578 lines
16 KiB
C
578 lines
16 KiB
C
#include "clapack.h"
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/* Subroutine */ int dsytf2_(char *uplo, integer *n, doublereal *a, integer *
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lda, integer *ipiv, integer *info)
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{
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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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Purpose
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=======
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DSYTF2 computes the factorization of a real symmetric matrix A using
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the Bunch-Kaufman diagonal pivoting method:
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A = U*D*U' or A = L*D*L'
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where U (or L) is a product of permutation and unit upper (lower)
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triangular matrices, U' is the transpose of U, and D is symmetric and
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block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
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This is the unblocked version of the algorithm, calling Level 2 BLAS.
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Arguments
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=========
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UPLO (input) CHARACTER*1
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Specifies whether the upper or lower triangular part of the
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symmetric matrix A is stored:
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= 'U': Upper triangular
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= 'L': Lower triangular
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N (input) INTEGER
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The order of the matrix A. N >= 0.
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A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
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On entry, the symmetric matrix A. If UPLO = 'U', the leading
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n-by-n upper triangular part of A contains the upper
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triangular part of the matrix A, and the strictly lower
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triangular part of A is not referenced. If UPLO = 'L', the
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leading n-by-n lower triangular part of A contains the lower
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triangular part of the matrix A, and the strictly upper
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triangular part of A is not referenced.
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On exit, the block diagonal matrix D and the multipliers used
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to obtain the factor U or L (see below for further details).
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LDA (input) INTEGER
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The leading dimension of the array A. LDA >= max(1,N).
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IPIV (output) INTEGER array, dimension (N)
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Details of the interchanges and the block structure of D.
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If IPIV(k) > 0, then rows and columns k and IPIV(k) were
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interchanged and D(k,k) is a 1-by-1 diagonal block.
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If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
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columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
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is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
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IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
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interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
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INFO (output) INTEGER
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= 0: successful exit
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< 0: if INFO = -k, the k-th argument had an illegal value
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> 0: if INFO = k, D(k,k) is exactly zero. The factorization
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has been completed, but the block diagonal matrix D is
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exactly singular, and division by zero will occur if it
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is used to solve a system of equations.
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Further Details
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===============
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09-29-06 - patch from
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Bobby Cheng, MathWorks
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Replace l.204 and l.372
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IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
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by
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IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN
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01-01-96 - Based on modifications by
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J. Lewis, Boeing Computer Services Company
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A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
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1-96 - Based on modifications by J. Lewis, Boeing Computer Services
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Company
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If UPLO = 'U', then A = U*D*U', where
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U = P(n)*U(n)* ... *P(k)U(k)* ...,
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i.e., U is a product of terms P(k)*U(k), where k decreases from n to
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1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
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and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
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defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
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that if the diagonal block D(k) is of order s (s = 1 or 2), then
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( I v 0 ) k-s
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U(k) = ( 0 I 0 ) s
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( 0 0 I ) n-k
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k-s s n-k
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If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
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If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
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and A(k,k), and v overwrites A(1:k-2,k-1:k).
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If UPLO = 'L', then A = L*D*L', where
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L = P(1)*L(1)* ... *P(k)*L(k)* ...,
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i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
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n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
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and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
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defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
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that if the diagonal block D(k) is of order s (s = 1 or 2), then
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( I 0 0 ) k-1
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L(k) = ( 0 I 0 ) s
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( 0 v I ) n-k-s+1
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k-1 s n-k-s+1
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If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
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If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
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and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
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=====================================================================
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Test the input parameters.
