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Open Source Computer Vision Library
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604 lines
14 KiB
604 lines
14 KiB
#include "clapack.h" |
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/* Table of constant values */ |
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static real c_b9 = 0.f; |
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static real c_b10 = 1.f; |
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static integer c__0 = 0; |
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static integer c__1 = 1; |
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static integer c__2 = 2; |
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/* Subroutine */ int ssteqr_(char *compz, integer *n, real *d__, real *e, |
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real *z__, integer *ldz, real *work, integer *info) |
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{ |
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/* System generated locals */ |
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integer z_dim1, z_offset, i__1, i__2; |
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real r__1, r__2; |
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/* Builtin functions */ |
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double sqrt(doublereal), r_sign(real *, real *); |
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/* Local variables */ |
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real b, c__, f, g; |
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integer i__, j, k, l, m; |
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real p, r__, s; |
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integer l1, ii, mm, lm1, mm1, nm1; |
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real rt1, rt2, eps; |
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integer lsv; |
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real tst, eps2; |
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integer lend, jtot; |
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extern /* Subroutine */ int slae2_(real *, real *, real *, real *, real *) |
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; |
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extern logical lsame_(char *, char *); |
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real anorm; |
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extern /* Subroutine */ int slasr_(char *, char *, char *, integer *, |
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integer *, real *, real *, real *, integer *), sswap_(integer *, real *, integer *, real *, integer *); |
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integer lendm1, lendp1; |
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extern /* Subroutine */ int slaev2_(real *, real *, real *, real *, real * |
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, real *, real *); |
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extern doublereal slapy2_(real *, real *); |
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integer iscale; |
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extern doublereal slamch_(char *); |
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real safmin; |
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extern /* Subroutine */ int xerbla_(char *, integer *); |
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real safmax; |
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extern /* Subroutine */ int slascl_(char *, integer *, integer *, real *, |
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real *, integer *, integer *, real *, integer *, integer *); |
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integer lendsv; |
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extern /* Subroutine */ int slartg_(real *, real *, real *, real *, real * |
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), slaset_(char *, integer *, integer *, real *, real *, real *, |
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integer *); |
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real ssfmin; |
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integer nmaxit, icompz; |
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real ssfmax; |
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extern doublereal slanst_(char *, integer *, real *, real *); |
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extern /* Subroutine */ int slasrt_(char *, integer *, real *, integer *); |
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/* -- LAPACK routine (version 3.1) -- */ |
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/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */ |
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/* November 2006 */ |
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/* .. Scalar Arguments .. */ |
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/* .. */ |
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/* .. Array Arguments .. */ |
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/* .. */ |
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/* Purpose */ |
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/* ======= */ |
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/* SSTEQR computes all eigenvalues and, optionally, eigenvectors of a */ |
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/* symmetric tridiagonal matrix using the implicit QL or QR method. */ |
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/* The eigenvectors of a full or band symmetric matrix can also be found */ |
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/* if SSYTRD or SSPTRD or SSBTRD has been used to reduce this matrix to */ |
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/* tridiagonal form. */ |
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/* Arguments */ |
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/* ========= */ |
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/* COMPZ (input) CHARACTER*1 */ |
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/* = 'N': Compute eigenvalues only. */ |
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/* = 'V': Compute eigenvalues and eigenvectors of the original */ |
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/* symmetric matrix. On entry, Z must contain the */ |
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/* orthogonal matrix used to reduce the original matrix */ |
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/* to tridiagonal form. */ |
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/* = 'I': Compute eigenvalues and eigenvectors of the */ |
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/* tridiagonal matrix. Z is initialized to the identity */ |
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/* matrix. */ |
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/* N (input) INTEGER */ |
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/* The order of the matrix. N >= 0. */ |
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/* D (input/output) REAL array, dimension (N) */ |
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/* On entry, the diagonal elements of the tridiagonal matrix. */ |
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/* On exit, if INFO = 0, the eigenvalues in ascending order. */ |
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/* E (input/output) REAL array, dimension (N-1) */ |
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/* On entry, the (n-1) subdiagonal elements of the tridiagonal */ |
