qr

Purpose

Orthogonal-triangular decomposition.

Synopsis

[Q,R] = qr(X)
[Q,R,E] = qr(X)
A = qr(X)

Description

qr performs the orthogonal-triangular decomposition of a matrix. This factorization is useful for both square and rectangular matrices. It expresses the matrix as the product of a real orthonormal or complex unitary matrix and an upper triangular matrix.

[Q,R] = qr(X) produces an upper triangular matrix R of the same dimension as X and a unitary matrix Q so that X = Q*R.

[Q,R,E] = qr(X) produces a permutation matrix E, an upper triangular matrix R with decreasing diagonal elements, and a unitary matrix Q so that X*E = Q*R.

A = qr(X) returns the output of the LINPACK subroutine ZQRDC. triu(qr(X)) is R.

Examples

Start with

A =
     1     2     3
     4     5     6
     7     8     9
    10    11    12
This is a rank-deficient matrix; the middle column is the average of the other two columns. The rank deficiency is revealed by the factorization:

[Q,R] = qr(A)
          
Q =
    -0.0776    -0.8331     0.5444     0.0605
    -0.3105    -0.4512    -0.7709     0.3251
    -0.5433    -0.0694    -0.0913    -0.8317
    -0.7762     0.3124     0.3178     0.4461
          
R =
    -12.8841    -14.5916    -16.2992
           0     -1.0413     -2.0826
           0           0      0.0000
           0           0           0
The triangular structure of R gives it zeros below the diagonal; the zero on the diagonal in R(3,3) implies that R, and consequently A, does not have full rank.

The QR factorization is used to solve linear systems with more equations than unknowns. For example

b =
    1
    3
    5
    7
The linear system Ax = b represents four equations in only three unknowns. The best solution in a least squares sense is computed by

x = A\b
which produces

Warning: Rank deficient, rank = 2, tol = 1.4594E-014
          
x =
    0.5000
    0.0000
    0.1667
The quantity tol is a tolerance used in deciding that a diagonal element of R is negligible. If [Q,R,E] = qr(A), then 

tol = max(size(A))*eps*abs(R(1,1))
The solution x was computed using the factorization and the two steps

y = Q'*b;
x = R\y
The computed solution can be checked by forming Ax. This equals b to within roundoff error, which indicates that even though the simultaneous equations Ax = b are overdetermined and rank deficient, they happen to be consistent. There are infinitely many solution vectors x; the QR factorization has found just one of them.

Algorithm

qr uses the LINPACK routines ZQRDC and ZQRSL. ZQRDC computes the QR decomposition, while ZQRSL applies the decomposition.

See Also

\, /, lu, null, orth, qrdelete, qrinsert

References

J.J. Dongarra, J.R. Bunch, C.B. Moler, and G.W. Stewart, LINPACK User's Guide, SIAM, Philadelphia, 1979.

(c) Copyright 1994 by The MathWorks, Inc.