Factor square Hermitian positive definite matrices into lower, upper, and diagonal components
Math Functions / Matrices and Linear Algebra / Matrix Factorizations
dspfactors
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, 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^{*}. The output format is shown below for a 5by5 matrix.
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.
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 Nonpositive definite input parameter.
The following 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 as discussed below.
The output of the second multiplier is in the product output data type when the input is real. 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.
LDL decomposition of a 3by3 Hermitian positive definite matrix:
The Main pane of the LDL Factorization block dialog appears as follows.
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 and terminate the simulation.
The Data Types pane of the LDL Factorization block dialog appears as follows.
Select the rounding mode for fixedpoint operations.
Select the overflow mode for fixedpoint operations.
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 it 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)
Click the Show data type assistant button to display the Data Type Assistant, which helps you set the Product output data type parameter.
See Specify Data Types Using Data Type Assistant for more information.
Specify the product output data type. See FixedPoint Data Types and Multiplication Data Types for illustrations depicting the use of the product output data type in this block. You can set it to:
A rule that inherits a data type, for example, Inherit:
Inherit via internal rule
An expression that evaluates to a valid data type,
for example, fixdt(1,16,0)
Click the Show data type assistant button to display the Data Type Assistant, which helps you set the Product output data type parameter.
See Specify Data Types Using Data Type Assistant for more information.
Specify the accumulator data type. See FixedPoint Data Types for illustrations depicting the use of the accumulator data type in this block. You can set this parameter to:
A rule that inherits a data type, for example, Inherit:
Inherit via internal rule
An expression that evaluates to a valid data type,
for example, fixdt(1,16,0)
Click the Show data type assistant button to display the Data Type Assistant, which helps you set the Accumulator data type parameter.
See Specify Data Types Using Data Type Assistant for more information.
Specify the output data type. See FixedPoint Data Types for illustrations depicting the use of the output data type in this block. You can set it 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)
Click the Show data type assistant button to display the Data Type Assistant, which helps you set the Output data type parameter.
See Control Signal Data Types for more information.
Select this parameter to prevent the fixedpoint tools from overriding the data types you specify on the block mask.
Golub, G. H., and C. F. Van Loan. Matrix Computations. 3rd ed. Baltimore, MD: Johns Hopkins University Press, 1996.
Port  Supported Data Types 

S 

LDL 

Cholesky Factorization  DSP System Toolbox 
LDL Inverse  DSP System Toolbox 
LDL Solver  DSP System Toolbox 
LU Factorization  DSP System Toolbox 
QR Factorization  DSP System Toolbox 
See Matrix Factorizations for related information.