DSP Blockset

QR Factorization

Factor a rectangular matrix into unitary and upper triangular components

Library

Math Functions / Matrices and Linear Algebra / Matrix Factorizations

Description

The QR Factorization block uses a modified Gram-Schmidt iteration to factor a column permutation of the M-by-N input matrix A as

where Q is an M-by-min(M,N) unitary matrix, and R is a min(M,N)-by-N upper-triangular matrix. A length-M vector input is treated as an M-by-1 matrix, and is always sample based.

The column-pivoted matrix A_e contains the columns of A permuted as indicated by the contents of length-N permutation vector E.

• Ae = A(:,E)          % Equivalent MATLAB code
  

The block selects a column permutation vector E, which ensures that the diagonal elements of matrix R are arranged in order of decreasing magnitude.

QR factorization is an important tool for solving linear systems of equations because of good error propagation properties and the invertability of unitary matrices.

Unlike LU and Cholesky factorizations, the matrix A does not need to be square for QR factorization. Note, however, that QR factorization requires twice as many operations as Gaussian elimination.

Example

A sample factorization is shown below. The input to the block is matrix A, which is permuted according to vector E to produce matrix A_e. Matrix A_e is factored to produce the Q and R output matrices.

Dialog Box

References

Golub, G. H., and C. F. Van Loan. Matrix Computations. 3rd ed. Baltimore, MD: Johns Hopkins University Press, 1996.

Supported Data Types

• Double-precision floating point
• Single-precision floating point

To learn how to convert your data types to the above data types in MATLAB and Simulink, see Supported Data Types and How to Convert to Them.

See Also

Cholesky Factorization	DSP Blockset
LU Factorization	DSP Blockset
QR Solver	DSP Blockset
Singular Value Decomposition	DSP Blockset
qr	MATLAB

See Factoring Matrices for related information.

Pseudoinverse

QR Solver