LUPDT Call :: SAS/IML(R) 13.1 User's Guide

LUPDT Call

CALL LUPDT (lup, bup, sup, L, z <*>, b <*>, y <*>, ssq ) ;

The LUPDT subroutine provides updating and downdating for rank deficient linear least squares solutions, complete orthogonal factorization, and Moore-Penrose inverses.

The LUPDT subroutine returns the following values:

lup: is an $n \times n$ lower triangular matrix $L$ that is updated or downdated by using the $q$ rows in $Z$ .
bup: is an $n \times p$ matrix $B$ of right-hand sides that is updated or downdated by using the $q$ rows in $Y$ . If b is not specified, bup is not accessible.
sup: is a $p$ vector of square roots of residual sum of squares that is updated or downdated by using the $q$ rows in $Y$ . If ssq is not specified, sup is not accessible.

The input arguments to the LUPDT subroutine are as follows:

L: specifies an $n \times n$ lower triangular matrix $\mathbf{L}$ to be updated or downdated by $q$ row vectors $z$ stored in the $q \times n$ matrix $Z$ . Only the lower triangle of L is used; the upper triangle can contain any information.
z: is a $q \times n$ matrix $Z$ used rowwise to update or downdate the matrix $L$ .
b: specifies an optional $n \times p$ matrix $\mathbf{B}$ of right-hand sides that have to be updated or downdated simultaneously with L. If b is specified, the argument y must be specified.
y: specifies an optional $q \times p$ matrix $Y$ used rowwise to update or downdate the right-hand-side matrix b.
ssq: specifies an optional $p \times 1$ vector that, if b is specified, specifies the square root of the error sum of squares that should be updated or downdated simultaneously with L and b.

The relevant formula for the LUPDT call is $\mb {\tilde{L}\tilde{L}^{\prime } = LL^{\prime } + ZZ^{\prime }}$ . See the section Complete QR Decomposition with LUPDT in the documentation for the RZLIND call.