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Parameter adjustments */
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/* Table of constant values */
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static integer c__1 = 1;
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/* System generated locals */
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integer a_dim1, a_offset, i__1, i__2;
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doublereal d__1, d__2, d__3;
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/* Builtin functions */
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double sqrt(doublereal);
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/* Local variables */
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static integer i__, j, k;
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static doublereal t, r1, d11, d12, d21, d22;
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static integer kk, kp;
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static doublereal wk, wkm1, wkp1;
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static integer imax, jmax;
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extern /* Subroutine */ int dsyr_(char *, integer *, doublereal *,
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doublereal *, integer *, doublereal *, integer *);
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static doublereal alpha;
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extern /* Subroutine */ int dscal_(integer *, doublereal *, doublereal *,
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integer *);
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extern logical lsame_(char *, char *);
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extern /* Subroutine */ int dswap_(integer *, doublereal *, integer *,
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doublereal *, integer *);
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static integer kstep;
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static logical upper;
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static doublereal absakk;
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extern integer idamax_(integer *, doublereal *, integer *);
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extern logical disnan_(doublereal *);
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extern /* Subroutine */ int xerbla_(char *, integer *);
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static doublereal colmax, rowmax;
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a_dim1 = *lda;
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a_offset = 1 + a_dim1;
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a -= a_offset;
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--ipiv;
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/* Function Body */
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*info = 0;
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upper = lsame_(uplo, "U");
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if (! upper && ! lsame_(uplo, "L")) {
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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 (*lda < max(1,*n)) {
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*info = -4;
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}
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if (*info != 0) {
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i__1 = -(*info);
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xerbla_("DSYTF2", &i__1);
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return 0;
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}
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/* Initialize ALPHA for use in choosing pivot block size. */
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alpha = (sqrt(17.) + 1.) / 8.;
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if (upper) {
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/* Factorize A as U*D*U' using the upper triangle of A
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K is the main loop index, decreasing from N to 1 in steps of
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1 or 2 */
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k = *n;
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L10:
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/* If K < 1, exit from loop */
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if (k < 1) {
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goto L70;
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}
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kstep = 1;
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/* Determine rows and columns to be interchanged and whether
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a 1-by-1 or 2-by-2 pivot block will be used */
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absakk = (d__1 = a[k + k * a_dim1], abs(d__1));
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/* IMAX is the row-index of the largest off-diagonal element in
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column K, and COLMAX is its absolute value */
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if (k > 1) {
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i__1 = k - 1;
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imax = idamax_(&i__1, &a[k * a_dim1 + 1], &c__1);
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colmax = (d__1 = a[imax + k * a_dim1], abs(d__1));
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} else {
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colmax = 0.;
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}
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if (max(absakk,colmax) == 0. || disnan_(&absakk)) {
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/* Column K is zero or contains a NaN: set INFO and continue */
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if (*info == 0) {
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*info = k;
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}
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kp = k;
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} else {
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if (absakk >= alpha * colmax) {
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/* no interchange, use 1-by-1 pivot block */
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kp = k;
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} else {
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/* JMAX is the column-index of the largest off-diagonal
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element in row IMAX, and ROWMAX is its absolute value */
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i__1 = k - imax;
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jmax = imax + idamax_(&i__1, &a[imax + (imax + 1) * a_dim1],
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lda);
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rowmax = (d__1 = a[imax + jmax * a_dim1], abs(d__1));
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if (imax > 1) {
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i__1 = imax - 1;
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jmax = idamax_(&i__1, &a[imax * a_dim1 + 1], &c__1);
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/* Computing MAX */
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d__2 = rowmax, d__3 = (d__1 = a[jmax + imax * a_dim1],
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abs(d__1));
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rowmax = max(d__2,d__3);
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}
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if (absakk >= alpha * colmax * (colmax / rowmax)) {
