# LU Factorization

Factor square matrix into lower and upper triangular components

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

• ## Description

The LU Factorization block factors a row-permuted version of the square input matrix A as Ap = L*U, where L is a unit-lower triangular matrix, U is an upper triangular matrix, and Ap contains the rows of A permuted as indicated by the permutation index vector P. The block uses the pivot matrix Ap instead of the exact input matrix A because it improves the numerical accuracy of the factorization.

## Ports

### Input

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Specify the input matrix as an M-by-M square matrix through this port. If the input matrix is fixed point, it must be signed integer or signed fixed point with power-of-two slope and zero bias.

Data Types: `single` | `double` | `int8` | `int16` | `int32` | `fixed point`
Complex Number Support: Yes

### Output

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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

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 unit-lower 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 Ap 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`

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

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### Main Tab

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

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.

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, binary-point, scaled, fixed-point 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 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 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, binary-point scaled, fixed-point 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 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 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, binary-point scaled, fixed-point 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 button .

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

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 fixed-point data types.

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 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`

 Golub, G. H., and C. F. Van Loan. Matrix Computations. 3rd ed. Baltimore, MD: Johns Hopkins University Press, 1996.