%Polinomio de interpolaci�n de Newton... 533
format short

formula_interpolacion_lineal = 'f(x0) + ((f(x1) - f(x0)) * x - x(0))/(x1 - x0)';

formula_interpolacion_cuadratica = 'b0 + b1(x-x0)+b2(x-x0)(x-x1)';

n = 4;
evaluado_en = 2;
fx = 'log(x)';
func = str2func(['@(x) ' vectorize(fx)]);
puntos  = [1 4 6 5];
evals   = zeros(1, n - 1);

for c=1:n
   evals(c) = func(puntos(c));
end

N=length(puntos)-1;

M=NaN(N+1,N+2);
M(:,1)=puntos;
M(:,2)=evals;
for i=2:N+1
    for j=i:N+1
    M(j,i+1)=(M(j,i) - M(j-1,i)) / (M(j,1)   - M(j-i+1,1)  );
    end
end

% Creamos un bucle para completar hasta el polinomio de n-�simo grado
vals = '';
for c=0:n-1
    other = '';
    for j=0:c-1
       other = strcat(other, '*(x-x', string(j), ')');
    end
    vals = strcat(vals, 'b', string(c), other);
    if c ~= n-1
       vals = strcat(vals, '+');
    end
end

%Remplazamos los valores calculados en la funcion final.
for i=1:n
    for j=1:n+1
        vals = strrep(vals, strcat('b',string(i-1)), string(M(i,i+1)));
        vals = strrep(vals, strcat('x',string(i-1)), string(puntos(i)));
    end
    vals = strrep(vals, '+-', '-');
end

func_final = str2func(strcat("@(x)", vals));

disp(strcat('La ecuaci�n de grado ', " ", string(n), " es: ", vals));

val_real = func(2);
val_aprox = func_final(2);

disp(strcat("El error relativo es: ", string((val_real-val_aprox)*100/val_real), "%"));

fplot(matlabFunction(str2sym(fx)), 'r--o');
hold on
fplot(matlabFunction(str2sym(vals)), [0,n], 'b--*');
hold off