UNM-MATLAB kolokvijum 22.12.2018.- GRUPA 3:  http://www.matf.bg.ac.rs/p/files/82-matlab_klk_grupa_3.pdf

%podaci.m
X = [0.5, 1, 1.5, 2, 2.5, 3];
f=@(x) cos(x);

%Prvi zadatak
%poli.m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function L=poli(n,a)

L0=[0 1];
L1=[-1 1+a];

for j=1:n-1
    L2=(conv(L1,[-1 2*j+a+1])-conv(L0,[0 j+a]))/(j+1);
    L0=[0 L1];
    L1=L2;
end

L=L2;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Drugi zadatak
%skrati.m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [T, maks]=skrati(A,k)
n=size(A,1);
T=zeros(n);
iter=1;
for j=1:n
    suma=0;
    for i=1:n-j+1
        suma=suma+abs(A(i,j));
    end
    if suma<=k*(n-j+1)
            for i=1:n
                T(i,iter)=A(i,j);
            end
        iter=iter+1;
    end
end
T=T(1:n,1:iter-1);
maks=max(T,[],2);
maks=maks';
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Treci zadatak
%Njutn1.m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function y=Njutn1(x,tol)
podaci;
n=length(X);
kr=zeros(n,n-1);
Y=f(X);
for i=1:n-1
    kr(i,1)=Y(i+1)-Y(i);
end
for j=2:n-1
    for i=1:n-j
        kr(i,j)=kr(i+1,j-1)-kr(i,j-1);
    end
end

disp([kr]);

maks=max(abs(kr),[],1);

y=Y(1);
q=(x-X(1))/(X(2)-X(1));
Q=q;
for i=1:n-1
   if abs(maks(i))>power(2,i)*tol
    y=y+q*kr(1,i)/factorial(i);
    q=q*(Q-i);
    end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Cetvrti zadatak
%koef.m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function C=koef(L,p,a,b)
n=length(L);
M=zeros(n);
v=zeros(1,n);
for i = 1:n
    for j = 1:n
        M(i,j)=power(L(j),i-1);
    end
    t=@(x) p(x).*(x.^(i-1));
    v(i)=integral(t,a,b);
end
v=v';
C=M\v;

x=linspace(0,1);
y=p(x);
plot(x,y.^2,'b')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%