# LDL Inverse

Compute inverse of Hermitian positive definite matrix using LDL factorization

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

## Description

The LDL Inverse block computes the inverse of the Hermitian positive definite input matrix S by performing an LDL factorization.

`${S}^{-1}={\left(LD{L}^{*}\right)}^{-1}$`

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.

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

## Ports

### Input

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Specify the input as a square matrix S that is Hermitian positive definite. When the input is not positive definite, the block reacts with the behavior specified by the Non-positive definite input parameter. For more details, see the description of the Non-positive definite input parameter.

Data Types: `single` | `double`
Complex Number Support: Yes

### Output

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Inverse of the Hermitian positive definite input matrix S, returned as a square matrix. Here is the equation for LDL factorization.

`${S}^{-1}={\left(LD{L}^{*}\right)}^{-1}$`

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.

Data Types: `single` | `double`
Complex Number Support: Yes

## Parameters

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Specify the response to nonpositive definite matrix input as `Ignore`, `Warning`, or `Error`.

The algorithm requires that the input be 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 inverse.

• `Warning` — Display a warning message in the MATLAB® command window, and continue the simulation. The output is not a valid inverse.

• `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, 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.

## Version History

Introduced before R2006a