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/* no interchange, use 1-by-1 pivot block */
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kp = k;
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} else if ((d__1 = a[imax + imax * a_dim1], abs(d__1)) >=
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alpha * rowmax) {
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/* interchange rows and columns K and IMAX, use 1-by-1
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pivot block */
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kp = imax;
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} else {
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/* interchange rows and columns K-1 and IMAX, use 2-by-2
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pivot block */
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kp = imax;
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kstep = 2;
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}
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}
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kk = k - kstep + 1;
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if (kp != kk) {
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/* Interchange rows and columns KK and KP in the leading
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submatrix A(1:k,1:k) */
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i__1 = kp - 1;
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dswap_(&i__1, &a[kk * a_dim1 + 1], &c__1, &a[kp * a_dim1 + 1],
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&c__1);
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i__1 = kk - kp - 1;
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dswap_(&i__1, &a[kp + 1 + kk * a_dim1], &c__1, &a[kp + (kp +
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1) * a_dim1], lda);
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t = a[kk + kk * a_dim1];
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a[kk + kk * a_dim1] = a[kp + kp * a_dim1];
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a[kp + kp * a_dim1] = t;
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if (kstep == 2) {
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t = a[k - 1 + k * a_dim1];
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a[k - 1 + k * a_dim1] = a[kp + k * a_dim1];
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a[kp + k * a_dim1] = t;
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}
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}
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/* Update the leading submatrix */
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if (kstep == 1) {
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/* 1-by-1 pivot block D(k): column k now holds
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W(k) = U(k)*D(k)
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where U(k) is the k-th column of U
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Perform a rank-1 update of A(1:k-1,1:k-1) as
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A := A - U(k)*D(k)*U(k)' = A - W(k)*1/D(k)*W(k)' */
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r1 = 1. / a[k + k * a_dim1];
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i__1 = k - 1;
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d__1 = -r1;
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dsyr_(uplo, &i__1, &d__1, &a[k * a_dim1 + 1], &c__1, &a[
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a_offset], lda);
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/* Store U(k) in column k */
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i__1 = k - 1;
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dscal_(&i__1, &r1, &a[k * a_dim1 + 1], &c__1);
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} else {
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/* 2-by-2 pivot block D(k): columns k and k-1 now hold
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( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
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where U(k) and U(k-1) are the k-th and (k-1)-th columns
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of U
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Perform a rank-2 update of A(1:k-2,1:k-2) as
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A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )'
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= A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )' */
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if (k > 2) {
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d12 = a[k - 1 + k * a_dim1];
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d22 = a[k - 1 + (k - 1) * a_dim1] / d12;
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d11 = a[k + k * a_dim1] / d12;
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t = 1. / (d11 * d22 - 1.);
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d12 = t / d12;
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for (j = k - 2; j >= 1; --j) {
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wkm1 = d12 * (d11 * a[j + (k - 1) * a_dim1] - a[j + k
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* a_dim1]);
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wk = d12 * (d22 * a[j + k * a_dim1] - a[j + (k - 1) *
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a_dim1]);
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for (i__ = j; i__ >= 1; --i__) {
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a[i__ + j * a_dim1] = a[i__ + j * a_dim1] - a[i__
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+ k * a_dim1] * wk - a[i__ + (k - 1) *
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a_dim1] * wkm1;
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/* L20: */
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}
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a[j + k * a_dim1] = wk;
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a[j + (k - 1) * a_dim1] = wkm1;
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/* L30: */
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}
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}
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}
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}
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/* Store details of the interchanges in IPIV */
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if (kstep == 1) {
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ipiv[k] = kp;
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} else {
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ipiv[k] = -kp;
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ipiv[k - 1] = -kp;
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}
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/* Decrease K and return to the start of the main loop */
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k -= kstep;
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goto L10;
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} else {
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/* Factorize A as L*D*L' using the lower triangle of A
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K is the main loop index, increasing from 1 to N in steps of
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1 or 2 */
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k = 1;
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L40:
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/* If K > N, exit from loop */
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if (k > *n) {
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goto L70;
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}
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kstep = 1;
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/* Determine rows and columns to be interchanged and whether
