.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:

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}

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;