Access previous iteration in while loop to derive stopping criterion for differential equation correction-prediction method

Hi all,

I have been trying to write a prediction-correction method for matrix differential equations, however I have been unsuccessful, as I do not know how to access the previous iteration in the while loop in order to derive the stopping criterion. In specific, my method is the following:

Lets assume:

Then we make an initial guess via Euler:

Next we improve the intial guess in an iterative process which stops if , where e is a user defined tolerance:

Hence, the above implies that the previous step has to be accessed in a while loop such that if the condition for e is achieved, then the iteration stops. I have attempted to use the answer from this question, however I must be doing something wrong as there is no improvement in the answer as I change the tolerance:

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EDIT: The actual equation that I have to solve is way more complex than the differential equation in the code below. I just used the differential equation below in order to make it easier for myself to identify a solution to my problem.

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clear 
clc

% Inputs
tmax = 5 ;              % Max solution time
y0 = 0 ;                % Initial conditions
Dt = 0.1 ;              % Time step (for Euler)
e = 1e-3 ;              % Tolerance (for Euler) 
odefun = @(t, y) 2*t ;  % Function to be integrated 

% Solution

    % ODE45
    [tODE, yODE] = ode45(odefun, [0 tmax], y0) ;

    % Euler
    tEUL = transpose(0:Dt:tmax) ;
    yEUL = [y0 ; zeros(length(tEUL)-1, 1)] ; 
    
    for i = 2:length(tEUL) 
        
        yEUL(i) = yEUL(i-1) + Dt * odefun(Dt*i, yEUL(i-1)) ;
        
        y_pred = yEUL(i) ; 
        y_corr = Inf ;
        
        while true
            
            y_pred = yEUL(i-1) + Dt / 2 * (odefun(Dt*i, yEUL(i-1)) + odefun(Dt*(i+1), y_pred)) ;
            
            if abs(y_pred - y_corr) < e
                break
            end
            
            y_corr = y_pred ; 
            
        end
        
        yEUL(i) = y_corr ;
                        
    end

% Plot
figure
plot(tODE, yODE, '-ob', tEUL, yEUL, '-or')
legend('ODE45', 'EULER', 'Location', 'West')
grid on

Thanks for your help in advance,

KMT.

Answer 1

[Bob Thompson]

In MATLAB, you cannot access the previous iteration of a loop directly. If you want to utilize that information you need to either index your results, or record the previous iteration as a separate variable.

% Index method
i = 0;
while true
    i = i + 1;
    y_pred(i) = ...
    
    if abs(y_pred(i)-y_corr) < e
        break
    end
    y_corr = y_pred(i);
end

% Variable method
while true
    i = i + 1;
    y_pred = ...
    
    if abs(y_pred-y_corr) < e
        break
    end
    y_prev = y_corr;
    y_corr = y_pred;
end