Factor arbitrary matrix into unitary and upper triangular components
Math Functions / Matrices and Linear Algebra / Matrix Factorizations
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
Economyfor 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
Fullfor 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 invertibility of unitary matrices:
Q –1 =
' is the complex conjugate transpose of
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).
- 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.
- Simulate using
Simulate model using the MATLAB® interpreter. This option shortens startup time and has faster simulation speed compared to
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.
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|
C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.
Generated code relies on the
memset function (
string.h) under certain
Introduced before R2006a