WOLFRAM SYSTEM MODELER

# LU

LU decomposition of square or rectangular matrix

# Wolfram Language

In[1]:=
`SystemModel["Modelica.Math.Matrices.LU"]`
Out[1]:=

# Information

This information is part of the Modelica Standard Library maintained by the Modelica Association.

#### Syntax

```(LU, pivots)       = Matrices.LU(A);
(LU, pivots, info) = Matrices.LU(A);
```

#### Description

This function call returns the LU decomposition of a "Real[m,n]" matrix A, i.e.,

P*L*U = A

where P is a permutation matrix (implicitly defined by vector `pivots`), L is a lower triangular matrix with unit diagonal elements (lower trapezoidal if m > n), and U is an upper triangular matrix (upper trapezoidal if m < n). Matrices L and U are stored in the returned matrix `LU` (the diagonal of L is not stored). With the companion function Matrices.LU_solve, this decomposition can be used to solve linear systems (P*L*U)*x = b with different right hand side vectors b. If a linear system of equations with just one right hand side vector b shall be solved, it is more convenient to just use the function Matrices.solve.

The optional third (Integer) output argument has the following meaning:

 info = 0: successful exit info > 0: if info = i, U[i,i] is exactly zero. The factorization has been completed, but the factor U is exactly singular, and division by zero will occur if it is used to solve a system of equations.

The LU factorization is computed with the LAPACK function "dgetrf", i.e., by Gaussian elimination using partial pivoting with row interchanges. Vector "pivots" are the pivot indices, i.e., for 1 ≤ i ≤ min(m,n), row i of matrix A was interchanged with row pivots[i].

#### Example

```  Real A[3,3] = [1,2,3;
3,4,5;
2,1,4];
Real b1[3] = {10,22,12};
Real b2[3] = { 7,13,10};
Real    LU[3,3];
Integer pivots[3];
Real    x1[3];
Real    x2[3];
algorithm
(LU, pivots) := Matrices.LU(A);
x1 := Matrices.LU_solve(LU, pivots, b1);  // x1 = {3,2,1}
x2 := Matrices.LU_solve(LU, pivots, b2);  // x2 = {1,0,2}
```