LU Factorization
Factor square matrix into lower and upper triangular components
Libraries:
DSP System Toolbox /
Math Functions /
Matrices and Linear Algebra /
Matrix Factorizations
Description
The LU Factorization block factors a rowpermuted version of the square input matrix A as A_{p} = L*U, where L is a unitlower triangular matrix, U is an upper triangular matrix, and A_{p} contains the rows of A permuted as indicated by the permutation index vector P. The block uses the pivot matrix A_{p} instead of the exact input matrix A because it improves the numerical accuracy of the factorization.
Examples
Ports
Input
A — Input matrix
MbyM square matrix
Specify the input matrix as an MbyM square matrix through this port. If the input matrix is fixed point, it must be signed integer or signed fixed point with poweroftwo slope and zero bias.
Data Types: single
 double
 int8
 int16
 int32
 fixed point
Complex Number Support: Yes
Output
LU — Lower and upper triangular factors
MbyM square matrix
Composite matrix containing both the lower triangle elements of L and
the upper triangle elements of U. The composite
matrix improves the efficiency of the block. The matrix output is in a
different format than the output of the lu
function, which
returns L and U as separate
matrices. To convert the output from the LU port of
the block to separate L and U
matrices, use the following code:
L = tril(LU,1)+eye(size(LU)); U = triu(LU);
If you compare the results produced by these equations to the actual output of the
lu
function, you might see slightly different
values. These differences are due to a rounding error, and are expected.
For more information on the LU factorizations, see the lu
function
page.
This composite matrix has the same size, data type, and complexity as the input matrix A.
Data Types: single
 double
 int8
 int16
 int32
 fixed point
Complex Number Support: Yes
P — Permutation vector
vector
Permutation vector, returned as a vector of length M. The block permutes the input square matrix A using the permutation vector P and factors the permuted matrix into unitlower triangular L and upper triangular matrix U.
Consider the following input matrix.
$$A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 0& 6\\ 2& 1& 3\end{array}\right]$$
With a permutation index vector P that is equal to $$P=\left[\begin{array}{c}2\\ 1\\ 3\end{array}\right]$$, the permuted matrix A_{p} becomes:
$${A}_{p}=\left[\begin{array}{ccc}4& 0& 6\\ 1& 2& 3\\ 2& 1& 3\end{array}\right]$$
The block factors this matrix into upper and lower triangular matrices. For a more detailed version of this example, see Factorize Matrix Using LU Factorization Block.
Data Types: single
 double
 uint32
S — Singularity of input matrix
logical scalar
You can determine the singularity of the input matrix
A by enabling the optional output port
S. When A is singular, the
block outputs a 1
at port S.
When A is nonsingular, the block outputs a
0
.
Data Types: Boolean
Parameters
Main Tab
Show singularity status — Show singularity status
off
(default)  on
Select to output the singularity of the input at port S, which outputs Boolean data type values of 1 or 0. An output of 1 indicates that the current input is singular, and an output of 0 indicates the current input is nonsingular.
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.
Product output — Product output data type
Inherit: Inherit via internal
rule
(default)  Inherit: Same as input
 fixdt(1,16,0)
Product output specifies the data type of the output of a product operation in the LU Factorization block. For more information on the product output data type, see Fixed Point and Multiplication Data Types.
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(1,16,0)
— The block specifies a signed, binarypoint, scaled, fixedpoint data type with a word length of 16 bits and a fraction length of 0.
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(1,16,0)
Accumulator specifies the data type of the output of an accumulation operation in the LU Factorization block. For more information on how to use the accumulator data type in this block, see the Fixed Point section.
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(1,16,0)
— The block specifies a signed, binarypoint scaled, fixedpoint data type with a word length of 16 bits and a fraction length of 0.
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(1,16,0)
Output specifies the data type of the output of the LU Factorization block. For more information on the output data type, see the Fixed Point section.
Inherit: Same as input
— The block specifies the output data type to be the same as the input data type.fixdt(1,16,0)
— The block specifies a signed, binarypoint scaled, fixedpoint data type with 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 the block can output
[]
(default)  scalar
Specify the minimum value 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 block can output
[]
(default)  scalar
Specify the maximum value 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 

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™.
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.
The following diagram shows the data types used within the LU Factorization block for fixedpoint signals.
You can set the product output, accumulator, and output data types in the block dialog as discussed below.
The output of the 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.
Version History
Introduced before R2006a
See Also
Functions
Blocks
 Cholesky Factorization  Autocorrelation LPC  LU Solver  LDL Factorization  LU Inverse  Permute Matrix (Simulink)  QR Factorization
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