#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 { doublereal ops, itcnt; } latime_; #define latime_1 latime_ /* Table of constant values */ static doublereal c_b3 = -1.; static integer c__1 = 1; /* Subroutine */ int zlaed8_(integer *k, integer *n, integer *qsiz, doublecomplex *q, integer *ldq, doublereal *d__, doublereal *rho, integer *cutpnt, doublereal *z__, doublereal *dlamda, doublecomplex * q2, integer *ldq2, doublereal *w, integer *indxp, integer *indx, integer *indxq, integer *perm, integer *givptr, integer *givcol, doublereal *givnum, integer *info) { /* System generated locals */ integer q_dim1, q_offset, q2_dim1, q2_offset, i__1; doublereal d__1; /* Builtin functions */ double sqrt(doublereal); /* Local variables */ static integer jlam, imax, jmax; static doublereal c__; static integer i__, j; static doublereal s, t; extern /* Subroutine */ int dscal_(integer *, doublereal *, doublereal *, integer *), dcopy_(integer *, doublereal *, integer *, doublereal *, integer *); static integer k2, n1, n2; extern /* Subroutine */ int zdrot_(integer *, doublecomplex *, integer *, doublecomplex *, integer *, doublereal *, doublereal *), zcopy_( integer *, doublecomplex *, integer *, doublecomplex *, integer *) ; extern doublereal dlapy2_(doublereal *, doublereal *), dlamch_(char *); static integer jp; extern integer idamax_(integer *, doublereal *, integer *); extern /* Subroutine */ int dlamrg_(integer *, integer *, doublereal *, integer *, integer *, integer *), xerbla_(char *, integer *), zlacpy_(char *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, integer *); static integer n1p1; static doublereal eps, tau, tol; #define q_subscr(a_1,a_2) (a_2)*q_dim1 + a_1 #define q_ref(a_1,a_2) q[q_subscr(a_1,a_2)] #define q2_subscr(a_1,a_2) (a_2)*q2_dim1 + a_1 #define q2_ref(a_1,a_2) q2[q2_subscr(a_1,a_2)] #define givcol_ref(a_1,a_2) givcol[(a_2)*2 + a_1] #define givnum_ref(a_1,a_2) givnum[(a_2)*2 + a_1] /* -- LAPACK routine (instrumented to count operations, version 3.0) -- Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab, Courant Institute, NAG Ltd., and Rice University September 30, 1994 Common block to return operation count and iteration count ITCNT is unchanged, OPS is only incremented Purpose ======= ZLAED8 merges the two sets of eigenvalues together into a single sorted set. Then it tries to deflate the size of the problem. There are two ways in which deflation can occur: when two or more eigenvalues are close together or if there is a tiny element in the Z vector. For each such occurrence the order of the related secular equation problem is reduced by one. Arguments ========= K (output) INTEGER Contains the number of non-deflated eigenvalues. This is the order of the related secular equation. N (input) INTEGER The dimension of the symmetric tridiagonal matrix. N >= 0. QSIZ (input) INTEGER The dimension of the unitary matrix used to reduce the dense or band matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1. Q (input/output) COMPLEX*16 array, dimension (LDQ,N) On entry, Q contains the eigenvectors of the partially solved system which has been previously updated in matrix multiplies with other partially solved eigensystems. On exit, Q contains the trailing (N-K) updated eigenvectors (those which were deflated) in its last N-K columns. LDQ (input) INTEGER The leading dimension of the array Q. LDQ >= max( 1, N ). D (input/output) DOUBLE PRECISION array, dimension (N) On entry, D contains the eigenvalues of the two submatrices to be combined. On exit, D contains the trailing (N-K) updated eigenvalues (those which were deflated) sorted into increasing order. RHO (input/output) DOUBLE PRECISION Contains the off diagonal element associated with the