qr (MATLAB Functions) (2024)

Orthogonal-triangular decomposition

Syntax

• [Q,R] = qr(A) (full and sparse matrices)[Q,R] = qr(A,0) (full and sparse matrices)[Q,R,E] = qr(A) (full matrices)[Q,R,E] = qr(A,0) (full matrices)X = qr(A) (full matrices)R = qr(A) (sparse matrices)[C,R] = qr(A,B) (sparse matrices)R = qr(A,0) (sparse matrices)[C,R] = qr(A,B,0) (sparse matrices)

Description

The qr function 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(A) produces an upper triangular matrix R of the same dimension as A and a unitary matrix Q so that A=Q*R. For sparse matrices, Q is often nearly full. If [mn]=size(A), then Q is m-by-m and R is m-by-n.

[Q,R] = qr(A,0) produces an "economy-size" decomposition. If [mn]=size(A), and m > n, then qr computes only the first n columns of of Q and R is n-by-n. If m<=n, it is the same as [Q,R]=qr(A).

[Q,R,E] = qr(A) for full matrix A, produces a permutation matrix E, an upper triangular matrix R with decreasing diagonal elements, and a unitary matrix Q so that A*E=Q*R. The column permutation E is chosen so that abs(diag(R)) is decreasing.

[Q,R,E] = qr(A,0) for full matrix A, produces an "economy-size" decomposition in which E is a permutation vector, so that A(:,E)=Q*R. The column permutation E is chosen so that abs(diag(R)) is decreasing.

X = qr(A) for full matrix A, returns the output of the LAPACK subroutine DGEQRF or ZGEQRF. triu(qr(A)) is R.

R = qr(A) for sparse matrix A, produces only an upper triangular matrix, R. The matrix R provides a Cholesky factorization for the matrix associated with the normal equations,

• R'*R = A'*A

This approach avoids the loss of numerical information inherent in the computation of A'*A. It may be preferred to [Q,R] = qr(A) since Q is always nearly full.

[C,R] = qr(A,B) for sparse matrix A, applies the orthogonal transformations to B, producing C = Q'*B without computing Q. B and A must have the same number of rows.

R = qr(A,0) and [C,R] = qr(A,B,0) for sparse matrix A, produce "economy-size" results.

For sparse matrices, the Q-less QR factorization allows the solution of sparse least squares problems

with two steps

• [C,R] = qr(A,b)x = R\c

If A is sparse but not square, MATLAB uses the two steps above for the linear equation solving backslash operator, i.e., x = A\b.

Examples

Example 1. 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 0.4461R = -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.

Example 2. This examples uses matrix A from the first example. The QR factorization is used to solve linear systems with more equations than unknowns. For example, let

• b = [1;3;5;7]

The linear system 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-014x = 0.5000 0 0.1667

The quantity tol is a tolerance used to decide if 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 . This equals to within roundoff error, which indicates that even though the simultaneous equations 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

The qr function uses LAPACK routines to compute the QR decomposition:

See Also

lu, null, orth, qrdelete, qrinsert, qrupdate

The arithmetic operators \ and /

References

[1] Anderson, E., Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J.Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK User's Guide (http://www.netlib.org/lapack/lug/lapack_lug.html), Third Edition, SIAM, Philadelphia, 1999.