LDL Factorization

Factor square Hermitian positive definite matrices into lower, upper, and diagonal components

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

Description

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

`$S=LD{L}^{*}$`

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. 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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Input signal, specified as a square matrix.

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

If the input is fixed point, it must be signed fixed point with a power-of-two slope and zero bias.

Data Types: `single` | `double` | `int8` | `int16` | `int32` | `fixed point`

Output

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The output is a composite matrix with lower triangle elements lij from L, diagonal elements dij from D and upper triangle elements uij from L*, where L is a lower triangular square matrix that has unity diagonal elements, D is a diagonal matrix, and L* is the Hermitian (complex conjugate) transpose of L.

The output format is shown below for a 5-by-5 matrix.

Data Types: `single` | `double` | `int8` | `int16` | `int32` | `fixed point`

Parameters

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Main Tab

Specify the action when nonpositive definite matrix inputs occur:

• `Ignore` — Proceed with the computation and do not issue an alert. The output is not a valid factorization. A partial factorization is 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 is present in the upper-left corner of the output.

• `Error` — Display an error dialog box and terminate the simulation.

Data Types Tab

Specify the rounding mode for fixed-point operations as one of the following:

• `Floor`

• `Ceiling`

• `Convergent`

• `Nearest`

• `Round`

• `Simplest`

• `Zero`

For more details, see rounding mode.

When you select this parameter, the block saturates the result of its fixed-point operation. When you clear this parameter, the block wraps the result of its fixed-point operation. For details on `saturate` and `wrap`, see overflow mode for fixed-point operations.

Specify the intermediate product data type. As shown in Fixed-Point Conversion section in Extended Capabilities, the output of the multiplier is cast to the intermediate product data type before the next element of the input is multiplied into it. You can set the data type to:

• A rule that inherits a data type, for example, `Inherit: Same as input`

• An expression that evaluates to a valid data type, for example, `fixdt(1,16,0)`

Alternatively, you can set the Intermediate product data type by using the Data Type Assistant. To use the assistant, click the button.

Product output specifies the data type of the output of a product operation in the LDL Factorization block.

• `Inherit: Inherit via internal rule` — The block inherits the product output data type based on an internal rule. For more information on this rule, see Inherit via Internal Rule.

• `Inherit: Same as input` — The block specifies the product output data type to be the same as the input data type.

• `fixdt([],16,0)` — The block specifies an autosigned, binary-point, scaled, fixed-point data type that has a word length of 16 bits and a fraction length of 0.

For more information, see Multiplication Data Types and Fixed-Point Conversion in Extended Capabilities.

Alternatively, you can set the Product output data type by using the Data Type Assistant. To use the assistant, click the button.

For more information on the data type assistant, see Specify Data Types Using Data Type Assistant (Simulink).

Accumulator specifies the data type of the output of an accumulation operation in the LDL Factorization block.

• `Inherit: Inherit via internal rule` — The block inherits the accumulator data type based on an internal rule. For more information on this rule, see Inherit via Internal Rule.

• `Inherit: Same as input` — The block specifies the accumulator data type to be the same as the input data type.

• `Inherit: Same as product output` — The block specifies the accumulator data type to be the same as the product output data type.

• `fixdt([],16,0)` — The block specifies an autosigned, binary-point, scaled, fixed-point data type that has a word length of 16 bits and a fraction length of 0.

For illustrations on how to use the accumulator data type in this block, see Fixed-Point Conversion in Extended Capabilities.

Alternatively, you can set the Accumulator data type by using the Data Type Assistant. To use the assistant, click the button.

For more information on the data type assistant, see Specify Data Types Using Data Type Assistant (Simulink).

Output specifies the data type of the output of the LDL Factorization block. For more information on the output data type, see the 'Fixed-Point Conversion' section in Extended Capabilities.

• `Inherit: Same as input` — The block specifies the output data type to be the same as the input data type.

• `fixdt([],16,0)` — The block specifies an autosigned, binary-point, scaled, fixed-point data type that has a word length of 16 bits and a fraction length of 0.

Alternatively, you can set the Output data type by using the Data Type Assistant. To use the assistant, click the button.

For more information on the data type assistant, see Specify Data Types Using Data Type Assistant (Simulink).

Specify the minimum value that the block can output. Simulink® software uses this minimum value to perform:

• Simulation range checking. See Specify Signal Ranges (Simulink).

• Automatic scaling of fixed-point data types.

Specify the maximum value that the block can output. Simulink software uses this maximum value to perform:

• Simulation range checking. See Specify Signal Ranges (Simulink).

• Automatic scaling of fixed-point data types.

Select this parameter to prevent the fixed-point tools from overriding the data types you specify in the block dialog box.

Block Characteristics

 Data Types `double` | `fixed point` | `integer` | `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