SLATEC Routines --- DCHEX ---
*DECK DCHEX
SUBROUTINE DCHEX (R, LDR, P, K, L, Z, LDZ, NZ, C, S, JOB)
C***BEGIN PROLOGUE DCHEX
C***PURPOSE Update the Cholesky factorization A=TRANS(R)*R of a
C positive definite matrix A of order P under diagonal
C permutations of the form TRANS(E)*A*E, where E is a
C permutation matrix.
C***LIBRARY SLATEC (LINPACK)
C***CATEGORY D7B
C***TYPE DOUBLE PRECISION (SCHEX-S, DCHEX-D, CCHEX-C)
C***KEYWORDS CHOLESKY DECOMPOSITION, EXCHANGE, LINEAR ALGEBRA, LINPACK,
C MATRIX, POSITIVE DEFINITE
C***AUTHOR Stewart, G. W., (U. of Maryland)
C***DESCRIPTION
C
C DCHEX updates the Cholesky factorization
C
C A = TRANS(R)*R
C
C of a positive definite matrix A of order P under diagonal
C permutations of the form
C
C TRANS(E)*A*E
C
C where E is a permutation matrix. Specifically, given
C an upper triangular matrix R and a permutation matrix
C E (which is specified by K, L, and JOB), DCHEX determines
C an orthogonal matrix U such that
C
C U*R*E = RR,
C
C where RR is upper triangular. At the users option, the
C transformation U will be multiplied into the array Z.
C If A = TRANS(X)*X, so that R is the triangular part of the
C QR factorization of X, then RR is the triangular part of the
C QR factorization of X*E, i.e. X with its columns permuted.
C For a less terse description of what DCHEX does and how
C it may be applied, see the LINPACK guide.
C
C The matrix Q is determined as the product U(L-K)*...*U(1)
C of plane rotations of the form
C
C ( C(I) S(I) )
C ( ) ,
C ( -S(I) C(I) )
C
C where C(I) is double precision. The rows these rotations operate
C on are described below.
C
C There are two types of permutations, which are determined
C by the value of JOB.
C
C 1. Right circular shift (JOB = 1).
C
C The columns are rearranged in the following order.
C
C 1,...,K-1,L,K,K+1,...,L-1,L+1,...,P.
C
C U is the product of L-K rotations U(I), where U(I)
C acts in the (L-I,L-I+1)-plane.
C
C 2. Left circular shift (JOB = 2).
C The columns are rearranged in the following order
C
C 1,...,K-1,K+1,K+2,...,L,K,L+1,...,P.
C
C U is the product of L-K rotations U(I), where U(I)
C acts in the (K+I-1,K+I)-plane.
C
C On Entry
C
C R DOUBLE PRECISION(LDR,P), where LDR .GE. P.
C R contains the upper triangular factor
C that is to be updated. Elements of R
C below the diagonal are not referenced.
C
C LDR INTEGER.
C LDR is the leading dimension of the array R.
C
C P INTEGER.
C P is the order of the matrix R.
C
C K INTEGER.
C K is the first column to be permuted.
C
C L INTEGER.
C L is the last column to be permuted.
C L must be strictly greater than K.
C
C Z DOUBLE PRECISION(LDZ,N)Z), where LDZ .GE. P.
C Z is an array of NZ P-vectors into which the
C transformation U is multiplied. Z is
C not referenced if NZ = 0.
C
C LDZ INTEGER.
C LDZ is the leading dimension of the array Z.
C
C NZ INTEGER.
C NZ is the number of columns of the matrix Z.
C
C JOB INTEGER.
C JOB determines the type of permutation.
C JOB = 1 right circular shift.
C JOB = 2 left circular shift.
C
C On Return
C
C R contains the updated factor.
C
C Z contains the updated matrix Z.
C
C C DOUBLE PRECISION(P).
C C contains the cosines of the transforming rotations.
C
C S DOUBLE PRECISION(P).
C S contains the sines of the transforming rotations.
C
C***REFERENCES J. J. Dongarra, J. R. Bunch, C. B. Moler, and G. W.
C Stewart, LINPACK Users' Guide, SIAM, 1979.
C***ROUTINES CALLED DROTG
C***REVISION HISTORY (YYMMDD)
C 780814 DATE WRITTEN
C 890531 Changed all specific intrinsics to generic. (WRB)
C 890831 Modified array declarations. (WRB)
C 890831 REVISION DATE from Version 3.2
C 891214 Prologue converted to Version 4.0 format. (BAB)
C 900326 Removed duplicate information from DESCRIPTION section.
C (WRB)
C 920501 Reformatted the REFERENCES section. (WRB)
C***END PROLOGUE DCHEX