diff options
Diffstat (limited to 'homeworks/hw-8.org')
| -rw-r--r-- | homeworks/hw-8.org | 30 |
1 files changed, 25 insertions, 5 deletions
diff --git a/homeworks/hw-8.org b/homeworks/hw-8.org index 42d9dac..f4f4ebd 100644 --- a/homeworks/hw-8.org +++ b/homeworks/hw-8.org @@ -4,14 +4,34 @@ #+LATEX: \setlength\parindent{0pt} #+OPTIONS: toc:nil -TODO: Update LIZFCM org file with jacobi solve, format_matrix_into, rand +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 -A problem arises when using the Jacobi method to solve for the previous population -distribution, $n_k$, from $Ln_{k} = n_{k+1}$, because a Leslie matrix is not diagonally -dominant and will cause a division by zero. Likewise, we cannot factor it into $L$ -and $U$ terms and apply back substitution because pivot points are zero. +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 + |