Array indices must be positive integers or logical values.

nMax = 100; % Maximum value of n
tol = 1e-5; % Tolerance value

% Initialize arrays to store the values of eta
eta1 = zeros(nMax, 1);
eta2 = zeros(nMax, 1);

% Compute the values of eta for different values of n
for n = 1:nMax
    % Compute eta for the first sequence
    if n == 1
        eta1(n) = 0; % Initial condition for eta1
    else
        eta1(n) = (1/(n+1))*(eta1(n-1)/2 + (n-1)/(n+1)*eta1(n-1) + 1/(n+1)*(1/(n*(n+1)))*1*(eta1(n-1)/4));
    end
    
    % Compute eta for the second sequence
    if n == 1
        eta2(n) = 0; % Initial condition for eta2
    else
        eta2(n) = (1/(n+1))*(eta2(n-1)/2 + (n-1)/(n+1)*((eta2(n-1)/2)/2) + 1/(n+1)*(1/(n*(n+1)))*1*(eta2(n-1)/4));
    end
    
    % Check convergence condition
    if norm(eta1(n) - eta1(n-1)) < tol
        break;
    if norm(eta2(n) - eta2(n-1)) < tol
        break; % Convergence reached, stop the loop
    end
    end
end

% Plot the convergence behavior
plot(1:n, eta1(1:n), '-o', 'LineWidth', 2, 'DisplayName', 'Sequence 1');
hold on;
plot(1:n, eta2(1:n), '-o', 'LineWidth', 2, 'DisplayName', 'Sequence 2');
hold off;

% Add labels and legend
xlabel('n');
ylabel('\eta');
title('Convergence Behavior of Sequences');
legend('Location', 'best');

Answer 1

[Dyuman Joshi]

if norm(eta1(n) - eta1(n-1)) < tol
        break;
    if norm(eta2(n) - eta2(n-1)) < tol
        break; % Convergence reached, stop the loop
    end
    end

When n==1, n-1==0, which can not be used as an index in MATLAB.

Modified code -

nMax = 100; % Maximum value of n
tol = 1e-5; % Tolerance value

% Initialize arrays to store the values of eta
eta1 = zeros(nMax, 1);
eta2 = zeros(nMax, 1);

%%eta1 and eta2 are preallocated as zeros array, thus assigning 0 for n==1 is redundant
%%Start the loop from n==2, it also takes care of the error 
for n = 2:nMax
    
    %%Combine the conditions
    % Compute eta for the first sequence
        eta1(n) = (1/(n+1))*(eta1(n-1)/2 + (n-1)/(n+1)*eta1(n-1) + 1/(n+1)*(1/(n*(n+1)))*1*(eta1(n-1)/4));
        eta2(n) = (1/(n+1))*(eta2(n-1)/2 + (n-1)/(n+1)*((eta2(n-1)/2)/2) + 1/(n+1)*(1/(n*(n+1)))*1*(eta2(n-1)/4));

    % Check convergence condition
    
    %%Combine the check for convergence
    if norm(eta1(n) - eta1(n-1)) < tol || norm(eta2(n) - eta2(n-1)) < tol
        break;
    end
end
n
% Plot the convergence behavior
plot(1:n, eta1(1:n), '-o', 'LineWidth', 2, 'DisplayName', 'Sequence 1');
hold on;
plot(1:n, eta2(1:n), '-o', 'LineWidth', 2, 'DisplayName', 'Sequence 2');
hold off;

% Add labels and legend
xlabel('n');
ylabel('\eta');
title('Convergence Behavior of Sequences');
legend('Location', 'best');

However, as the 1st values of eta1 and eta2 are zero, the rest of the values will also be zero, as the value of term n is directly proportional to the value of term (n-1) (observed from the formula/expression above).

Are you sure the inital values of eta1 and eta2 are zero?

Answer 2

[Davide Masiello]

The problem is in the convergence check step, where you try to evaluate eta at n-1 even when n = 1.

Adding n > 1 as additional statement solves the problem. See below.

nMax = 100; % Maximum value of n
tol = 1e-5; % Tolerance value
% Initialize arrays to store the values of eta
eta1 = zeros(nMax, 1);
eta2 = zeros(nMax, 1);
% Compute the values of eta for different values of n
for n = 1:nMax
    % Compute eta for the first sequence
    if n == 1
        eta1(n) = 0; % Initial condition for eta1
    else
        eta1(n) = (1/(n+1))*(eta1(n-1)/2 + (n-1)/(n+1)*eta1(n-1) + 1/(n+1)*(1/(n*(n+1)))*1*(eta1(n-1)/4));
    end
    
    % Compute eta for the second sequence
    if n == 1
        eta2(n) = 0; % Initial condition for eta2
    else
        eta2(n) = (1/(n+1))*(eta2(n-1)/2 + (n-1)/(n+1)*((eta2(n-1)/2)/2) + 1/(n+1)*(1/(n*(n+1)))*1*(eta2(n-1)/4));
    end
    
    % Check convergence condition
    if n > 1 && norm(eta1(n) - eta1(n-1)) < tol
        break;
    if n > 1 && norm(eta2(n) - eta2(n-1)) < tol
        break; % Convergence reached, stop the loop
    end
    end
end
% Plot the convergence behavior
plot(1:n, eta1(1:n), '-o', 'LineWidth', 2, 'DisplayName', 'Sequence 1');
hold on;
plot(1:n, eta2(1:n), '-o', 'LineWidth', 2, 'DisplayName', 'Sequence 2');
hold off;
% Add labels and legend
xlabel('n');
ylabel('\eta');
title('Convergence Behavior of Sequences');
legend('Location', 'best');