RDODT and RUPDT Calls
downdate and update QR and Cholesky decompositions
 CALL RDODT( def, rup, bup, sup, r, z<, b, y<,
ssq>>);
 CALL RUPDT( rup, bup, sup, r, z<, b, y<,
ssq>>);
The RDODT and RUPDT subroutines return the values:
 def
 is only used for downdating, and it specifies
whether the downdating of matrix R by using
the q rows in argument z has been successful.
The result def=2 means that the downdating
of R by at least one row of Z leads to a
singular matrix and cannot be completed successfully
(since the result of downdating is not unique).
In that case, the results rup, bup,
and sup contain missing values only.
The result def=1 means that the residual sum of
squares, ssq, could not be downdated successfully
and the result sup contains missing values only.
The result def=0 means that the downdating
of R by Z was completed successfully.
 rup
 is the n ×n upper triangular matrix R that has
been updated or downdated by using the q rows in Z.
 bup
 is the n ×p matrix B of righthand sides that has
been updated or downdated by using the q rows in argument y.
If the argument b is not specified, bup is not computed.
 sup
 is a p vector of square roots of residual sum of squares that
is updated or downdated by using the q rows of argument y.
If ssq is not specified, sup is not computed.
The inputs to the RDODT and RUPDT subroutines are as follows:
 r
 specifies an n ×n upper triangular matrix R
to be updated or downdated by the q rows in Z.
Only the upper triangle of R is used; the
lower triangle can contain any information.
 z
 specifies a q ×n matrix Z used
rowwise to update or downdate the matrix R.
 b
 specifies an optional n ×p matrix
B of righthand sides that have to be
updated or downdated simultaneously with R.
If b is specified, the argument y must also be specified.
 y
 specifies an optional q ×p matrix Y used rowwise
to update or downdate the righthand side matrix B.
If b is specified, the argument y must also be specified.
 ssq
 is an optional p vector that, if b is specified, specifies
the square root of the error sum of squares that should be
updated or downdated simultaneously with R and B.
The upper triangular matrix R of the QR
decomposition of an m ×n matrix A,

A = QR, where Q' Q = QQ' = I_{m}
is recomputed efficiently in two cases:
 update: An n vector z is added to matrix A.
 downdate: An n vector
z is deleted from matrix A.
Computing the whole QR decomposition of matrix A by
Householder transformations requires 4mn^{2}  4n^{3}/3
floating point operations, whereas updating or downdating
the QR decomposition (by Givens rotations) of one row
vector z requires only 2n^{2} floating point operations.
If the QR decomposition is used to solve
the full rank linear leastsquares problem

min_{x}  Ax b^{2} = ssq
by solving the nonsingular upper triangular system

x = R^{1} Q' b
then the RUPDT and RDODT subroutines can be used to update or
downdate the ptransformed righthand sides Q' B and the residual sumofsquares p vector ssq
provided that for each n vector z added to or deleted
from A there is also a p vector y added to or
deleted from the m ×p righthandside matrix B.
If the arguments z and y of the subroutines RUPDT and
RDODT contain q > 1 row vectors for which R
(and Q' B, and eventually ssq) is
to be updated or downdated, the process is performed
stepwise by processing the rows z_{k} (and y_{k}),
k = 1, ... ,q, in the order in which they are stored.
The QR decomposition of an m ×n
matrix A, , rank(A) = n,

A = QR, where Q' Q = QQ' = I_{m}
corresponds to the Cholesky factorization

C = R' R, where C = A' A
of the positive definite n ×n
crossproduct matrix C = A' A.
In the case where and rank(A) = n, the upper
triangular matrix R computed by the QR decomposition
(with positive diagonal elements) is the same as the one
computed by Cholesky factorization except for numerical error,

A' A = (QR)' (QR) = R' R
Adding a row vector z to matrix A corresponds to
the rank1 modification of the crossproduct matrix C
and the (m+1) ×n matrix
contains all rows of A with the row z added.
Deleting a row vector z from matrix
A corresponds to the rank1 modification

C^{*} = C z' z, where C^{*} = A^{*'} A^{*}
and the (m1) ×n matrix A^{*} contains
all rows of A with the row z deleted.
Thus, you can also use the subroutines RUPDT and
RDODT to update or downdate the Cholesky factor R of a
positive definite crossproduct matrix C of A.
The process of downdating an upper triangular matrix
R (and eventually a residual sumofsquares
vector ssq) is not always successful.
First of all, the downdated matrix R could be rank deficient.
Even if the downdated matrix R is of full rank,
the process of downdating can be ill conditioned
and does not work well if the downdated matrix is
close (by rounding errors) to a rankdeficient one.
In these cases, the downdated matrix R is not
unique and cannot be computed by subroutine RDODT.
If R cannot be computed, def returns 2, and the results
rup, bup, and sup return missing values.
The downdating of the residual sumofsquares
vector ssq can be a problem, too.
In practice, the downdate formula
cannot always be computed because, due to
rounding errors, the radicand can be negative.
In this case, the result vector sup
returns missing values, and def returns 1.
You can use various methods to compute the p columns x_{k}
of the n ×p matrix X that minimize the p linear
leastsquares problems with an m ×n coefficient matrix
A, , rank(A) = n, and p righthandside vectors
b_{k} (stored columnwise in the m ×p matrix B).
The first of the following methods solves the
normal equations and cannot be applied to
the example with the 6 ×5 Hilbert matrix
since too much rounding error is introduced.
Therefore, use the following simple example:
proc iml;
a = { 1 3 ,
2 2 ,
3 1 };
b = { 1, 1, 1};
m = nrow(a);
n = ncol(a);
p = 1;
 Cholesky Decomposition of Crossproduct Matrix:
aa = a` * a; ab = a` * b;
r = root(aa);
x = trisolv(2,r,ab);
x = trisolv(1,r,x);
 QR Decomposition by Householder Transformations:
call qr(qtb,r,piv,lindep,a, ,b);
x = trisolv(1,r[,piv],qtb[1:n,]);
 Stepwise Update by Givens Rotations:
r = j(n,n,0.); qtb = j(n,p,0.); ssq = j(1,p,0.);
do i = 1 to m;
z = a[i,];
y = b[i,];
call rupdt(rup,bup,sup,r,z,qtb,y,ssq);
r = rup;
qtb = bup;
ssq = sup;
end;
x = trisolv(1,r,qtb);
Or equivalently:
r = j(n,n,0.);
qtb = j(n,p,0.);
ssq = j(1,p,0.);
call rupdt(rup,bup,sup,r,a,qtb,b,ssq);
x = trisolv(1,rup,bup);
 Singular Value Decomposition:
call svd(u,d,v,a);
d = diag(1 / d);
x = v * d * u` * b;
For the preceding 3 ×2 example matrix A,
each method obtains the unique LS estimator:
ss = ssq(a * x  b);
print ss x;
To compute the (transposed) matrix Q,
you can use the following specification:
r = shape(0,n,n);
y = i(m);
qt = shape(0,n,m);
call rupdt(rup,qtup,sup,r,a,qt,y);
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.