LULU decomposition of square or rectangular matrix |
This information is part of the Modelica Standard Library maintained by the Modelica Association.
(LU, pivots) = Matrices.LU(A); (LU, pivots, info) = Matrices.LU(A);
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].
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}
A |
Type: Real[:,:] Description: Square or rectangular matrix |
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LU |
Default Value: A Type: Real[size(A, 1),size(A, 2)] Description: L,U factors (used with LU_solve(..)) |
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pivots |
Type: Integer[min(size(A, 1), size(A, 2))] Description: Pivot indices (used with LU_solve(..)) |
info |
Type: Integer Description: Information |