; /* Builtin functions */ double sqrt(doublereal), r_sign(real *, real *); /* Local variables */ static real h43h34, disc, unfl, ovfl, work[1], opst; extern /* Subroutine */ int srot_(integer *, real *, integer *, real *, integer *, real *, real *); static integer i__, j, k, l, m; static real s, v[3]; static integer i1, i2; extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *, integer *); static real t1, t2, t3, v1, v2, v3; extern /* Subroutine */ int slanv2_(real *, real *, real *, real *, real * , real *, real *, real *, real *, real *); static real h00, h10, h11, h12, h21, h22, h33, h44; static integer nh; static real cs; extern /* Subroutine */ int slabad_(real *, real *); static integer nr; static real sn; extern doublereal slamch_(char *); static integer nz; extern /* Subroutine */ int slarfg_(integer *, real *, real *, integer *, real *); extern doublereal slanhs_(char *, integer *, real *, integer *, real *); static real smlnum, ave, h33s, h44s; static integer itn, its; static real ulp, sum, tst1; #define h___ref(a_1,a_2) h__[(a_2)*h_dim1 + a_1] #define z___ref(a_1,a_2) z__[(a_2)*z_dim1 + a_1] /* -- LAPACK auxiliary routine (instrum. to count ops. version 3.0) -- 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 ======= 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 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 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). 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 > 0: SLAHQR failed to compute all the eigenvalues ILO to IHI in a total of 30*(IHI-ILO+1) iterations; if INFO = i, elements i+1:ihi of WR and WI contain those eigenvalues which have been successfully computed. Further Details =============== 2-96 Based on modifications by David Day, Sandia National Laboratory, USA ===================================================================== Parameter adjustments */ h_dim1 = *ldh; h_offset = 1 + h_dim1 * 1; h__ -= h_offset; --wr; --wi; 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) { wr[*ilo] = h___ref(*ilo, *ilo); wi[*ilo] = 0.f; 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. */ unfl = slamch_("Safe minimum"); ovfl = 1.f / unfl; slabad_(&unfl, &ovfl); ulp = slamch_("Precision"); smlnum = unfl * (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; } /* 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 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; L10: l = *ilo; if (i__ < *ilo) { goto L150; } /* 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. */ 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) { tst1 = (r__1 = h___ref(k - 1, k - 1), dabs(r__1)) + (r__2 = h___ref(k, k), dabs(r__2)); if (tst1 == 0.f) { i__3 = i__ - l + 1; tst1 = slanhs_("1", &i__3, &h___ref(l, l), ldh, work); /* ** Increment op count */ latime_1.ops += (i__ - l + 1) * (i__ - l + 2) / 2; /* ** */ } /* Computing MAX */ r__2 = ulp * tst1; if ((r__1 = h___ref(k, k - 1), dabs(r__1)) <= dmax(r__2,smlnum)) { goto L30; } /* L20: */ } L30: l = k; /* ** Increment op count */ opst += (i__ - l + 1) * 3; /* ** */ if (l > *ilo) { /* H(L,L-1) is negligible */ h___ref(l, l - 1) = 0.f; } /* Exit from loop if a submatrix of order 1 or 2 has split off. */ if (l >= i__ - 1) { goto L140; } /* 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. */ s = (r__1 = h___ref(i__, i__ - 1), dabs(r__1)) + (r__2 = h___ref( i__ - 1, i__ - 2), dabs(r__2)); h44 = s * .75f + h___ref(i__, i__); h33 = h44; h43h34 = s * -.4375f * s; /* ** Increment op count */ opst += 5; /* ** */ } else { /* Prepare to use Francis' double shift (i.e. 2nd degree generalized Rayleigh quotient) */ h44 = h___ref(i__, i__); h33 = h___ref(i__ - 1, i__ - 1); h43h34 = h___ref(i__, i__ - 1) * h___ref(i__ - 1, i__); s = h___ref(i__ - 1, i__ - 2) * h___ref(i__ - 1, i__ - 2); disc = (h33 - h44) * .5f; disc = disc * disc + h43h34; /* ** Increment op count */ opst += 6; /* ** */ if (disc > 0.f) { /* Real roots: use Wilkinson's shift twice */ disc = sqrt(disc); ave = (h33 + h44) * .5f; /* ** Increment op count */ opst += 2; /* ** */ if (dabs(h33) - dabs(h44) > 0.f) { h33 = h33 * h44 - h43h34; h44 = h33 / (r_sign(&disc, &ave) + ave); /* ** Increment op count */ opst += 4; /* ** */ } else { h44 = r_sign(&disc, &ave) + ave; /* ** Increment op count */ opst += 1; /* ** */ } h33 = h44; h43h34 = 0.f; } } /* Look for two consecutive small subdiagonal elements. */ i__2 = l; for (m = i__ - 2; m >= i__2; --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. */ h11 = h___ref(m, m); h22 = h___ref(m + 1, m + 1); h21 = h___ref(m + 1, m); h12 = h___ref(m, m + 1); h44s = h44 - h11; h33s = h33 - h11; v1 = (h33s * h44s - h43h34) / h21 + h12; v2 = h22 - h11 - h33s - h44s; v3 = h___ref(m + 2, m + 1); s = dabs(v1) + dabs(v2) + dabs(v3); v1 /= s; v2 /= s; v3 /= s; v[0] = v1; v[1] = v2; v[2] = v3; if (m == l) { goto L50; } h00 = h___ref(m - 1, m - 1); h10 = h___ref(m, m - 1); tst1 = dabs(v1) * (dabs(h00) + dabs(h11) + dabs(h22)); if (dabs(h10) * (dabs(v2) + dabs(v3)) <= ulp * tst1) { goto L50; } /* L40: */ } L50: /* ** Increment op count */ opst += (i__ - m - 1) * 20; /* ** Double-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. NR is the order of G. Computing MIN */ i__3 = 3, i__4 = i__ - k + 1; nr = min(i__3,i__4); if (k > m) { scopy_(&nr, &h___ref(k, k - 1), &c__1, v, &c__1); } slarfg_(&nr, v, &v[1], &c__1, &t1); /* ** Increment op count */ opst = opst + nr * 3 + 9; /* ** */ if (k > m) { h___ref(k, k - 1) = v[0]; h___ref(k + 1, k - 1) = 0.f; if (k < i__ - 1) { h___ref(k + 2, k - 1) = 0.f; } } else if (m > l) { h___ref(k, k - 1) = -h___ref(k, k - 1); } 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__3 = i2; for (j = k; j <= i__3; ++j) { sum = h___ref(k, j) + v2 * h___ref(k + 1, j) + v3 * h___ref(k + 2, j); h___ref(k, j) = h___ref(k, j) - sum * t1; h___ref(k + 1, j) = h___ref(k + 1, j) - sum * t2; h___ref(k + 2, j) = h___ref(k + 2, j) - sum * t3; /* L60: */ } /* Apply G from the right to transform the columns of the matrix in rows I1 to min(K+3,I). Computing MIN */ i__4 = k + 3; i__3 = min(i__4,i__); for (j = i1; j <= i__3; ++j) { sum = h___ref(j, k) + v2 * h___ref(j, k + 1) + v3 * h___ref(j, k + 2); h___ref(j, k) = h___ref(j, k) - sum * t1; h___ref(j, k + 1) = h___ref(j, k + 1) - sum * t2; h___ref(j, k + 2) = h___ref(j, k + 2) - sum * t3; /* L70: */ } /* ** Increment op count Computing MIN */ i__3 = 3, i__4 = i__ - k; latime_1.ops += (i2 - i1 + 2 + min(i__3,i__4)) * 10; /* ** */ if (*wantz) { /* Accumulate transformations in the matrix Z */ i__3 = *ihiz; for (j = *iloz; j <= i__3; ++j) { sum = z___ref(j, k) + v2 * z___ref(j, k + 1) + v3 * z___ref(j, k + 2); z___ref(j, k) = z___ref(j, k) - sum * t1; z___ref(j, k + 1) = z___ref(j, k + 1) - sum * t2; z___ref(j, k + 2) = z___ref(j, k + 2) - sum * t3; /* L80: */ } /* ** Increment op count */ latime_1.ops += nz * 10; /* ** */ } } else if (nr == 2) { /* 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) { sum = h___ref(k, j) + v2 * h___ref(k + 1, j); h___ref(k, j) = h___ref(k, j) - sum * t1; h___ref(k + 1, j) = h___ref(k + 1, j) - sum * t2; /* L90: */ } /* Apply G from the right to transform the columns of the matrix in rows I1 to min(K+3,I). */ i__3 = i__; for (j = i1; j <= i__3; ++j) { sum = h___ref(j, k) + v2 * h___ref(j, k + 1); h___ref(j, k) = h___ref(j, k) - sum * t1; h___ref(j, k + 1) = h___ref(j, k + 1) - sum * t2; /* L100: */ } /* ** Increment op count */ latime_1.ops += (i2 - i1 + 3) * 6; /* ** */ if (*wantz) { /* Accumulate transformations in the matrix Z */ i__3 = *ihiz; for (j = *iloz; j <= i__3; ++j) { sum = z___ref(j, k) + v2 * z___ref(j, k + 1); z___ref(j, k) = z___ref(j, k) - sum * t1; z___ref(j, k + 1) = z___ref(j, k + 1) - sum * t2; /* L110: */ } /* ** Increment op count */ latime_1.ops += nz * 6; /* ** */ } } /* L120: */ } /* L130: */ } /* Failure to converge in remaining number of iterations */ *info = i__; return 0; L140: if (l == i__) { /* H(I,I-1) is negligible: one eigenvalue has converged. */ wr[i__] = h___ref(i__, i__); 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___ref(i__ - 1, i__ - 1), &h___ref(i__ - 1, i__), &h___ref( i__, i__ - 1), &h___ref(i__, i__), &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___ref(i__ - 1, i__ + 1), ldh, &h___ref(i__, i__ + 1), ldh, &cs, &sn); } i__1 = i__ - i1 - 1; srot_(&i__1, &h___ref(i1, i__ - 1), &c__1, &h___ref(i1, i__), & c__1, &cs, &sn); /* ** Increment op count */ latime_1.ops += (i2 - i1 - 1) * 6; /* ** */ } if (*wantz) { /* Apply the transformation to Z. */ srot_(&nz, &z___ref(*iloz, i__ - 1), &c__1, &z___ref(*iloz, i__), &c__1, &cs, &sn); /* ** Increment op count */ latime_1.ops += nz * 6; /* ** */ } } /* 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; L150: /* ** Compute final op count */ latime_1.ops += opst; /* ** */ return 0; /* End of SLAHQR */ } /* slahqr_ */ #undef z___ref #undef h___ref