Question

assume that the data are stored in variable names x and y. write a matlab code to plot the quadratic spline and along with the data points

x= 1 2 3 4 5 6
y= 40 78 125 256 348 425

Answer 1

% This code solves for quadratic interpolation
x   =   1:6
y   =   [40 78 125 256 348 425]
N   =   length(x)-1;
% The unknowns are 3*N with ao=0 "Linear Spline"
% Filling Matrix from point matching
V   =   [0;zeros(2*N,1);zeros(N-1,1)];
Z   =   zeros(length(V),length(V));
j=1;
f=1;
for i=2:2:2*N    
    Z(i,f:f+2)          =   [x(j)^2 x(j) 1];
    V(i)                =   y(j);
    j                   =   j+1;
    Z(i+1,f:f+2)        =   [x(j)^2 x(j) 1];  
    V(i+1)              =   y(j);
    f                   =   f+3;
end
% Filling Matrix from smoothing condition
j=1;
l=2;
for i=2*N+2:3*N
    
    Z(i,j:j+1)            =   [2*x(l) 1];
    Z(i,j+3:j+4)          =   [-2*x(l) -1];
    j                     =   j+3;
    l                     =   l+1;
end
% Adjusting the value of a1 to be zero "Linear Spline"
Z(1,1)=1;
% Inverting and obtaining the coeffiecients, Plotting
Coeff       =       Z\V;
j=1;
hold on;
for i=1:N
    curve=@(l) Coeff(j)*l.^2+Coeff(j+1)*l+Coeff(j+2);
    ezplot(curve,[x(i),x(i+1)]);
    hndl=get(gca,'Children');
    set(hndl,'LineWidth',2);
    hold on
    j=j+3;
end
scatter(x,y,50,'r','filled')
grid on;
xlim([min(x)-2 max(x)+2]);
ylim([min(y)-2 max(y)+2]);
xlabel('x');
ylabel('y');
title('Quadratic Spline')