`qrinsert`

Updates a QR factorization after a column or row is inserted into the matrix.

Syntax

[Q1, R1] = qrinsert(Q, R, j, x)

Returns the factorization of A with the column x inserted before column j.

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

Inserts the row x before row j.

Description

The new column is transformed by Q' and spliced into R; Givens rotations then remove the unwanted entries below the diagonal. The cost is much lower than refactoring the whole matrix.

It is the inverse operation of qrdelete: deleting the column just inserted gives back the original factors, up to signs.

Input arguments

Q, R: Factors of a QR factorization A = Q*R.
j: Position at which the new column or row goes.
x: The column (or row) vector to insert, with the matching length.

Output arguments

Q1, R1: Factors of the enlarged matrix.

Example

Insert a column

A = magic(3);
[Q, R] = qr(A);
x = [1; 2; 3];
[Q1, R1] = qrinsert(Q, R, 2, x);
A1 = [A(:,1) x A(:,2:3)];
norm(Q1*R1 - A1) < 1e-10

Output

ans = logical
   1

qrinsert