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# QR Factorization

Factor arbitrary matrix into unitary and upper triangular components

## Library

Math Functions / Matrices and Linear Algebra / Matrix Factorizations

dspfactors

## Description

The QR Factorization block uses a sequence of Householder transformations to triangularize the input matrix A. The block factors a column permutation of the M-by-N input matrix A as

Ae = QR

The column-pivoted matrix Ae 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.

The size of matrices Q and R depends on the setting of the Output size parameter:

• When you select Economy for the output size, Q is an M-by-min(M,N) unitary matrix, and R is a min(M,N)-by-N upper-triangular matrix.

`[Q R E] = qr(A,0)   % Equivalent MATLAB code`
• When you select Full for the output size, Q is an M-by-M unitary matrix, and R is a M-by-N upper-triangular matrix.

`[Q R E] = qr(A)      % Equivalent MATLAB code`

The block treats length-M unoriented vector input as an M-by-1 matrix.

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

Q –1 = Q'

where Q' is the complex conjugate transpose of Q.

Unlike LU and Cholesky factorizations, the matrix A does not need to be square for QR factorization. However, QR factorization requires twice as many operations as LU Factorization (Gaussian elimination).

## Examples

The Output size parameter of the QR factorization block has two settings: Economy and Full. When the M-by-N input matrix A has dimensions such that M > N, the dimensions of output matrices Q and R differ depending on the setting of the Output size parameter. If, however, the size of the input matrix A is such that MN, output matrices Q and R have the same dimensions, regardless of whether the Output size is set to Economy or Full.

The input to the QR Factorization block in the following model is a 5-by-2 matrix A. When you change the setting of the Output size parameter from Economy to Full, the dimensions of the output given by the QR Factorization block also change.

1. Open the model by typing ex_qrfactorization_refex_qrfactorization_ref at the MATLAB® command line.

2. Double-click the QR Factorization block, set the Output size parameter to Economy, and run the model.

The QR Factorization block outputs a 5-by-2 matrix Q and a 2-by-2 matrix R.

3. Change the Output size parameter of the QR Factorization block to Full and rerun the model.

The QR Factorization block outputs a 5-by-5 matrix Q and a 5-by-2 matrix R.

## Dialog Box

Output size

Specify the size of output matrices Q and R:

• Economy — When this output size is selected, the block outputs an M-by-min(M,N) unitary matrix Q and a min(M,N)-by-N upper-triangular matrix R.

• Full — When this output size is selected, the block outputs an M-by-M unitary matrix Q and a M-by-N upper-triangular matrix R.

## References

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

## Supported Data Types

PortSupported Data Types

Input

• Double-precision floating point

• Single-precision floating point

Output

• Double-precision floating point

• Single-precision floating point

 Cholesky Factorization DSP System Toolbox LU Factorization DSP System Toolbox QR Solver DSP System Toolbox Singular Value Decomposition DSP System Toolbox qr MATLAB

See Matrix Factorizations for related information.