rank-1 cut which originally split the two submatrices which are now being recombined. RHO is modified during the computation to the value required by DLAED3. CUTPNT (input) INTEGER Contains the location of the last eigenvalue in the leading sub-matrix. MIN(1,N) <= CUTPNT <= N. Z (input) DOUBLE PRECISION array, dimension (N) On input this vector contains the updating vector (the last row of the first sub-eigenvector matrix and the first row of the second sub-eigenvector matrix). The contents of Z are destroyed during the updating process. DLAMDA (output) DOUBLE PRECISION array, dimension (N) Contains a copy of the first K eigenvalues which will be used by DLAED3 to form the secular equation. Q2 (output) COMPLEX*16 array, dimension (LDQ2,N) If ICOMPQ = 0, Q2 is not referenced. Otherwise, Contains a copy of the first K eigenvectors which will be used by DLAED7 in a matrix multiply (DGEMM) to update the new eigenvectors. LDQ2 (input) INTEGER The leading dimension of the array Q2. LDQ2 >= max( 1, N ). W (output) DOUBLE PRECISION array, dimension (N) This will hold the first k values of the final deflation-altered z-vector and will be passed to DLAED3. INDXP (workspace) INTEGER array, dimension (N) This will contain the permutation used to place deflated values of D at the end of the array. On output INDXP(1:K) points to the nondeflated D-values and INDXP(K+1:N) points to the deflated eigenvalues. INDX (workspace) INTEGER array, dimension (N) This will contain the permutation used to sort the contents of D into ascending order. INDXQ (input) INTEGER array, dimension (N) This contains the permutation which separately sorts the two sub-problems in D into ascending order. Note that elements in the second half of this permutation must first have CUTPNT added to their values in order to be accurate. PERM (output) INTEGER array, dimension (N) Contains the permutations (from deflation and sorting) to be applied to each eigenblock. GIVPTR (output) INTEGER Contains the number of Givens rotations which took place in this subproblem. GIVCOL (output) INTEGER array, dimension (2, N) Each pair of numbers indicates a pair of columns to take place in a Givens rotation. GIVNUM (output) DOUBLE PRECISION array, dimension (2, N) Each number indicates the S value to be used in the corresponding Givens rotation. INFO (output) INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. ===================================================================== Test the input parameters. Parameter adjustments */ q_dim1 = *ldq; q_offset = 1 + q_dim1 * 1; q -= q_offset; --d__; --z__; --dlamda; q2_dim1 = *ldq2; q2_offset = 1 + q2_dim1 * 1; q2 -= q2_offset; --w; --indxp; --indx; --indxq; --perm; givcol -= 3; givnum -= 3; /* Function Body */ *info = 0; if (*n < 0) { *info = -2; } else if (*qsiz < *n) { *info = -3; } else if (*ldq < max(1,*n)) { *info = -5; } else if (*cutpnt < min(1,*n) || *cutpnt > *n) { *info = -8; } else if (*ldq2 < max(1,*n)) { *info = -12; } if (*info != 0) { i__1 = -(*info); xerbla_("ZLAED8", &i__1); return 0; } /* Quick return if possible */ if (*n == 0) { return 0; } n1 = *cutpnt; n2 = *n - n1; n1p1 = n1 + 1; if (*rho < 0.) { latime_1.ops += n2; dscal_(&n2, &c_b3, &z__[n1p1], &c__1); } /* Normalize z so that norm(z) = 1 */ latime_1.ops = latime_1.ops + *n + 6; t = 1. / sqrt(2.); i__1 = *n; for (j = 1; j <= i__1; ++j) { indx[j] = j; /* L10: */ } dscal_(n, &t, &z__[1], &c__1); *rho = (d__1 = *rho * 2., abs(d__1)); /* Sort the eigenvalues into increasing order */ i__1 = *n; for (i__ = *cutpnt + 1; i__ <= i__1; ++i__) { indxq[i__] += *cutpnt; /* L20: */ } i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { dlamda[i__] = d__[indxq[i__]]; w[i__] = z__[indxq[i__]]; /* L30: */ } i__ = 1; j = *cutpnt + 1; dlamrg_(&n1, &n2, &dlamda[1], &c__1, &c__1, &indx[1]); i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { d__[i__] = dlamda[indx[i__]]; z__[i__] = w[indx[i__]]; /* L40: */ } /* Calculate the allowable deflation tolerance */ imax = idamax_(n, &z__[1], &c__1); jmax = idamax_(n, &d__[1], &c__1); eps = dlamch_("Epsilon"); tol = eps * 8. * (d__1 = d__[jmax], abs(d__1)); /* If the rank-1 modifier is small enough, no more needs to be done - except to reorganize Q so that its columns correspond with the elements in D. */ if (*rho * (d__1 = z__[imax], abs(d__1)) <= tol) { *k = 0; i__1 = *n; for (j = 1; j <= i__1; ++j) { perm[j] = indxq[indx[j]]; zcopy_(qsiz, &q_ref(1, perm[j]), &c__1, &q2_ref(1, j), &c__1); /* L50: */ } zlacpy_("A", qsiz, n, &q2_ref(1, 1), ldq2, &q_ref(1, 1), ldq); return 0; } /* If there are multiple eigenvalues then the problem deflates. Here the number of equal eigenvalues are found. As each equal eigenvalue is found, an elementary reflector is computed to rotate the corresponding eigensubspace so that the corresponding components of Z are zero in this new basis. */ *k = 0; *givptr = 0; k2 = *n + 1; i__1 = *n; for (j = 1; j <= i__1; ++j) { latime_1.ops += 1; if (*rho * (d__1 = z__[j], abs(d__1)) <= tol) { /* Deflate due to small z component. */ --k2; indxp[k2] = j; if (j == *n) { goto L100; } } else { jlam = j; goto L70; } /* L60: */ } L70: ++j; if (j > *n) { goto L90; } latime_1.ops += 1; if (*rho * (d__1 = z__[j], abs(d__1)) <= tol) { /* Deflate due to small z component. */ --k2; indxp[k2] = j; } else { /* Check if eigenvalues are close enough to allow deflation. */ s = z__[jlam]; c__ = z__[j]; /* Find sqrt(a**2+b**2) without overflow or destructive underflow. */ latime_1.ops += 10; tau = dlapy2_(&c__, &s); t = d__[j] - d__[jlam]; c__ /= tau; s = -s / tau; if ((d__1 = t * c__ * s, abs(d__1)) <= tol) { /* Deflation is possible. */ z__[j] = tau; z__[jlam] = 0.; /* Record the appropriate Givens rotation */ ++(*givptr); givcol_ref(1, *givptr) = indxq[indx[jlam]]; givcol_ref(2, *givptr) = indxq[indx[j]]; givnum_ref(1, *givptr) = c__; givnum_ref(2, *givptr) = s; latime_1.ops += *qsiz * 12; zdrot_(qsiz, &q_ref(1, indxq[indx[jlam]]), &c__1, &q_ref(1, indxq[ indx[j]]), &c__1, &c__, &s); latime_1.ops += 10; t = d__[jlam] * c__ * c__ + d__[j] * s * s; d__[j] = d__[jlam] * s * s + d__[j] * c__ * c__; d__[jlam] = t; --k2; i__ = 1; L80: if (k2 + i__ <= *n) { if (d__[jlam] < d__[indxp[k2 + i__]]) { indxp[k2 + i__ - 1] = indxp[k2 + i__]; indxp[k2 + i__] = jlam; ++i__; goto L80; } else { indxp[k2 + i__ - 1] = jlam; } } else { indxp[k2 + i__ - 1] = jlam; } jlam = j; } else { ++(*k); w[*k] = z__[jlam]; dlamda[*k] = d__[jlam]; indxp[*k] = jlam; jlam = j; } } goto L70; L90: /* Record the last eigenvalue. */ ++(*k); w[*k] = z__[jlam]; dlamda[*k] = d__[jlam]; indxp[*k] = jlam; L100: /* Sort the eigenvalues and corresponding eigenvectors into DLAMDA and Q2 respectively. The eigenvalues/vectors which were not deflated go into the first K slots of DLAMDA and Q2 respectively, while those which were deflated go into the last N - K slots. */ i__1 = *n; for (j = 1; j <= i__1; ++j) { jp = indxp[j]; dlamda[j] = d__[jp]; perm[j] = indxq[indx[jp]]; zcopy_(qsiz, &q_ref(1, perm[j]), &c__1, &q2_ref(1, j), &c__1); /* L110: */ } /* The deflated eigenvalues and their corresponding vectors go back into the last N - K slots of D and Q respectively. */ if (*k < *n) { i__1 = *n - *k; dcopy_(&i__1, &dlamda[*k + 1], &c__1, &d__[*k + 1], &c__1); i__1 = *n - *k; zlacpy_("A", qsiz, &i__1, &q2_ref(1, *k + 1), ldq2, &q_ref(1, *k + 1), ldq); } return 0; /* End of ZLAED8 */ } /* zlaed8_ */ #undef givnum_ref #undef givcol_ref #undef q2_ref #undef q2_subscr #undef q_ref #undef q_subscr