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Compute LU Factorization

MuPAD® notebooks are not recommended. Use MATLAB® live scripts instead.

MATLAB live scripts support most MuPAD functionality, though there are some differences. For more information, see Convert MuPAD Notebooks to MATLAB Live Scripts.

The LU factorization expresses an m×n matrix A as follows: P*A = L*U. Here L is an m×m lower triangular matrix that contains 1s on the main diagonal, U is an m×n matrix upper triangular matrix, and P is a permutation matrix. To compute the LU decomposition of a matrix, use the linalg::factorLU function. For example, compute the LU decomposition of the following square matrix:

A := matrix([[0, 0, 1], [1, 2, 3], [0, 1, 2]]):
[L, U, p] := linalg::factorLU(A)

Instead of returning the permutation matrix P, MuPAD® returns the list p with numbers corresponding to row exchanges in the matrix A. For an n×n matrix, the list p represents the following permutation matrix with indices i and j ranging from 1 to n:


Using this expression, restore the permutation matrix P from the list p:

P := matrix(3, 3):
for i from 1 to 3 do
   P[i, p[i]] := 1

More efficiently, compute the result of applying the permutation matrix to A without restoring the permutation matrix itself:

PA := matrix(3, 3):
for i from 1 to 3 do
  PA[i, 1..3] := A[p[i], 1..3]

The product of the lower triangular matrix L and the upper triangular matrix U is the original matrix A with the rows interchanged according to the permutation matrix P:

testeq(PA = L*U)

Now, compute the LU decomposition for the 3 ×2 matrix B:

B := matrix([[1, 2], [3, 4], [5, 6]]):
[L, U, p] := linalg::factorLU(B)

The permutation matrix for this LU factorization shows that the order of the rows does not change. Therefore, the product of the lower triangular matrix L and the upper triangular matrix U gives the original matrix A:

testeq(B = L*U)

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