#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int zstedc_(char *compz, integer *n, doublereal *d__, doublereal *e, doublecomplex *z__, integer *ldz, doublecomplex *work, integer *lwork, doublereal *rwork, integer *lrwork, integer *iwork, integer *liwork, integer *info) { /* -- LAPACK routine (version 3.0) -- Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., Courant Institute, Argonne National Lab, and Rice University June 30, 1999 Purpose ======= ZSTEDC computes all eigenvalues and, optionally, eigenvectors of a symmetric tridiagonal matrix using the divide and conquer method. The eigenvectors of a full or band complex Hermitian matrix can also be found if ZHETRD or ZHPTRD or ZHBTRD has been used to reduce this matrix to tridiagonal form. This code makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none. See DLAED3 for details. Arguments ========= COMPZ (input) CHARACTER*1 = 'N': Compute eigenvalues only. = 'I': Compute eigenvectors of tridiagonal matrix also. = 'V': Compute eigenvectors of original Hermitian matrix also. On entry, Z contains the unitary matrix used to reduce the original matrix to tridiagonal form. N (input) INTEGER The dimension of the symmetric tridiagonal matrix. N >= 0. D (input/output) DOUBLE PRECISION array, dimension (N) On entry, the diagonal elements of the tridiagonal matrix. On exit, if INFO = 0, the eigenvalues in ascending order. E (input/output) DOUBLE PRECISION array, dimension (N-1) On entry, the subdiagonal elements of the tridiagonal matrix. On exit, E has been destroyed. Z (input/output) COMPLEX*16 array, dimension (LDZ,N) On entry, if COMPZ = 'V', then Z contains the unitary matrix used in the reduction to tridiagonal form. On exit, if INFO = 0, then if COMPZ = 'V', Z contains the orthonormal eigenvectors of the original Hermitian matrix, and if COMPZ = 'I', Z contains the orthonormal eigenvectors of the symmetric tridiagonal matrix. If COMPZ = 'N', then Z is not referenced. LDZ (input) INTEGER The leading dimension of the array Z. LDZ >= 1. If eigenvectors are desired, then LDZ >= max(1,N). WORK (workspace/output) COMPLEX*16 array, dimension (LWORK) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK (input) INTEGER The dimension of the array WORK. If COMPZ = 'N' or 'I', or N <= 1, LWORK must be at least 1. If COMPZ = 'V' and N > 1, LWORK must be at least N*N. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. RWORK (workspace/output) DOUBLE PRECISION array, dimension (LRWORK) On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK. LRWORK (input) INTEGER The dimension of the array RWORK. If COMPZ = 'N' or N <= 1, LRWORK must be at least 1. If COMPZ = 'V' and N > 1, LRWORK must be at least 1 + 3*N + 2*N*lg N + 3*N**2 , where lg( N ) = smallest integer k such that 2**k >= N. If COMPZ = 'I' and N > 1, LRWORK must be at least 1 + 4*N + 2*N**2 . If LRWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the RWORK array, returns this value as the first entry of the RWORK array, and no error message related to LRWORK is issued by XERBLA. IWORK (workspace/output) INTEGER array, dimension (LIWORK) On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. LIWORK (input) INTEGER The dimension of the array IWORK. If COMPZ = 'N' or N <= 1, LIWORK must be at least 1. If COMPZ = 'V' or N > 1, LIWORK must be at least 6 + 6*N + 5*N*lg N. If COMPZ = 'I' or N > 1, LIWORK must be at least 3 + 5*N . If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the IWORK array, returns this value as the first entry of the IWORK array, and no error message related to LIWORK is issued by XERBLA. INFO (output) INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: The algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns INFO/(N+1) through mod(INFO,N+1). Further Details =============== Based on contributions by Jeff Rutter, Computer Science Division, University of California at Berkeley, USA ===================================================================== Test the input parameters. Parameter adjustments */ /* Table of constant values */ static integer c__2 = 2; static integer c__9 = 9; static integer c__0 = 0; static doublereal c_b18 = 0.; static doublereal c_b19 = 1.; static integer c__1 = 1; /* System generated locals */ integer z_dim1, z_offset, i__1, i__2, i__3, i__4; doublereal d__1, d__2; /* Builtin functions */ double log(doublereal); integer pow_ii(integer *, integer *); double sqrt(doublereal); /* Local variables */ static doublereal tiny; static integer i__, j, k, m; static doublereal p; extern logical lsame_(char *, char *); static integer lwmin, start; extern /* Subroutine */ int zswap_(integer *, doublecomplex *, integer *, doublecomplex *, integer *), zlaed0_(integer *, integer *, doublereal *, doublereal *, doublecomplex *, integer *, doublecomplex *, integer *, doublereal *, integer *, integer *); static integer ii, ll; extern doublereal dlamch_(char *); extern /* Subroutine */ int dlascl_(char *, integer *, integer *, doublereal *, doublereal *, integer *, integer *, doublereal *, integer *, integer *), dstedc_(char *, integer *, doublereal *, doublereal *, doublereal *, integer *, doublereal *, integer *, integer *, integer *, integer *), dlaset_( char *, integer *, integer *, doublereal *, doublereal *, doublereal *, integer *), xerbla_(char *, integer *); extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *, ftnlen, ftnlen); extern doublereal dlanst_(char *, integer *, doublereal *, doublereal *); extern /* Subroutine */ int dsterf_(integer *, doublereal *, doublereal *, integer *), zlacrm_(integer *, integer *, doublecomplex *, integer *, doublereal *, integer *, doublecomplex *, integer *, doublereal *); static integer liwmin, icompz; extern /* Subroutine */ int dsteqr_(char *, integer *, doublereal *, doublereal *, doublereal *, integer *, doublereal *, integer *), zlacpy_(char *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, integer *); static doublereal orgnrm; static integer lrwmin; static logical lquery; static integer smlsiz; extern /* Subroutine */ int zsteqr_(char *, integer *, doublereal *, doublereal *, doublecomplex *, integer *, doublereal *, integer *); static integer end, lgn; static doublereal eps; #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)] --d__; --e; z_dim1 = *ldz; z_offset = 1 + z_dim1 * 1; z__ -= z_offset; --work; --rwork; --iwork; /* Function Body */ *info = 0; lquery = *lwork == -1 || *lrwork == -1 || *liwork == -1; if (lsame_(compz, "N")) { icompz = 0; } else if (lsame_(compz, "V")) { icompz = 1; } else if (lsame_(compz, "I")) { icompz = 2; } else { icompz = -1; } if (*n <= 1 || icompz <= 0) { lwmin = 1; liwmin = 1; lrwmin = 1; } else { lgn = (integer) (log((doublereal) (*n)) / log(2.)); if (pow_ii(&c__2, &lgn) < *n) { ++lgn; } if (pow_ii(&c__2, &lgn) < *n) { ++lgn; } if (icompz == 1) { lwmin = *n * *n; /* Computing 2nd power */ i__1 = *n; lrwmin = *n * 3 + 1 + (*n << 1) * lgn + i__1 * i__1 * 3; liwmin = *n * 6 + 6 + *n * 5 * lgn; } else if (icompz == 2) { lwmin = 1; /* Computing 2nd power */ i__1 = *n; lrwmin = (*n << 2) + 1 + (i__1 * i__1 << 1); liwmin = *n * 5 + 3; } } if (icompz < 0) { *info = -1; } else if (*n < 0) { *info = -2; } else if (*ldz < 1 || icompz > 0 && *ldz < max(1,*n)) { *info = -6; } else if (*lwork < lwmin && ! lquery) { *info = -8; } else if (*lrwork < lrwmin && ! lquery) { *info = -10; } else if (*liwork < liwmin && ! lquery) { *info = -12; } if (*info == 0) { work[1].r = (doublereal) lwmin, work[1].i = 0.; rwork[1] = (doublereal) lrwmin; iwork[1] = liwmin; } if (*info != 0) { i__1 = -(*info); xerbla_("ZSTEDC", &i__1); return 0; } else if (lquery) { return 0; } /* Quick return if possible */ if (*n == 0) { return 0; } if (*n == 1) { if (icompz != 0) { i__1 = z___subscr(1, 1); z__[i__1].r = 1., z__[i__1].i = 0.; } return 