1.7 KiB
Homework 7
TODO: Update LIZFCM org file with jacobi solve
Question One
See UTEST(jacobi, solve_jacobi) in test/jacobi.t.c and the entry
Jacobi -> solve_jacobi in the LIZFCM API documentation.
Question Two
We cannot just perform the Jacobi algorithm on a Leslie matrix since
it is obviously not diagonally dominant - which is a requirement. It is
certainly not always the case, but, if a Leslie matrix $L$ is invertible, we can
first perform gaussian elimination on $L$ augmented with $n_{k+1}$
to obtain $n_k$ with the Jacobi method. See UTEST(jacobi, leslie_solve)
in test/jacobi.t.c for an example wherein this method is tested on a Leslie
matrix to recompute a given initial population distribution.
In terms of accuracy, an LU factorization and back substitution approach will always be as correct as possible within the limits of computation; it's a direct solution method. It's simply the nature of the Jacobi algorithm being a convergent solution that determines its accuracy.
LU factorization also performs in order $O(n^3)$ runtime for an $n \times n$ matrix, whereas the Jacobi algorithm runs in order $O(k n^2) = O(n^2)$ but with the con that $k$ is given by the convergence criteria, which might end up worse in some cases, than LU.
Question Three
See UTEST(jacobi, gauss_siedel_solve) in test/jacobi.t.c which runs the same
unit test as UTEST(jacobi, solve_jacobi) but using the
Jacobi -> gauss_siedel_solve method as documented in the LIZFCM API reference.