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