#include "blaswrap.h" /* dlaed8.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" /* Table of constant values */ static doublereal c_b3 = -1.; static integer c__1 = 1; /* Subroutine */ int dlaed8_(integer *icompq, integer *k, integer *n, integer *qsiz, doublereal *d__, doublereal *q, integer *ldq, integer *indxq, doublereal *rho, integer *cutpnt, doublereal *z__, doublereal *dlamda, doublereal *q2, integer *ldq2, doublereal *w, integer *perm, integer *givptr, integer *givcol, doublereal *givnum, integer *indxp, integer *indx, 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 doublereal c__; static integer i__, j; static doublereal s, t; static integer k2, n1, n2, jp, n1p1; static doublereal eps, tau, tol; static integer jlam, imax, jmax; extern /* Subroutine */ int drot_(integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *), dscal_( integer *, doublereal *, doublereal *, integer *), dcopy_(integer *, doublereal *, integer *, doublereal *, integer *); extern doublereal dlapy2_(doublereal *, doublereal *), dlamch_(char *); extern integer idamax_(integer *, doublereal *, integer *); extern /* Subroutine */ int dlamrg_(integer *, integer *, doublereal *, integer *, integer *, integer *), dlacpy_(char *, integer *, integer *, doublereal *, integer *, doublereal *, integer *), xerbla_(char *, integer *); /* -- LAPACK routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 Purpose ======= DLAED8 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 ========= ICOMPQ (input) INTEGER = 0: Compute eigenvalues only. = 1: Compute eigenvectors of original dense symmetric matrix also. On entry, Q contains the orthogonal matrix used to reduce the original matrix to tridiagonal form. K (output) INTEGER The number of non-deflated eigenvalues, and 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 orthogonal matrix used to reduce the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1. D (input/output) DOUBLE PRECISION array, dimension (N) On entry, the eigenvalues of the two submatrices to be combined. On exit, the trailing (N-K) updated eigenvalues (those which were deflated) sorted into increasing order. Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N) If ICOMPQ = 0, Q is not referenced. Otherwise, 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). INDXQ (input) INTEGER array, dimension (N) 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. RHO (input/output) DOUBLE PRECISION On entry, the off-diagonal element associated with the rank-1 cut which originally split the two submatrices which are now being recombined. On exit, RHO has been modified to the value required by DLAED3. CUTPNT (input) INTEGER The location of the last eigenvalue in the leading sub-matrix. min(1,N) <= CUTPNT <= N. Z (input) DOUBLE PRECISION array, dimension (N) On entry, Z contains the updating vector (the last row of the first sub-eigenvector matrix and the first row of the second sub-eigenvector matrix). On exit, the contents of Z are destroyed by the updating process. DLAMDA (output) DOUBLE PRECISION array, dimension (N) A copy of the first K eigenvalues which will be used by DLAED3 to form the secular equation. Q2 (output) DOUBLE PRECISION array, dimension (LDQ2,N) If ICOMPQ = 0, Q2 is not referenced. Otherwise, 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) The first k values of the final deflation-altered z-vector and will be passed to DLAED3. PERM (output) INTEGER array, dimension (N) The permutations (from deflation and sorting) to be applied to each eigenblock. GIVPTR (output) INTEGER 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. INDXP (workspace) INTEGER array, dimension (N) The permutation used to place deflated values of D at the end of the array. 