Chapter 11, Problem 28P

A pentadiagonal system with a bandwidth of five can be expressed generally as

$\left[\begin{array}{cccccccc}{f}_1& g_1& h_1& & & & & \\ e_2& f_2& g_2& h_2& & & & \\ d_3& e_3& f_3& g_3& h_3& & & \\ & & \cdot & \cdot & \cdot & & & \\ & & & \cdot & \cdot & \cdot & & \\ & & & & \cdot & \cdot & \cdot & \\ & & & & d_{n-1}& e_{n-1}& f_{n-1}& g_{n-1}\\ & & & & & d_n& e_n& f_n\end{array}\right]\left\{\begin{array}{c}{x}_1\\ x_2\\ x_3\\ \cdot \\ \cdot \\ \cdot \\ x_{n-1}\\ x_n\end{array}\right\}=\left\{\begin{array}{c}{r}_1\\ r_2\\ r_3\\ \cdot \\ \cdot \\ \cdot \\ r_{n-1}\\ r_n\end{array}\right\}$

Develop a program to efficiently solve such systems without pivoting in a similar fashion to the algorithm used for tridiagonal matrices in Sec. 11.1.1. Test it for the following case:

$\left[\begin{array}{ccccc}8& -2& -1& 0& 0\\ -2& 9& -4& -1& 0\\ -1& -3& 7& -1& -2\\ 0& -4& -2& 12& -5\\ 0& 0& -7& -3& 15\end{array}\right]\left\{\begin{array}{c}\begin{array}{l}{x}_1\\ x_2\end{array}\\ x_3\\ x_4\\ x_5\end{array}\right\}=\left\{\begin{array}{c}\begin{array}{l}5\\ 2\end{array}\\ 0\\ 1\\ 5\end{array}\right\}$