SCHEX(3F) SCHEX(3F)
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(LK)*...*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,...,K1,L,K,K+1,...,L1,L+1,...,P.
U is the product of LK rotations U(I), where U(I)
acts in the (LI,LI+1)plane.
2. Left circular shift (JOB = 2).
The columns are rearranged in the following order
1,...,K1,K+1,K+2,...,L,K,L+1,...,P.
U is the product of LK rotations U(I), where U(I)
acts in the (K+I1,K+I)plane.
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SCHEX(3F) SCHEX(3F)
SUBROUTINE SCHEX(R,LDR,P,K,L,Z,LDZ,NZ,C,S,JOB)
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 Pvectors 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
PPPPaaaaggggeeee 3333 [ Back ]
