Consider a linear system described by a banded coefficient matrix of width 5, or a pentdiagonal system: f1 g1 h1 e2 f2 g2 h2 d3 e3 f3 g3 h3 ? ? ? ? ? ? ? ? ? ? dn ?2 en ?2 fn ?2 gn ?2 hn ?2 dn ?1 en ?1 fn ?1 gn ?1 dn en fn # $ % % % % % % % % % % & ( ( ( ( ( ( ( ( ( ( x1 x2 x3 ? ? ? xn ?1 xn # $ % % % % % % % % % % & ( ( ( ( ( ( ( ( ( ( = r1 r2 r3 ? ? ? rn ?1 rn # $ % % % % % % % % % % & ( ( ( ( ( ( ( ( ( ( Write an algorithm to solve this system that is more efficient than Gauss elimination, in the manner of the algorithm for tridiagonal systems that we discussed. (1) Write a program to implement this algorithm. Use this program to solve the system represented by the pentdiagonal_coefficients.mat and pentdiagonal_constants.mat files available on the course Piazza site. Find the solution. (2) Write a program that can solve the same system using naïve Gauss elimination. Find the solution. (3) Compare the number of operations used to find the solution in part (1) to those used to find the solution in part (2). Explain how this difference will scale with the size of the system. Note: The data files are both in the standard MATLAB .mat format. You can use the load function to read the data in these files into your program.
