Cholesky Factorization

Factor square Hermitian positive definite matrix into triangular components

Library

Math Functions / Matrices and Linear Algebra / Matrix Factorizations

dspfactors

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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. The block outputs a 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.

Here, LL' is the output of the Cholesky Factorization block. Due to roundoff error, these equations do not produce a result that is exactly the same as the MATLAB result.

Block Output Composed of L and L^*

Input Requirements for Valid Output

The block output is valid only when its input has the following characteristics:

• Hermitian — The block does not check whether the input is Hermitian; it 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's diagonal entries.

• Positive definite — Set the block to notify you when the input is not positive definite as described in Response to Nonpositive Definite Input.

Response to Nonpositive Definite Input

To generate a valid output, the block algorithm requires a positive definite input (see Input Requirements for Valid Output). Set the Non-positive definite input parameter to determine how the block responds to a nonpositive definite input:

• 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.

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.

Performance Comparisons with Other Blocks

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.

Parameters

Non-positive definite input

Response to nonpositive definite matrix inputs: Ignore, Warning, or Error. See Response to Nonpositive Definite Input.

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
S	Double-precision floating point Single-precision floating point
LL'	Double-precision floating point Single-precision floating point