### A MATLAB Algorithm for a Pentadiagonal Matrix

Problem Statement: Develop a computer algorithm for the pentadiagonal matrix with band of b and solve following equations as well.

MATLAB Program to Solve a Pentadiagonal Matrix with a band b:

% A function to solve equations which coefficients have form of penta
% diagonal matrix with band b

function x=penta_diagonal(A,b)

[M,N]=size(A);
% Dimension checking

if M~=N

error('Matrix must be square');

end

if length(b)~=M

error('Matrix and vector must have the same number of rows');

end

x=zeros(N,1);

if A==A'    % Matrix symmetry checking

% Band Extractions

d=diag(A);
f=diag(A,1);
e=diag(A,2);

disp(d);
disp(f);
disp(e);

alpha=zeros(N,1);
gamma=zeros(N-1,1);
delta=zeros(N-2,1);
c=zeros(N,1);
z=zeros(N,1);

% Factor A=LDL'

alpha(1)=d(1);
gamma(1)=f(1)/alpha(1);
delta(1)=e(1)/alpha(1);

alpha(2)=d(2)-f(1)*gamma(1);
gamma(2)=(f(2)-e(1)*gamma(1))/alpha(2);
delta(2)=e(2)/alpha(2);

for k=3:N-2

alpha(k)=d(k)-e(k-2)*delta(k-2)-alpha(k-1)*gamma(k-1)^2;
gamma(k)=(f(k)-e(k-1)*gamma(k-1))/alpha(k);
delta(k)=e(k)/alpha(k);
end

alpha(N-1)=d(N-1)-e(N-3)*delta(N-3)-alpha(N-2)*gamma(N-2)^2;
gamma(N-1)=(f(N-1)-e(N-2)*gamma(N-2))/alpha(N-1);
alpha(N)=d(N)-e(N-2)*delta(N-2)-alpha(N-1)*gamma(N-1)^2;

% Updating Lx=b, Dc=z

z(1)=b(1);
z(2)=b(2)-gamma(1)*z(1);

for k=3:N
z(k)=b(k)-gamma(k-1)*z(k-1)-delta(k-2)*z(k-2);
end

c=z./alpha;

% Back Substitution L'x=c

x(N)=c(N);
x(N-1)=c(N-1)-gamma(N-1)*x(N);

for k=N-2:-1:1

x(k)=c(k)-gamma(k)*x(k+1)-delta(k)*x(k+2);
end

end

x;
end

Program Outputs:

ans =

0.673740053050398
-1.021220159151194
1.389920424403183
-1.148541114058356
2.055702917771884
-2.018567639257295