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/* matrix. */ |
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/* On exit, E has been destroyed. */ |
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/* Z (input/output) REAL array, dimension (LDZ, N) */ |
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/* On entry, if COMPZ = 'V', then Z contains the orthogonal */ |
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/* matrix used in the reduction to tridiagonal form. */ |
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/* On exit, if INFO = 0, then if COMPZ = 'V', Z contains the */ |
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/* orthonormal eigenvectors of the original symmetric matrix, */ |
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/* and if COMPZ = 'I', Z contains the orthonormal eigenvectors */ |
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/* of the symmetric tridiagonal matrix. */ |
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/* If COMPZ = 'N', then Z is not referenced. */ |
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/* LDZ (input) INTEGER */ |
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/* The leading dimension of the array Z. LDZ >= 1, and if */ |
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/* eigenvectors are desired, then LDZ >= max(1,N). */ |
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/* WORK (workspace) REAL array, dimension (max(1,2*N-2)) */ |
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/* If COMPZ = 'N', then WORK is not referenced. */ |
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/* INFO (output) INTEGER */ |
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/* = 0: successful exit */ |
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/* < 0: if INFO = -i, the i-th argument had an illegal value */ |
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/* > 0: the algorithm has failed to find all the eigenvalues in */ |
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/* a total of 30*N iterations; if INFO = i, then i */ |
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/* elements of E have not converged to zero; on exit, D */ |
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/* and E contain the elements of a symmetric tridiagonal */ |
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/* matrix which is orthogonally similar to the original */ |
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/* matrix. */ |
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/* ===================================================================== */ |
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/* .. Parameters .. */ |
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/* .. */ |
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/* .. Local Scalars .. */ |
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/* .. */ |
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/* .. External Functions .. */ |
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/* .. */ |
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/* .. External Subroutines .. */ |
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/* .. */ |
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/* .. Intrinsic Functions .. */ |
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/* .. */ |
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/* .. Executable Statements .. */ |
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/* Test the input parameters. */ |
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/* Parameter adjustments */ |
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--d__; |
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--e; |
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z_dim1 = *ldz; |
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z_offset = 1 + z_dim1; |
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z__ -= z_offset; |
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--work; |
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/* Function Body */ |
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*info = 0; |
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if (lsame_(compz, "N")) { |
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icompz = 0; |
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} else if (lsame_(compz, "V")) { |
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icompz = 1; |
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} else if (lsame_(compz, "I")) { |
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icompz = 2; |
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} else { |
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icompz = -1; |
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} |
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if (icompz < 0) { |
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*info = -1; |
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} else if (*n < 0) { |
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*info = -2; |
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} else if (*ldz < 1 || icompz > 0 && *ldz < max(1,*n)) { |
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*info = -6; |
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} |
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if (*info != 0) { |
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i__1 = -(*info); |
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xerbla_("SSTEQR", &i__1); |
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return 0; |
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} |
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/* Quick return if possible */ |
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if (*n == 0) { |
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return 0; |
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} |
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if (*n == 1) { |
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if (icompz == 2) { |
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z__[z_dim1 + 1] = 1.f; |
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} |
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return 0; |
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} |
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/* Determine the unit roundoff and over/underflow thresholds. */ |
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eps = slamch_("E"); |
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/* Computing 2nd power */ |
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r__1 = eps; |
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eps2 = r__1 * r__1; |
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safmin = slamch_("S"); |
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safmax = 1.f / safmin; |
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ssfmax = sqrt(safmax) / 3.f; |
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ssfmin = sqrt(safmin) / eps2; |
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/* Compute the eigenvalues and eigenvectors of the tridiagonal */ |
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/* matrix. */ |
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if (icompz == 2) { |
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slaset_("Full", n, n, &c_b9, &c_b10, &z__[z_offset], ldz); |
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} |
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nmaxit = *n * 30; |
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jtot = 0; |
