WOLFRAM SYSTEM MODELER

# LU_solve

Solve real system of linear equations P*L*U*x=b with a b vector and an LU decomposition (from LU(..)) # Wolfram Language

In:= `SystemModel["Modelica.Math.Matrices.LU_solve"]`
Out:= # Information

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

#### Syntax

```Matrices.LU_solve(LU, pivots, b);
```

#### Description

This function call returns the solution x of the linear systems of equations

P*L*U*x = b;

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). The matrices of this decomposition are computed with function Matrices.LU that returns arguments `LU` and `pivots` used as input arguments of `Matrices.LU_solve`. With `Matrices.LU` and `Matrices.LU_solve` it is possible to efficiently solve linear systems with different right hand side vectors. If a linear system of equations with just one right hand side vector shall be solved, it is more convenient to just use the function Matrices.solve.

If a unique solution x does not exist (since the LU decomposition is singular), an exception is raised.

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 = {10,22,12};
Real b2 = { 7,13,10};
Real    LU[3,3];
Integer pivots;
Real    x1;
Real    x2;
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}
```

Matrices.LU, Matrices.solve,

# Syntax

x = LU_solve(LU, pivots, b)

# Inputs (3)

LU Type: Real[:,size(LU, 1)] Description: L,U factors of Matrices.LU(..) for a square matrix Type: Integer[size(LU, 1)] Description: Pivots indices of Matrices.LU(..) Type: Real[size(LU, 1)] Description: Right hand side vector of P*L*U*x=b

# Outputs (1)

x Type: Real[size(b, 1)] Description: Solution vector such that P*L*U*x = b