lizfcm/homeworks/hw-8.org

#+TITLE: Homework 7
#+AUTHOR: Elizabeth Hunt
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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.

* Question Four, Five