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/* Determine where the matrix splits and choose QL or QR iteration */ |
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/* for each block, according to whether top or bottom diagonal */ |
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/* element is smaller. */ |
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l1 = 1; |
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nm1 = *n - 1; |
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L10: |
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if (l1 > *n) { |
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goto L160; |
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} |
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if (l1 > 1) { |
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e[l1 - 1] = 0.f; |
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} |
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if (l1 <= nm1) { |
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i__1 = nm1; |
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for (m = l1; m <= i__1; ++m) { |
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tst = (r__1 = e[m], dabs(r__1)); |
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if (tst == 0.f) { |
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goto L30; |
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} |
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if (tst <= sqrt((r__1 = d__[m], dabs(r__1))) * sqrt((r__2 = d__[m |
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+ 1], dabs(r__2))) * eps) { |
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e[m] = 0.f; |
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goto L30; |
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} |
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/* L20: */ |
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} |
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} |
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m = *n; |
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L30: |
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l = l1; |
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lsv = l; |
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lend = m; |
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lendsv = lend; |
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l1 = m + 1; |
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if (lend == l) { |
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goto L10; |
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} |
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/* Scale submatrix in rows and columns L to LEND */ |
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i__1 = lend - l + 1; |
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anorm = slanst_("I", &i__1, &d__[l], &e[l]); |
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iscale = 0; |
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if (anorm == 0.f) { |
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goto L10; |
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} |
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if (anorm > ssfmax) { |
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iscale = 1; |
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i__1 = lend - l + 1; |
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slascl_("G", &c__0, &c__0, &anorm, &ssfmax, &i__1, &c__1, &d__[l], n, |
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info); |
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i__1 = lend - l; |
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slascl_("G", &c__0, &c__0, &anorm, &ssfmax, &i__1, &c__1, &e[l], n, |
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info); |
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} else if (anorm < ssfmin) { |
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iscale = 2; |
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i__1 = lend - l + 1; |
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slascl_("G", &c__0, &c__0, &anorm, &ssfmin, &i__1, &c__1, &d__[l], n, |
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info); |
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i__1 = lend - l; |
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slascl_("G", &c__0, &c__0, &anorm, &ssfmin, &i__1, &c__1, &e[l], n, |
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info); |
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} |
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/* Choose between QL and QR iteration */ |
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if ((r__1 = d__[lend], dabs(r__1)) < (r__2 = d__[l], dabs(r__2))) { |
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lend = lsv; |
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l = lendsv; |
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} |
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if (lend > l) { |
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/* QL Iteration */ |
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/* Look for small subdiagonal element. */ |
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L40: |
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if (l != lend) { |
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lendm1 = lend - 1; |
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i__1 = lendm1; |
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for (m = l; m <= i__1; ++m) { |
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/* Computing 2nd power */ |
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r__2 = (r__1 = e[m], dabs(r__1)); |
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tst = r__2 * r__2; |
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if (tst <= eps2 * (r__1 = d__[m], dabs(r__1)) * (r__2 = d__[m |
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+ 1], dabs(r__2)) + safmin) { |
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goto L60; |
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} |
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/* L50: */ |
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} |
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} |
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m = lend; |
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L60: |
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if (m < lend) { |
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e[m] = 0.f; |
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} |
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p = d__[l]; |
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if (m == l) { |
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goto L80; |
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} |
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/* If remaining matrix is 2-by-2, use SLAE2 or SLAEV2 */ |
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/* to compute its eigensystem. */ |
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if (m == l + 1) { |
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if (icompz > 0) { |
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slaev2_(&d__[l], &e[l], &d__[l + 1], &rt1, &rt2, &c__, &s); |
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work[l] = c__; |
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work[*n - 1 + l] = s; |
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slasr_("R", "V", "B", n, &c__2, &work[l], &work[*n - 1 + l], & |
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z__[l * z_dim1 + 1], ldz); |
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} else { |
