%% Problem 5. Method of simple iteration
clear all,format ,clc, format compact
A=[4, 4, 3;3, 7, -1;3, -1, 35];b=[7; 19; -2];
tau=0.08; x=zeros(3,1);
itermax=80;
ni=1; % is the coefficient matrix positive definite?
for i=1:3
 if det(A(1:i,1:i))>0
 else ni=2; break
 end
end
if ni==2
 disp('the coefficient matrix is not positive definite'),
 return
end
check=isequal(A,A');
if check==0
 disp('the coefficient matrix is not symmetric'),
return
end
disp('A is symmetric and positive definite ')

disp('Answer:')
disp(' k_iter=251, X_approx=[x1=1.003,x2=1.999,x3=2.999]')

%% Problem 6 interpolating polynomial
clear all, clc ,close all, format compact, format
xnodes=[2.1, 2.6, 3.1, 3.6, 4.1, 4.6, 5.1, 5.6];
ynodes=[2.37,3.36, 3.97, 4.11, 3.84, 3.30, 2.71, 2.33];
coef=ynodes; % determination of 𝒂𝟏, 𝒂𝟐, …
for k = 2:8
 coef(k:8) = (coef(k:8) - coef(k-1:7))./(xnodes(k:8) - xnodes(1:9-k));
end
syms x
pol= coef(7); % coefficients of the polynomial
for k = 7:-1:1
 pol=pol*(x-xnodes(k))+coef(k);
end
polyn=collect(pol)
coefpol=sym2poly(polyn)
% coefficients of the polynomial (format double)

graph1 = ezplot(polyn,[1,8])
hold on
plot(xnodes,ynodes,'o','LineWidth',3)
hold off
set(graph1,'LineWidth',3,'Color','r','LineStyle','-')
title('Newton interpolating polynomial')
legend('polynomial','nodes')
ylim([1 5]);