what's the difference

I want to piecewise plot a interlolation function. When I input the nodes like this

X = -1:0.1:1;
N = length(X);

the graph is continuous like this

However When I input the nodes like this :

X = zeros(1,21);
N = 21;
for i = 1: 21
X(i) = -1 + 2*(i-1)/(N-1);

there is an obvious gap in the graph like this :

I suppose these two codes are exactly the same, why the graphs are different??

the codes are as follows:

tic
clc
clear
f = @(x)1./(1+25.*x.^2);
syms x 
f_dri = diff(f(x));
%X = -1:0.1:1;
%N = length(X);
 X = zeros(1,21);
 N = 21;
    F = zeros(1, N);
    F_dri = zeros(1, N);
    for i = 1 : N
       X(i) = -1 + 2*(i-1)/(N-1);
        F(i) = f(X(i));
        F_dri(i) = subs(f_dri, x, X(i));
    end
    for j = 1 : N-1
        M = zeros(4,4);
        M(1,1) = F(j);
        M(2,1) = F(j);
        M(3,1) = F(j+1);
        M(4,1) = F(j+1);
        M(2,2) = F_dri(j);
        M(3,2) = (F(j+1)-F(j))/(X(j+1)-X(j));
        M(4,2) = F_dri(j+1);
        M(3,3) = (M(3,2)-M(2,2))/(X(j+1)-X(j));
        M(4,3) = (M(4,2)-M(3,2))/(X(j+1)-X(j));
        M(4,4) = (M(4,3)-M(3,3))/(X(j+1)-X(j));
    f_inp = M(1,1)+(x-X(j))*M(2,2)+(x-X(j))^2*M(3,3)+(x-X(j+1))*(x-X(j))^2*M(4,4);
    a = X(j):0.01:X(j+1);
    b = subs(f_inp, x, a);
    G = plot(a, b, 'r', 'LineWidth', 2);
    hold on 
    end
     saveas(gcf, 'pic_2', 'jpg'); 
% r = -1:0.01:1;
% f_ori = f(r);
% G(5) = plot(r, f_ori, 'k', 'LineWidth', 2);
legend(G,'N=21');
toc

Answer 1

[Walter Roberson]

X(13) is eps/4 less than 0.2 . When you use X(j):0.01:X(j+1) that is just enough that the previous plot ends at 0.19 instead of 0.20

Always remember that when you calculate your endpoints using floating point numbers that they will usually not be **exactly* nice multiples of 1/10 or 1/100 .

[zilai si]

thanks, i undersatand. But how to solve such problems?

[Walter Roberson]

You have a fundamental mismatch of expecting that breaking up the range 0 to 1 into (N-1) subdivisions (N total points including both endpoints) will give you ranges that are nicely divisible into increments of 0.01

Instead of using X(j):0.01:X(j+1) consider using a = linspace(X(j), X(j+1), 11)