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slae2_(&d__[l], &e[l], &d__[l + 1], &rt1, &rt2); |
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} |
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d__[l] = rt1; |
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d__[l + 1] = rt2; |
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e[l] = 0.f; |
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l += 2; |
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if (l <= lend) { |
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goto L40; |
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} |
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goto L140; |
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} |
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if (jtot == nmaxit) { |
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goto L140; |
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} |
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++jtot; |
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/* Form shift. */ |
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g = (d__[l + 1] - p) / (e[l] * 2.f); |
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r__ = slapy2_(&g, &c_b10); |
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g = d__[m] - p + e[l] / (g + r_sign(&r__, &g)); |
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s = 1.f; |
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c__ = 1.f; |
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p = 0.f; |
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/* Inner loop */ |
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mm1 = m - 1; |
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i__1 = l; |
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for (i__ = mm1; i__ >= i__1; --i__) { |
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f = s * e[i__]; |
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b = c__ * e[i__]; |
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slartg_(&g, &f, &c__, &s, &r__); |
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if (i__ != m - 1) { |
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e[i__ + 1] = r__; |
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} |
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g = d__[i__ + 1] - p; |
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r__ = (d__[i__] - g) * s + c__ * 2.f * b; |
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p = s * r__; |
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d__[i__ + 1] = g + p; |
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g = c__ * r__ - b; |
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/* If eigenvectors are desired, then save rotations. */ |
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if (icompz > 0) { |
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work[i__] = c__; |
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work[*n - 1 + i__] = -s; |
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} |
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/* L70: */ |
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} |
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/* If eigenvectors are desired, then apply saved rotations. */ |
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if (icompz > 0) { |
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mm = m - l + 1; |
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slasr_("R", "V", "B", n, &mm, &work[l], &work[*n - 1 + l], &z__[l |
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* z_dim1 + 1], ldz); |
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} |
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d__[l] -= p; |
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e[l] = g; |
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goto L40; |
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/* Eigenvalue found. */ |
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L80: |
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d__[l] = p; |
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++l; |
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if (l <= lend) { |
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goto L40; |
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} |
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goto L140; |
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} else { |
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/* QR Iteration */ |
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/* Look for small superdiagonal element. */ |
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L90: |
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if (l != lend) { |
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lendp1 = lend + 1; |
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i__1 = lendp1; |
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for (m = l; m >= i__1; --m) { |
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/* Computing 2nd power */ |
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r__2 = (r__1 = e[m - 1], dabs(r__1)); |
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tst = r__2 * r__2; |
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if (tst <= eps2 * (r__1 = d__[m], dabs(r__1)) * (r__2 = d__[m |
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- 1], dabs(r__2)) + safmin) { |
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goto L110; |
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} |
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/* L100: */ |
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} |
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} |
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m = lend; |
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L110: |
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if (m > lend) { |
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e[m - 1] = 0.f; |
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} |
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p = d__[l]; |
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if (m == l) { |
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goto L130; |
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} |
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/* If remaining matrix is 2-by-2, use SLAE2 or SLAEV2 */ |
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/* to compute its eigensystem. */ |
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if (m == l - 1) { |
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if (icompz > 0) { |
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slaev2_(&d__[l - 1], &e[l - 1], &d__[l], &rt1, &rt2, &c__, &s) |
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; |
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work[m] = c__; |
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work[*n - 1 + m] = s; |
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slasr_("R", "V", "F", n, &c__2, &work[m], &work[*n - 1 + m], & |
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z__[(l - 1) * z_dim1 + 1], ldz); |
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} else { |
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slae2_(&d__[l - 1], &e[l - 1], &d__[l], &rt1, &rt2); |
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} |
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d__[l - 1] = rt1; |
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d__[l] = rt2; |
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e[l - 1] = 0.f; |
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l += -2; |
