Code covered by the BSD License

### Greg von Winckel (view profile)

23 Mar 2004 (Updated )

Solves symmetric and asymmetric pentadiagonal systems.

x=pentsolve(A,b)
```function x=pentsolve(A,b)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% pentsolve.m
%
% Solve a pentadiagonal system Ax=b where A is a strongly nonsingular matrix
%
% If A is not a pentadiagonal matrix, results will be wrong
%
% Reference: G. Engeln-Muellges, F. Uhlig, "Numerical Algorithms with C"
%               Chapter 4. Springer-Verlag Berlin (1996)
%
% Written by Greg von Winckel 3/15/04
% Contact: gregvw@chtm.unm.edu
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

[M,N]=size(A);

% Check dimensions
if M~=N
error('Matrix must be square');
return;
end

if length(b)~=M
error('Matrix and vector must have the same number of rows');
return;
end

x=zeros(N,1);

% Check for symmetry
if A==A'    % Symmetric Matrix Scheme

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

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;

% Update 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;

% Backsubstitution 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

else        % Non-symmetric Matrix Scheme

% Extract bands
d=diag(A);
e=diag(A,1);
f=diag(A,2);
h=[0;diag(A,-1)];
g=[0;0;diag(A,-2)];

alpha=zeros(N,1);
gam=zeros(N-1,1);
delta=zeros(N-2,1);
bet=zeros(N,1);

c=zeros(N,1);
z=zeros(N,1);

% Factor A=LR
alpha(1)=d(1);
gam(1)=e(1)/alpha(1);
delta(1)=f(1)/alpha(1);
bet(2)=h(2);
alpha(2)=d(2)-bet(2)*gam(1);
gam(2)=( e(2)-bet(2)*delta(1) )/alpha(2);
delta(2)=f(2)/alpha(2);

for k=3:N-2
bet(k)=h(k)-g(k)*gam(k-2);
alpha(k)=d(k)-g(k)*delta(k-2)-bet(k)*gam(k-1);
gam(k)=( e(k)-bet(k)*delta(k-1) )/alpha(k);
delta(k)=f(k)/alpha(k);
end

bet(N-1)=h(N-1)-g(N-1)*gam(N-3);
alpha(N-1)=d(N-1)-g(N-1)*delta(N-3)-bet(N-1)*gam(N-2);
gam(N-1)=( e(N-1)-bet(N-1)*delta(N-2) )/alpha(N-1);
bet(N)=h(N)-g(N)*gam(N-2);
alpha(N)=d(N)-g(N)*delta(N-2)-bet(N)*gam(N-1);

% Update b=Lc
c(1)=b(1)/alpha(1);
c(2)=(b(2)-bet(2)*c(1))/alpha(2);

for k=3:N
c(k)=( b(k)-g(k)*c(k-2)-bet(k)*c(k-1) )/alpha(k);
end

% Back substitution Rx=c
x(N)=c(N);
x(N-1)=c(N-1)-gam(N-1)*x(N);

for k=N-2:-1:1
x(k)=c(k)-gam(k)*x(k+1)-delta(k)*x(k+2);
end

end

```