Iteration for convergence in Matlab without using a while loop

Question

I have to iterate a process where I have an initial guess for the Mach number (M0). This initial guess will give me another guess for the Mach number by using two equations (Mn). Eventually, i want to iterate this process untill the error between M0 and Mn is small. I have the following piece of code and it actually works well with a while loop.

However, I am afraid that the while loop will take many iterations and computational time for certain inputs since this will be part of a bigger code which most likely will give unfeasible inputs for the while loop.

Therefore my question is the following. How can I iterate this process within Matlab without consulting a while loop? The code that I am implementing now is the following:

%% Input
gamma = 1.4;
theta = atan(0.315);
cpi = -0.732;
%% Loop
M0 = 0.2; %initial guess 
Err =  100;
iterations = 0;

while Err > 0.5E-3  
B = (1-(M0^2)*(1-M0*cpi))^0.5; 
Mn = (((gamma+1)/2) * ((B+((1-cpi)^0.5)*sec(theta)-1)^2/(B^2 + (tan(theta))^2)) - ((gamma-1)/2) )^-0.5;
Err = abs(M0 - Mn);
M0 = Mn;
iterations=iterations+1;
end

disp(iterations) disp(Mn)

Many thanks

Google tells me most everyone doing convergence iterations in matlab is using a while loop. You might want to add another condition to your while loop, namely a maximum number of iterations: "while Err > 0.5E-3 && iterations<1e3" or however many iterations you want. — Adriaan, Aug 23 '15 at 13:03

Chris · Answer 1 · 2015-08-23T13:31:21.153

I think you need to clarify the issue. If it the issue you want to solve is that some inputs take a long time to calculate, it is not the while loop that takes the time, it is the execution of the code multiple times that causes it. Any method that loops through will be restricted by the time the block of code takes to execute multiplied by the number of iterations required to converge.

You can introduce something to stop at a certain number of iterationtions, conceptually:

While ((err > tolerance) && (numIterations < limit))

If you want an answer which does not require iterating over the code, this is akin to finding a closed form solution, and I suspect this does not exist.

Edit to add: by not exist I mean in a practical form which can be implemented in a more efficient way then iterating to a solution.

score 0 · Accepted Answer · answered Aug 23 '15 at 14:11

Since M0 is calculated in each iteration and you have trigonometric functions, you cannot use another way than iteration structures (i.e. while).

If you had a specific increase or decrease at M0, then you could initialize a vector of M0 and do vector calculations for B and Err.

But, with sec and tan this is not possible.

Another wat would be to use the parallel processing. But, since you change the M0 at each iteration then you cannot use the parfor loop.

As for a for loop, in MATLAB you need an array for for "command" argument (e.g. 1:10 or 1:length(x) or i = A, where A = 1:10 or A = [1:10;11:20]). Since you evaluate a condition and depending on the result of the evaluation you judge if you continue the execution or not, it seems that the while loop (or do while in another language) is the only way to go.

Iteration for convergence in Matlab without using a while loop

2 Answers2