| QT {mgcv} | R Documentation |
If A is an r by c matrix, where r<c, then it can be factorised: AQ=[0,T] , where Q is an orthogonal matrix and T is reverse lower triangular. The first c-r columns of Q from an orthogonal basis for the null space of A. Q is returned as a sequence of r Householder rotations (from the right), as described below.
QT(A)
A |
is an R matrix having more columns than rows |
This function is primarily useful for providing the null space of the
linear constraint matrix C, from the linear constraints
Cp=0 , as a series of Householder rotations of the form
used internally by mgcv(). It does not need to be called to set up a problem
for solution by mgcv.
The returned matrix R, say, is of the same dimension as the input matrix A. Each of its rows contains a vector, u_i , defining one Householder rotation, H_i = (I - u_i u_i') . The orthogonal matrix Q is defined by: Q=H_1 H_2 ... H_r .
Simon N. Wood simon@stats.gla.ac.uk
Gill, Murray and Wright (1981) Practical Optimization, Academic Press
set.seed(0) y<-rnorm(12,1,1) A<-matrix(y,2,6) B<-QT(A)