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 MbyN input matrix A as
A_{e} = QR
The columnpivoted matrix A_{e} contains the columns of A permuted as indicated by the contents of lengthN permutation vector E.
A_{e} = 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.
$$\left{r}_{i+1,j+1}\right<\left{r}_{i,j}\right\text{}i=j$$
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 Mbymin(M,N) unitary matrix, and R is a min(M,N)byN uppertriangular matrix.[Q R E] = qr(A,0) % Equivalent MATLAB code
When you select
Full
for the output size, Q is an MbyM unitary matrix, and R is a MbyN uppertriangular matrix.[Q R E] = qr(A) % Equivalent MATLAB code
The block treats lengthM unoriented vector input as an Mby1 matrix.
QR factorization is an important tool for solving linear systems of equations because of good error propagation properties and the invertibility 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).
Parameters
 Output size
Specify the size of output matrices Q and R:
Economy
— When this output size is selected, the block outputs an Mbymin(M,N) unitary matrix Q and a min(M,N)byN uppertriangular matrix R.Full
— When this output size is selected, the block outputs an MbyM unitary matrix Q and a MbyN uppertriangular matrix R.
 Simulate using
Interpreted execution
(default)Simulate model using the MATLAB^{®} interpreter. This option shortens startup time and has faster simulation speed compared to
Code generation
.Code generation
Simulate model using generated C code. The first time you run a simulation, Simulink^{®} generates C code for the block. The C code is reused for subsequent simulations, as long as the model does not change. This option requires additional startup time but provides faster subsequent simulations.
References
Golub, G. H., and C. F. Van Loan. Matrix Computations. 3rd ed. Baltimore, MD: Johns Hopkins University Press, 1996.
Supported Data Types
Port  Supported Data Types 

Input 

Output 

Extended Capabilities
Version History
Introduced before R2006a