DSP Blockset

LDL Factorization

Factor a square Hermitian positive definite matrix into lower, upper, and diagonal components.

Library

Math Functions / Matrices and Linear Algebra / Matrix Factorizations

Description

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

where L is a lower triangular square matrix with unity diagonal elements, D is a diagonal matrix, and L^* is the Hermitian (complex conjugate) transpose of L. Only the diagonal and lower triangle of the input matrix are used, and any imaginary component of the diagonal entries is disregarded.

The block's output is a composite matrix with lower triangle elements l_ij from L, diagonal elements d_ij from D, and upper triangle elements u_ij from L^*. It is always sample-based. The output format is shown below for a 5-by-5 matrix.

LDL factorization requires half the computation of Gaussian elimination (LU decomposition), and is always stable. It is more efficient that Cholesky factorization because it avoids computing the square roots of the diagonal elements.

The algorithm requires that the input be square and Hermitian positive definite. When the input is not positive definite, the block reacts with the behavior specified by the Non-positive definite input parameter.

The following options are available:

• 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 ouput.
• 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 ouput.
• Error -- Display an error dialog box and terminate the simulation.

Example

LDL decomposition of a 3-by-3 Hermitian positive definite matrix:

Dialog Box

Non-positive definite input

Response to non-positive definite matrix inputs. Tunable.

References

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

Supported Data Types

• Double-precision floating point
• Single-precision floating point

To learn how to convert to the above data types in MATLAB and Simulink, see Supported Data Types and How to Convert to Them.

See Also

Cholesky Factorization	DSP Blockset
LDL Inverse	DSP Blockset
LDL Solver	DSP Blockset
LU Factorization	DSP Blockset
QR Factorization	DSP Blockset

See Factoring Matrices for related information. Also see Matrix Factorizations for a list of all the blocks in the Matrix Factorizations library.

Kalman Adaptive Filter

LDL Inverse