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# linalg::matlinsolveLU

Solving the linear system given by an LU decomposition

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

## Syntax

```linalg::matlinsolveLU(L, U, b)
linalg::matlinsolveLU(L, U, B)
```

## Description

linalg::matlinsolveLU(L, U, b) solves the linear system , where the matrices L and U form an LU-decomposition, as computed by linalg::factorLU.

If the third parameter is an n×k matrix B then the result is an n×k matrix X satisfying the matrix equation LUX = B.

The system to be solved always has a unique solution.

The diagonal entries of the lower diagonal matrix L must be equal to one (Doolittle-decomposition, see linalg::factorLU).

linalg::matlinsolveLU expects L and U to be nonsingular.

linalg::matlinsolveLU does not check on any of the required properties of L and U.

The component ring of the matrices L and U must be a field, i.e., a domain of category Cat::Field.

The parameters must be defined over the same component ring.

## Examples

### Example 1

We solve the system

:

```MatR := Dom::Matrix(Dom::Real):
A  := MatR([[2, -3, -1], [1, 1, -1], [0, 1, -1]]);
I3 := MatR::identity(3)```

We start by computing an LU-decomposition of A:

`LU := linalg::factorLU(A)`

Now we solve the system AX = I3, which gives us the inverse of A:

`Ai := linalg::matlinsolveLU(LU[1], LU[2], I3)`

`A * Ai, Ai * A`

## Parameters

 L An n×n lower triangular matrix of a domain of category Cat::Matrix U An n×n upper triangular form matrix of the same domain as L B An n×k matrix of a domain of category Cat::Matrix b An n-dimensional column vector, i.e., an n×1 matrix of a domain of category Cat::Matrix

## Return Values

n-dimensional solution vector or n×k dimensional solution matrix, respectively, of the domain type Dom::Matrix(R), where R is the component ring of A.