0; } smlsiz = ilaenv_(&c__9, "ZSTEDC", " ", &c__0, &c__0, &c__0, &c__0, ( ftnlen)6, (ftnlen)1); /* If the following conditional clause is removed, then the routine will use the Divide and Conquer routine to compute only the eigenvalues, which requires (3N + 3N**2) real workspace and (2 + 5N + 2N lg(N)) integer workspace. Since on many architectures DSTERF is much faster than any other algorithm for finding eigenvalues only, it is used here as the default. If COMPZ = 'N', use DSTERF to compute the eigenvalues. */ if (icompz == 0) { dsterf_(n, &d__[1], &e[1], info); return 0; } /* If N is smaller than the minimum divide size (SMLSIZ+1), then solve the problem with another solver. */ if (*n <= smlsiz) { if (icompz == 0) { dsterf_(n, &d__[1], &e[1], info); return 0; } else if (icompz == 2) { zsteqr_("I", n, &d__[1], &e[1], &z__[z_offset], ldz, &rwork[1], info); return 0; } else { zsteqr_("V", n, &d__[1], &e[1], &z__[z_offset], ldz, &rwork[1], info); return 0; } } /* If COMPZ = 'I', we simply call DSTEDC instead. */ if (icompz == 2) { dlaset_("Full", n, n, &c_b18, &c_b19, &rwork[1], n); ll = *n * *n + 1; i__1 = *lrwork - ll + 1; dstedc_("I", n, &d__[1], &e[1], &rwork[1], n, &rwork[ll], &i__1, & iwork[1], liwork, info); i__1 = *n; for (j = 1; j <= i__1; ++j) { i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { i__3 = z___subscr(i__, j); i__4 = (j - 1) * *n + i__; z__[i__3].r = rwork[i__4], z__[i__3].i = 0.; /* L10: */ } /* L20: */ } return 0; } /* From now on, only option left to be handled is COMPZ = 'V', i.e. ICOMPZ = 1. Scale. */ orgnrm = dlanst_("M", n, &d__[1], &e[1]); if (orgnrm == 0.) { return 0; } eps = dlamch_("Epsilon"); start = 1; /* while ( START <= N ) */ L30: if (start <= *n) { /* Let END be the position of the next subdiagonal entry such that E( END ) <= TINY or END = N if no such subdiagonal exists. The matrix identified by the elements between START and END constitutes an independent sub-problem. */ end = start; L40: if (end < *n) { tiny = eps * sqrt((d__1 = d__[end], abs(d__1))) * sqrt((d__2 = d__[end + 1], abs(d__2))); if ((d__1 = e[end], abs(d__1)) > tiny) { ++end; goto L40; } } /* (Sub) Problem determined. Compute its size and solve it. */ m = end - start + 1; if (m > smlsiz) { *info = smlsiz; /* Scale. */ orgnrm = dlanst_("M", &m, &d__[start], &e[start]); dlascl_("G", &c__0, &c__0, &orgnrm, &c_b19, &m, &c__1, &d__[start] , &m, info); i__1 = m - 1; i__2 = m - 1; dlascl_("G", &c__0, &c__0, &orgnrm, &c_b19, &i__1, &c__1, &e[ start], &i__2, info); zlaed0_(n, &m, &d__[start], &e[start], &z___ref(1, start), ldz, & work[1], n, &rwork[1], &iwork[1], info); if (*info > 0) { *info = (*info / (m + 1) + start - 1) * (*n + 1) + *info % (m + 1) + start - 1; return 0; } /* Scale back. */ dlascl_("G", &c__0, &c__0, &c_b19, &orgnrm, &m, &c__1, &d__[start] , &m, info); } else { dsteqr_("I", &m, &d__[start], &e[start], &rwork[1], &m, &rwork[m * m + 1], info); zlacrm_(n, &m, &z___ref(1, start), ldz, &rwork[1], &m, &work[1], n, &rwork[m * m + 1]); zlacpy_("A", n, &m, &work[1], n, &z___ref(1, start), ldz); if (*info > 0) { *info = start * (*n + 1) + end; return 0; } } start = end + 1; goto L30; } /* endwhile If the problem split any number of times, then the eigenvalues will not be properly ordered. Here we permute the eigenvalues (and the associated eigenvectors) into ascending order. */ if (m != *n) { /* Use Selection Sort to minimize swaps of eigenvectors */ i__1 = *n; for (ii = 2; ii <= i__1; ++ii) { i__ = ii - 1; k = i__; p = d__[i__]; i__2 = *n; for (j = ii; j <= i__2; ++j) { if (d__[j] < p) { k = j; p = d__[j]; } /* L50: */ } if (k != i__) { d__[k] = d__[i__]; d__[i__] = p; zswap_(n, &z___ref(1, i__), &c__1, &z___ref(1, k), &c__1); } /* L60: */ } } work[1].r = (doublereal) lwmin, work[1].i = 0.; rwork[1] = (doublereal) lrwmin; iwork[1] = liwmin; return 0; /* End of ZSTEDC */ } /* zstedc_ */ #undef z___ref #undef z___subscr