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a 1-by-1 or 2-by-2 pivot block will be used */
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absakk = (d__1 = a[k + k * a_dim1], abs(d__1));
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/* IMAX is the row-index of the largest off-diagonal element in
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column K, and COLMAX is its absolute value */
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if (k < *n) {
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i__1 = *n - k;
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imax = k + idamax_(&i__1, &a[k + 1 + k * a_dim1], &c__1);
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colmax = (d__1 = a[imax + k * a_dim1], abs(d__1));
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} else {
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colmax = 0.;
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}
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if (max(absakk,colmax) == 0. || disnan_(&absakk)) {
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/* Column K is zero or contains a NaN: set INFO and continue */
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if (*info == 0) {
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*info = k;
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}
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kp = k;
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} else {
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if (absakk >= alpha * colmax) {
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/* no interchange, use 1-by-1 pivot block */
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kp = k;
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} else {
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/* JMAX is the column-index of the largest off-diagonal
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element in row IMAX, and ROWMAX is its absolute value */
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i__1 = imax - k;
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jmax = k - 1 + idamax_(&i__1, &a[imax + k * a_dim1], lda);
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rowmax = (d__1 = a[imax + jmax * a_dim1], abs(d__1));
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if (imax < *n) {
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i__1 = *n - imax;
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jmax = imax + idamax_(&i__1, &a[imax + 1 + imax * a_dim1],
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&c__1);
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/* Computing MAX */
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d__2 = rowmax, d__3 = (d__1 = a[jmax + imax * a_dim1],
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abs(d__1));
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rowmax = max(d__2,d__3);
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}
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if (absakk >= alpha * colmax * (colmax / rowmax)) {
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/* no interchange, use 1-by-1 pivot block */
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kp = k;
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} else if ((d__1 = a[imax + imax * a_dim1], abs(d__1)) >=
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alpha * rowmax) {
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/* interchange rows and columns K and IMAX, use 1-by-1
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pivot block */
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kp = imax;
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} else {
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/* interchange rows and columns K+1 and IMAX, use 2-by-2
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pivot block */
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kp = imax;
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kstep = 2;
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}
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}
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kk = k + kstep - 1;
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if (kp != kk) {
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/* Interchange rows and columns KK and KP in the trailing
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submatrix A(k:n,k:n) */
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if (kp < *n) {
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i__1 = *n - kp;
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dswap_(&i__1, &a[kp + 1 + kk * a_dim1], &c__1, &a[kp + 1
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+ kp * a_dim1], &c__1);
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}
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i__1 = kp - kk - 1;
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dswap_(&i__1, &a[kk + 1 + kk * a_dim1], &c__1, &a[kp + (kk +
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1) * a_dim1], lda);
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t = a[kk + kk * a_dim1];
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a[kk + kk * a_dim1] = a[kp + kp * a_dim1];
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a[kp + kp * a_dim1] = t;
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if (kstep == 2) {
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t = a[k + 1 + k * a_dim1];
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a[k + 1 + k * a_dim1] = a[kp + k * a_dim1];
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a[kp + k * a_dim1] = t;
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}
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}
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/* Update the trailing submatrix */
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if (kstep == 1) {
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|
/* 1-by-1 pivot block D(k): column k now holds
|
|
|
|
W(k) = L(k)*D(k)
|
|
|
|
where L(k) is the k-th column of L */
|
|
|
|
if (k < *n) {
|
|
|
|
/* Perform a rank-1 update of A(k+1:n,k+1:n) as
|
|
|
|
A := A - L(k)*D(k)*L(k)' = A - W(k)*(1/D(k))*W(k)' */
|
|
|
|
d11 = 1. / a[k + k * a_dim1];
|
|
i__1 = *n - k;
|
|
d__1 = -d11;
|
|
dsyr_(uplo, &i__1, &d__1, &a[k + 1 + k * a_dim1], &c__1, &
|
|
a[k + 1 + (k + 1) * a_dim1], lda);
|
|
|
|
/* Store L(k) in column K */
|
|
|
|
i__1 = *n - k;
|
|
dscal_(&i__1, &d11, &a[k + 1 + k * a_dim1], &c__1);
|
|
}
|
|
} else {
|
|
|
|
/* 2-by-2 pivot block D(k) */
|
|
|
|
if (k < *n - 1) {
|
|
|
|
/* Perform a rank-2 update of A(k+2:n,k+2:n) as
|
|
|
|
A := A - ( (A(k) A(k+1))*D(k)**(-1) ) * (A(k) A(k+1))'
|
|
|
|
where L(k) and L(k+1) are the k-th and (k+1)-th
|
|
columns of L */
|
|
|
|
d21 = a[k + 1 + k * a_dim1];
|
|
d11 = a[k + 1 + (k + 1) * a_dim1] / d21;
|
|
d22 = a[k + k * a_dim1] / d21;
|
|
t = 1. / (d11 * d22 - 1.);
|
|
d21 = t / d21;
|
|
|
|
i__1 = *n;
|
|
for (j = k + 2; j <= i__1; ++j) {
|
|
|
|
wk = d21 * (d11 * a[j + k * a_dim1] - a[j + (k + 1) *
|
|
a_dim1]);
|
|
wkp1 = d21 * (d22 * a[j + (k + 1) * a_dim1] - a[j + k
|
|
* a_dim1]);
|
|
|
|
i__2 = *n;
|
|
for (i__ = j; i__ <= i__2; ++i__) {
|
|
a[i__ + j * a_dim1] = a[i__ + j * a_dim1] - a[i__
|
|
+ k * a_dim1] * wk - a[i__ + (k + 1) *
|
|
a_dim1] * wkp1;
|
|
/* L50: */
|
|
}
|
|
|
|
a[j + k * a_dim1] = wk;
|
|
a[j + (k + 1) * a_dim1] = wkp1;
|
|
|
|
/* L60: */
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
/* Store details of the interchanges in IPIV */
|
|
|
|
if (kstep == 1) {
|
|
ipiv[k] = kp;
|
|
} else {
|
|
ipiv[k] = -kp;
|
|
ipiv[k + 1] = -kp;
|
|
}
|
|
|
|
/* Increase K and return to the start of the main loop */
|
|
|
|
k += kstep;
|
|
goto L40;
|
|
|
|
}
|
|
|
|
L70:
|
|
|
|
return 0;
|
|
|
|
/* End of DSYTF2 */
|
|
|
|
} /* dsytf2_ */
|