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) The permutation used to sort the contents of D into ascending order. INFO (output) INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. Further Details =============== Based on contributions by Jeff Rutter, Computer Science Division, University of California at Berkeley, USA ===================================================================== Test the input parameters. Parameter adjustments */ --d__; q_dim1 = *ldq; q_offset = 1 + q_dim1; q -= q_offset; --indxq; --z__; --dlamda; q2_dim1 = *ldq2; q2_offset = 1 + q2_dim1; q2 -= q2_offset; --w; --perm; givcol -= 3; givnum -= 3; --indxp; --indx; /* Function Body */ *info = 0; if (*icompq < 0 || *icompq > 1) { *info = -1; } else if (*n < 0) { *info = -3; } else if (*icompq == 1 && *qsiz < *n) { *info = -4; } else if (*ldq < max(1,*n)) { *info = -7; } else if (*cutpnt < min(1,*n) || *cutpnt > *n) { *info = -10; } else if (*ldq2 < max(1,*n)) { *info = -14; } if (*info != 0) { i__1 = -(*info); xerbla_("DLAED8", &i__1); return 0; } /* Quick return if possible */ if (*n == 0) { return 0; } n1 = *cutpnt; n2 = *n - n1; n1p1 = n1 + 1; if (*rho < 0.) { dscal_(&n2, &c_b3, &z__[n1p1], &c__1); } /* Normalize z so that norm(z) = 1 */ 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 tolerence */ 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; if (*icompq == 0) { i__1 = *n; for (j = 1; j <= i__1; ++j) { perm[j] = indxq[indx[j]]; /* L50: */ } } else { i__1 = *n; for (j = 1; j <= i__1; ++j) { perm[j] = indxq[indx[j]]; dcopy_(qsiz, &q[perm[j] * q_dim1 + 1], &c__1, &q2[j * q2_dim1 + 1], &c__1); /* L60: */ } dlacpy_("A", qsiz, n, &q2[q2_dim1 + 1], ldq2, &q[q_dim1 + 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) { if (*rho * (d__1 = z__[j], abs(d__1)) <= tol) { /* Deflate due to small z component. */ --k2; indxp[k2] = j; if (j == *n) { goto L110; } } else { jlam = j; goto L80; } /* L70: */ } L80: ++j; if (j > *n) { goto L100; } 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. */ 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[(*givptr << 1) + 1] = indxq[indx[jlam]]; givcol[(*givptr << 1) + 2] = indxq[indx[j]]; givnum[(*givptr << 1) + 1] = c__; givnum[(*givptr << 1) + 2] = s; if (*icompq == 1) { drot_(qsiz, &q[indxq[indx[jlam]] * q_dim1 + 1], &c__1, &q[ indxq[indx[j]] * q_dim1 + 1], &c__1, &c__, &s); } 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; L90: if (k2 + i__ <= *n) { if (d__[jlam] < d__[indxp[k2 + i__]]) { indxp[k2 + i__ - 1] = indxp[k2 + i__]; indxp[k2 + i__] = jlam; ++i__; goto L90; } 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 L80; L100: /* Record the last eigenvalue. */ ++(*k); w[*k] = z__[jlam]; dlamda[*k] = d__[jlam]; indxp[*k] = jlam; L110: /* 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. */ if (*icompq == 0) { i__1 = *n; for (j = 1; j <= i__1; ++j) { jp = indxp[j]; dlamda[j] = d__[jp]; perm[j] = indxq[indx[jp]]; /* L120: */ } } else { i__1 = *n; for (j = 1; j <= i__1; ++j) { jp = indxp[j]; dlamda[j] = d__[jp]; perm[j] = indxq[indx[jp]]; dcopy_(qsiz, &q[perm[j] * q_dim1 + 1], &c__1, &q2[j * q2_dim1 + 1] , &c__1); /* L130: */ } } /* The deflated eigenvalues and their corresponding vectors go back into the last N - K slots of D and Q respectively. */ if (*k < *n) { if (*icompq == 0) { i__1 = *n - *k; dcopy_(&i__1, &dlamda[*k + 1], &c__1, &d__[*k + 1], &c__1); } else { i__1 = *n - *k; dcopy_(&i__1, &dlamda[*k + 1], &c__1, &d__[*k + 1], &c__1); i__1 = *n - *k; dlacpy_("A", qsiz, &i__1, &q2[(*k + 1) * q2_dim1 + 1], ldq2, &q[(* k + 1) * q_dim1 + 1], ldq); } } return 0; /* End of DLAED8 */ } /* dlaed8_ */