`qr`

QR factorization: an orthogonal factor Q and an upper-triangular factor R.

Syntax

R = qr(A)

Returns only the triangular factor.

[Q, R] = qr(A)

Returns Q and R with A = Q*R; for an m-by-n matrix Q is m-by-m.

[Q, R] = qr(A, 0)

Economy size: for m > n, Q is m-by-n and R is n-by-n.

[Q, R, P] = qr(A)

Adds column pivoting, so that A*P = Q*R and the diagonal of R is non-increasing in magnitude.

Description

The factorization is computed with Householder reflections, which is numerically stable. The signs of the columns of Q and the rows of R are a convention, so they may differ from other software even though the product is the same.

QR is the standard route to least-squares problems: for an overdetermined system, solve the triangular system R*x = Q'*b. Column pivoting additionally reveals the numerical rank.

Input arguments

A: Numeric matrix of any shape.
0 or 'econ': Requests the economy-size factorization.

Output arguments

Q: Orthogonal (unitary, if A is complex) matrix.
R: Upper-triangular (trapezoidal) matrix.
P: Permutation matrix of the column pivoting.

Example

Factor and rebuild

A = [1 2; 3 4];
[Q, R] = qr(A)
norm(Q*R - A) < 1e-12

Output

Q = 2×2

   -0.3162   -0.9487
   -0.9487    0.3162
R = 2×2

   -3.1623   -4.4272
         0   -0.6325
ans = logical
   1

qr