Computer demonstrations of solving dense linear systems and solving sparse linear systems :

---

1. Download and run the Matlab codes in
   http://www.cse.ucsc.edu/~hongwang/Codes/Dense_linear_sys
   "no_pivoting.m" in directory "well_conditioned" demonstrates that solving a well-conditioned system using LU decomposition without pivoting may result in large numerica errors.
   "pivoting.m" in directory "well_conditioned" demonstrates that solving a well-conditioned system using LU decomposition with pivoting yields good results.
   "perturbation.m" in directory "well_conditioned" demonstrates that for a well-conditioned system small perturbations in A and b result in a small change in x.
   "perturbation.m" in directory "ill_conditioned" demonstrates that for an ill-conditioned system small perturbations in A and b result in a large change in x.

2. Download and run the Matlab codes in
   http://www.cse.ucsc.edu/~hongwang/Codes/Sparse_linear_sys
   "sd_0.m" and "sd_1D.m" in directory "Steepest_Descent" use the steepest descent method to solve A*u=b.
   • "sd_0.m" solves a simple 2x2 linear system.
   • "sd_1D.m" solves a 100x100 tridiagonal linear system.
   "cg_0.m", "cg_1D.m" and "cg_2D.m" in directory "Conjugate_Gradient" use the conjugate gradient method to solve A*u=b.
   • "cg_0.m" solves a simple 2x2 linear system.
   • "cg_1D.m" solves a 100x100 tridiagonal linear system.
   • "cg_2D.m" solves a 10000x10000 sparse linear system obtained from discretizing a two-dimensional Poisson equation with N=101.

---