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
$$S=LD{L}^{*}$$
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.
Examples
LDL Factorization of 3by3 Hermitian Positive Definite Matrix
Simulink^{®} model example that shows how to use the LDL Factorization block.
Ports
Input
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 Nonpositive definite input parameter.
If the input is fixed point, it must be a signed integer or a signed fixed point value with a poweroftwo slope and zero bias.
Data Types: single
 double
 int8
 int16
 int32
 fixed point
Output
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 5by5 matrix.
Data Types: single
 double
 int8
 int16
 int32
 fixed point
Parameters
Main Tab
Nonpositive definite input — Nonpositive 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 upperleft 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 upperleft 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 fixedpoint 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
fixedpoint operation. When you clear this parameter, the block wraps
the result of its fixedpoint operation. For details on
saturate
and wrap
, see overflow
mode for fixedpoint 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 FixedPoint 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, binarypoint, scaled, fixedpoint data type that has a word length of 16 bits and a fraction length of 0.
For more information, see Multiplication Data Types and FixedPoint 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, binarypoint, scaled, fixedpoint 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 FixedPoint 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 FixedPoint 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, binarypoint, scaled, fixedpoint 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 fixedpoint 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 fixedpoint data types.
Lock data type settings against changes by the fixedpoint tools — Prevent fixedpoint tools from overriding data types
off
(default)  on
Select this parameter to prevent the fixedpoint tools from overriding the data types you specify in the block dialog box.
Block Characteristics
Data Types 

Direct Feedthrough 

Multidimensional Signals 

VariableSize Signals 

ZeroCrossing Detection 

More About
FixedPoint Data Types
This diagram shows the data types used within the LDL Factorization block for fixedpoint 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.
FixedPoint Conversion
Design and simulate fixedpoint systems using FixedPoint Designer™.
Version History
Introduced before R2006a
See Also
Blocks
Topics
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