clear all; close all; clc;

% solving the bvp example with direct solve
% solves eq. 5.5.3 from the notes
N=100;
alpha=1;
beta=1;
t0=0;
tN=10;

t=linspace(t0,tN,N+1);
dt=t(2)-t(1);
p=exp(-(t.^2));
q=cos(t);
r=sin(3*t)+cos(t);

for i=1:N-1
    A(i,i)=2+(dt^2)*q(i+1); 
end
% we use q(i+1) instead of q(i) because the vector in the notes 
% starts with q0
for i=1:N-2
    A(i,i+1)=-1+(dt/2)*p(i+1); %upper diagonal
    A(i+1,i)=-1-(dt/2)*p(i+2); %lower diagonal
end

for i=2:N-2
    b(i)=-dt^2*r(i+1);
end
b(1)=-dt^2*r(2)+(1+dt*p(2)/2)*alpha;
b(N-1)=-dt^2*r(N)+(1-dt*p(N)/2)*beta;
b=b.';

% spy the matrix
spy(A)
y=A\b;
Y(2:100)=y;
Y(1)=alpha;
Y(N+1)=beta;


% solving an eigenvalue problem
clear all; close all; clc;
n=200;
x=linspace(-1,1,n);
dx=x(2)-x(1);

n0=5;
m=n-2;
e=ones(m,1);
A=spdiags([e (-2+dx^2*n0)*e e],-1:1, m, m);
[V,D]=eigs(A);

Y1(2:n-1)=V(:,1);
Y1(1)=0;
Y1(n)=0;

Y2(2:n-1)=V(:,2);
Y2(1)=0;
Y2(n)=0;

Y3(2:n-1)=V(:,3);
Y3(1)=0;
Y3(n)=0;

Y4(2:n-1)=V(:,4);
Y4(1)=0;
Y4(n)=0;