.Modelica.Math.Matrices.LU

Information

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:

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}

See also

Matrices.LU_solve, Matrices.solve,

Interface

pure function LU
  extends Modelica.Icons.Function;
  input Real A[:, :] "Square or rectangular matrix";
  output Real LU[size(A, 1), size(A, 2)] = A "L,U factors (used with LU_solve(..))";
  output Integer pivots[min(size(A, 1), size(A, 2))] "Pivot indices (used with LU_solve(..))";
  output Integer info "Information";
end LU;