Cholesky Factorization

Factor square Hermitian positive definite matrix into triangular components

Libraries:
DSP System Toolbox / Math Functions / Matrices and Linear Algebra / Matrix Factorizations

Description

The Cholesky Factorization block uniquely factors the square Hermitian positive definite input matrix S as

$S = L L'$

where L is a lower triangular square matrix with positive diagonal elements and L^' is the Hermitian (complex conjugate) transpose of L.

Note that L and L^' share the same diagonal in the output matrix. Cholesky factorization requires half the computation of Gaussian elimination (LU decomposition), and is always stable.

Ports

Input

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S — Input matrix
square matrix

Specify the input matrix as a square matrix. The block output is valid only when its input has the following characteristics:

Hermitian — The block does not check whether the input is Hermitian. The block uses only the diagonal and upper triangle of the input to compute the output.
Real-valued diagonal entries — The block disregards any imaginary component of the input diagonal entries.
Positive definite — The block notifies you when the input is not positive definite as described in the Non-positive definite input parameter description.

Data Types: single | double
Complex Number Support: Yes

Output

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LL' — Lower and upper triangular factors
square matrix

The block outputs a square matrix with lower triangle elements from L and upper triangle elements from L^'. The output is not in the same form as the output of the MATLAB^® chol function. In order to convert the output of the Cholesky Factorization block to the MATLAB form, use the following equation:

R = triu(LL');

In order to extract the L matrix exclusively, pass the output of the Cholesky Factorization block, LL', to the Extract Triangular Matrix block. Setting the Extract parameter of the Extract Triangular Matrix to Lower extracts the L matrix. Setting the Extract parameter to Upper extracts the L^' matrix.

In this diagram, LL' is the output of the Cholesky Factorization block. Due to round off error, these equations do not produce a result that is exactly the same as the MATLAB result.

Block Output Composed of L and L ^*

Data Types: single | double
Complex Number Support: Yes

Parameters

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Non-positive definite input — Non-positive definite input
`Warning` (default) | `Ignore` | `Error`

Specify the response to nonpositive definite matrix inputs as one of these:

Ignore — Proceed with the computation and do not issue an alert. The output is not a valid factorization. A partial factorization will be present in the upper left corner of the output.
Warning — Display a warning message in the MATLAB Command Window, and continue the simulation. The output is not a valid factorization. A partial factorization will be present in the upper left corner of the output.
Error — Display an error dialog and terminate the simulation.

To generate a valid output, the block algorithm requires a positive definite input. See the S port description for more information.

Note

The Non-positive definite input parameter is a diagnostic parameter. Like all diagnostic parameters on the Configuration Parameters dialog box, it is set to Ignore in the code generated for this block by Simulink^® Coder™ code generation software.

Block Characteristics

Data Types	`double` \| `single`
Direct Feedthrough	`no`
Multidimensional Signals	`no`
Variable-Size Signals	`no`
Zero-Crossing Detection	`no`

References

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

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Generated code relies on the memcpy or memset function (string.h) under certain conditions.

Version History

Introduced before R2006a

Cholesky Factorization

Description