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lu

lu(A) -> L, U, p

Compute the LU factorization of A, such that A[p,:] = L*U.

Examples

1. Compute the LU factorization of a matrix:

   
   julia> A = [1 2 3; 4 5 6; 7 8 10];
   julia> L, U, p = lu(A);
   julia> L
   3×3 Array{Float64,2}:
   1.0  0.0  0.0
   0.571429  1.0  0.0
   0.142857  0.5  1.0

julia> U 3×3 Array{Float64,2}: 7.0 8.0 10.0 0.0 0.857143 1.71429 0.0 0.0 -0.0

julia> p 3-element Array{Int64,1}: 3 2 1

This example computes the LU factorization of matrix `A` and assigns the lower triangular matrix `L`, upper triangular matrix `U`, and permutation vector `p` to the respective variables.

2. **Solve a system of linear equations using LU factorization:**
```julia
julia> A = [1 2 3; 4 5 6; 7 8 10];
julia> b = [3, 6, 9];
julia> L, U, p = lu(A);
julia> x = U \ (L \ (b[p]));
julia> x
3-element Array{Float64,1}:
 -1.9999999999999996
  2.0
  1.0

In this example, LU factorization is used to solve a system of linear equations Ax = b. The solution vector x is computed by first permuting b using p, then applying forward and backward substitutions using L and U.

Common mistake example:

julia> A = [1 2 3; 4 5 6];
julia> L, U, p = lu(A);
ERROR: ArgumentError: input matrix must be square

Here, the LU factorization is attempted on a non-square matrix A. The lu function requires the input matrix to be square. Make sure to provide a square matrix to the lu function to avoid this error.

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