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if (l >= lend) { |
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goto L90; |
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} |
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goto L140; |
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} |
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if (jtot == nmaxit) { |
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goto L140; |
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} |
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++jtot; |
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/* Form shift. */ |
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g = (d__[l - 1] - p) / (e[l - 1] * 2.f); |
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r__ = slapy2_(&g, &c_b10); |
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g = d__[m] - p + e[l - 1] / (g + r_sign(&r__, &g)); |
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s = 1.f; |
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c__ = 1.f; |
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p = 0.f; |
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/* Inner loop */ |
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lm1 = l - 1; |
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i__1 = lm1; |
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for (i__ = m; i__ <= i__1; ++i__) { |
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f = s * e[i__]; |
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b = c__ * e[i__]; |
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slartg_(&g, &f, &c__, &s, &r__); |
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if (i__ != m) { |
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e[i__ - 1] = r__; |
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} |
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g = d__[i__] - p; |
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r__ = (d__[i__ + 1] - g) * s + c__ * 2.f * b; |
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p = s * r__; |
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d__[i__] = g + p; |
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g = c__ * r__ - b; |
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/* If eigenvectors are desired, then save rotations. */ |
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if (icompz > 0) { |
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work[i__] = c__; |
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work[*n - 1 + i__] = s; |
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} |
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/* L120: */ |
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} |
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/* If eigenvectors are desired, then apply saved rotations. */ |
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if (icompz > 0) { |
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mm = l - m + 1; |
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slasr_("R", "V", "F", n, &mm, &work[m], &work[*n - 1 + m], &z__[m |
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* z_dim1 + 1], ldz); |
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} |
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d__[l] -= p; |
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e[lm1] = g; |
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goto L90; |
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/* Eigenvalue found. */ |
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L130: |
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d__[l] = p; |
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--l; |
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if (l >= lend) { |
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goto L90; |
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} |
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goto L140; |
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} |
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/* Undo scaling if necessary */ |
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L140: |
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if (iscale == 1) { |
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i__1 = lendsv - lsv + 1; |
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slascl_("G", &c__0, &c__0, &ssfmax, &anorm, &i__1, &c__1, &d__[lsv], |
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n, info); |
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i__1 = lendsv - lsv; |
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slascl_("G", &c__0, &c__0, &ssfmax, &anorm, &i__1, &c__1, &e[lsv], n, |
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info); |
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} else if (iscale == 2) { |
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i__1 = lendsv - lsv + 1; |
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slascl_("G", &c__0, &c__0, &ssfmin, &anorm, &i__1, &c__1, &d__[lsv], |
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n, info); |
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i__1 = lendsv - lsv; |
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slascl_("G", &c__0, &c__0, &ssfmin, &anorm, &i__1, &c__1, &e[lsv], n, |
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info); |
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} |
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/* Check for no convergence to an eigenvalue after a total */ |
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/* of N*MAXIT iterations. */ |
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if (jtot < nmaxit) { |
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goto L10; |
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} |
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i__1 = *n - 1; |
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for (i__ = 1; i__ <= i__1; ++i__) { |
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if (e[i__] != 0.f) { |
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++(*info); |
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} |
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/* L150: */ |
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} |
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goto L190; |
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/* Order eigenvalues and eigenvectors. */ |
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L160: |
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if (icompz == 0) { |
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/* Use Quick Sort */ |
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slasrt_("I", n, &d__[1], info); |
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} else { |
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/* Use Selection Sort to minimize swaps of eigenvectors */ |
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i__1 = *n; |
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for (ii = 2; ii <= i__1; ++ii) { |
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i__ = ii - 1; |
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k = i__; |
|
p = d__[i__]; |
|
i__2 = *n; |
|
for (j = ii; j <= i__2; ++j) { |
|
if (d__[j] < p) { |
|
k = j; |
|
p = d__[j]; |
|
} |
|
/* L170: */ |
|
} |
|
if (k != i__) { |
|
d__[k] = d__[i__]; |
|
d__[i__] = p; |
|
sswap_(n, &z__[i__ * z_dim1 + 1], &c__1, &z__[k * z_dim1 + 1], |
|
&c__1); |
|
} |
|
/* L180: */ |
|
} |
|
} |
|
|
|
L190: |
|
return 0; |
|
|
|
/* End of SSTEQR */ |
|
|
|
} /* ssteqr_ */
|
|
|