SCHEX(3F)							     SCHEX(3F)

NAME    [Toc]    [Back]

     SCHEX   - SCHEX updates the Cholesky factorization

	A = TRANS(R)*R

     of	a positive definite matrix A of	order P	under diagonal permutations of
     the form

	TRANS(E)*A*E

     where E is	a permutation matrix.  Specifically, given an upper triangular
     matrix R and a permutation	matrix E (which	is specified by	K, L, and
     JOB), SCHEX determines an orthogonal matrix U such	that

	U*R*E =	RR,

     where RR is upper triangular.  At the users option, the transformation U
     will be multiplied	into the array Z.  If A	= TRANS(X)*X, so that R	is the
     triangular	part of	the QR factorization of	X, then	RR is the triangular
     part of the QR factorization of X*E, i.e.,	X with its columns permuted.
     For a less	terse description of what SCHEX	does and how it	may be
     applied, see the LINPACK guide.

     The matrix	Q is determined	as the product U(L-K)*...*U(1) of plane
     rotations of the form

	(    C(I)	S(I) )
	(		     ) ,
	(    -S(I)	C(I) )

     where C(I)	is real.  The rows these rotations operate on are described
     below.

     There are two types of permutations, which	are determined by the value of
     JOB.

     1.	Right circular shift (JOB = 1).

	The columns are	rearranged in the following order.

	1,...,K-1,L,K,K+1,...,L-1,L+1,...,P.

	U is the product of L-K	rotations U(I),	where U(I)
	acts in	the (L-I,L-I+1)-plane.

     2.	Left circular shift (JOB = 2).
	The columns are	rearranged in the following order

	1,...,K-1,K+1,K+2,...,L,K,L+1,...,P.

	U is the product of L-K	rotations U(I),	where U(I)
	acts in	the (K+I-1,K+I)-plane.



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SCHEX(3F)							     SCHEX(3F)

SYNOPSYS    [Toc]    [Back]

      SUBROUTINE SCHEX(R,LDR,P,K,L,Z,LDZ,NZ,C,S,JOB)

DESCRIPTION    [Toc]    [Back]

     On	Entry

     R REAL(LDR,P), where LDR .GE. P.
	R contains the upper triangular	factor
	that is	to be updated.	Elements of R
	below the diagonal are not referenced.

     LDR INTEGER.
	LDR is the leading dimension of	the array R.

     P INTEGER.
	P is the order of the matrix R.

     K INTEGER.
	K is the first column to be permuted.

     L INTEGER.
	L is the last column to	be permuted.
	L must be strictly greater than	K.

     Z REAL(LDZ,NZ), where LDZ.GE.P.
	Z is an	array of NZ P-vectors into which the
	transformation U is multiplied.	 Z is
	not referenced if NZ = 0.

     LDZ INTEGER.
	LDZ is the leading dimension of	the array Z.

     NZ	INTEGER.
	NZ is the number of columns of the matrix Z.

     JOB INTEGER.
	JOB determines the type	of permutation.
	JOB = 1	 right circular	shift.
	JOB = 2	 left circular shift.  On Return

     R contains	the updated factor.

     Z contains	the updated matrix Z.

     C REAL(P).
	C contains the cosines of the transforming rotations.

     S REAL(P).
	S contains the sines of	the transforming rotations.  LINPACK.  This
     version dated 08/14/78 .  G. W. Stewart, University of Maryland, Argonne
     National Lab.




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SCHEX(3F)							     SCHEX(3F)



     SCHEX uses	the following functions	and subroutines. BLAS SROTG Fortran
     MIN0


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