`qrdelete`

Updates a QR factorization after a column or row is removed from the matrix.

Syntax

[Q1, R1] = qrdelete(Q, R, j)

Returns the factorization of A with column j deleted, where A = Q*R.

[Q1, R1] = qrdelete(Q, R, j, 'row')

Returns the factorization of A with row j deleted.

Description

Recomputing a factorization from scratch costs on the order of n^3 operations. Deleting one column only disturbs the triangular shape of R near that column, so Givens rotations restore it at a much lower cost.

This matters in algorithms that add and remove variables repeatedly, such as active-set methods and stepwise regression.

Input arguments

Q, R: Factors of a QR factorization A = Q*R.
j: Index of the column (default) or row to delete.

Output arguments

Q1, R1: Factors of the reduced matrix.

Example

Delete the second column

A = magic(3);
[Q, R] = qr(A);
[Q1, R1] = qrdelete(Q, R, 2);
norm(Q1*R1 - A(:, [1 3])) < 1e-10

Output

ans = logical
   1

qrdelete