LDL Factorization

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

Libraries:
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

$S = L D L^{*}$

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.

Examples

LDL Factorization of 3-by-3 Hermitian Positive Definite Matrix

Simulink^® model example that shows how to use the LDL Factorization block.

Open Model

Ports

Input

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Port_1 — Input signal
matrix

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 a signed integer or a signed fixed point value with a power-of-two slope and zero bias.

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

Output

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Port_1 — Output signal
matrix

The 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^*, 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

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

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

Rounding mode — Method of rounding operation
`Floor` (default) | `Ceiling` | `Convergent` | `Nearest` | `Round` | `Simplest` | `Zero`

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.

Saturate on integer overflow — Method of overflow action
off (default) | on

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.

Intermediate product — Intermediate product data type
`Inherit: Same as input` (default) | `fixdt(1,16,0)`

Specify the intermediate product data type. As shown in Fixed-Point Data Types, 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 Show data type assistant button.

For more information, see Specify Data Types Using Data Type Assistant (Simulink).

Product output — Product output data type
`Inherit: Inherit via internal rule` (default) | `Inherit: Same as input` | `fixdt([],16,0)`

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 Data Types.

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

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

Accumulator — Accumulator data type
`Inherit: Inherit via internal rule` (default) | `Inherit: Same as input` | `Inherit: Same as product output` | `fixdt([],16,0)`

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 Data Types.

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

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

Output — Output data type
`Inherit: Same as input` (default) | `fixdt([],16,0)`

Output specifies the data type of the output of the LDL Factorization block. For more information on the output data type, see Fixed-Point Data Types.

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 Show data type assistant button.

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

Output Minimum — Minimum value that block can output
`[]` (default) | scalar

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.

Output Maximum — Maximum value that block can output
`[]` (default) | scalar

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.

Lock data type settings against changes by the fixed-point tools — Prevent fixed-point tools from overriding data types
`off` (default) | `on`

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`

More About

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Fixed-Point Data Types

This diagram shows the data types used within the LDL Factorization block for fixed-point signals.

You can set the intermediate product, product output, accumulator, and output data types in the block dialog box.

When the input is real, the output of the second multiplier is in the product output data type. When the input is complex, the result of the multiplication is in the accumulator data type. For details on the complex multiplication performed, see Multiplication Data Types.

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.

Fixed-Point Conversion
Design and simulate fixed-point systems using Fixed-Point Designer™.

Version History

Introduced before R2006a

LDL Factorization

Description

Examples

LDL Factorization of 3-by-3 Hermitian Positive Definite Matrix