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
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 =
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 M ≤ N, 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.
Open the model by typing
ex_qrfactorization_refat the MATLAB® command line.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-2matrix Q and a2-by-2matrix R.Change the Output size parameter of the QR Factorization block to
Fulland rerun the model.
The QR Factorization block outputs a
5-by-5matrix Q and a5-by-2matrix R.
Parameters
- 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
Interpreted execution(default)Simulate model using the MATLAB interpreter. This option shortens startup time and has faster simulation speed compared to
Code generation.Code generationSimulate 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
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Input |
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Output |
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