normalhuman

function cubic
a = 1992;
b = 2000;
y = [26 25 20 23 28 21 15 20 18];
n = length(y);
h = (b-a)/(n-1);
rhs = (-6/h^2)*[0; diff(y',2); 0];
A = diag(4*ones(1,n))+diag(ones(1,n-1),1)+diag(ones(1,n-1),-1);
A(1, 1:2) = [1 0];
A(n, n-1:n) = [1 0];

normalhuman

function naturalcubic
a = 1995;
b = 2000;
y = [30 25 20 28 15 31];
n = length(y);
h = (b-a)/(n-1);
rhs = (-6/h^2)*[0; diff(y',2); 0];
A = diag(4*ones(1,n))+diag(ones(1,n-1),1)+diag(ones(1,n-1),-1);
A(1, 1:2) = [1 0];
A(n, n-1:n) = [0 1];

normalhuman

function adaptivePL
    t = linspace(0, 7, 1000);          % the default number of points, 100, may be not enough here
    points = refine(t(1), t(end));     % determine the nodes of interpolation 
    plot(points, f(points), 'r-*')     % piecewise linear plot with asterisks for data points
    hold on
    plot(t, f(t))                      % the original function for comparison 
end

function points = refine(x1,x2)
    xm = (x1+x2)/2;

normalhuman

function chebyshev_interpolation
a = -5;
b = 5;
t = linspace(a,b);
plot(t, [chebyshev(a,b,5,t); chebyshev(a,b,10,t); chebyshev(a,b,15,t); chebyshev(a,b,30,t)]);
hold on
plot(t, f(t), 'k--');  % original function in dashed black curve
legend('degree 5', 'degree 10', 'degree 15', 'degree 30', 'function')  % describe the curves
end

normalhuman

function plot_newton
f = @(x) 1./(1+x.^2);   % function to interpolate
x = -5:5;               % nodes of interpolation
y = f(x);
c = coeff(x,y);         % calculate Newton polynomial coefficients
t = linspace(-5,5);
poly = newton(c,x,t);   % evaluate the polynomial 
plot(t,[f(t);poly]);    % plot function and its polynomial
hold on 
plot(x,y,'ro')          % also mark the interpolated values

normalhuman

x = [1 2 3 4]'; % transpose to make a column 
y = [3 1 4 1]'; 
n = length(x);  % number of elements 
M = zeros(n,n); % initialize a matrix n by n
for k=1:n
    M(:,k) = x.^(k-1);    % fill the matrix with powers of x-values
end
coeff = M\y;    % find the coefficients 
t = linspace(1,4);
poly = coeff(n)*ones(size(t));  % prepare for evaluation