MTH3051 Introduction to Computational Mathematics - User Web ...

School of Mathematical Sciences
Monash University
4.2.6 A Matlab program
Here is a rough Matlab program that does the job. You might like to look at this code
very carefully.
x_old = 0.5; % initial guess
loop = 0; % set loop to zero
loop_max = 50;
% maximum number of iterations
small_number = 0.0001; % target accuracy
looping = (loop < loop_max);
while looping
% start of iterations
x_new = exp(-x_old);
f_old = x_old - exp(-x_old);
f_new = x_new - exp(-x_new);
loop = loop + 1;
if loop > loop_max
looping = false;
end
if abs(x_new-x_old) < small_number
looping = false;
end
if abs(f_new) < small_number
looping = false;
end
x_old = x_new;
disp([loop x_new f_new ]);
% next iteration
% too many iterations, exit
% x values converged, exit
% f values very small, exit
% prepare for next iteration
% display current approximation
end;
4.3 Newton-Raphson iteration
Once again we have the problem of finding x such 0 = f(x). And once again we will
generate a sequence x 1 , x 2 , x 3 , · · · which we hope will converge to x.
How? Okay, let’s suppose we have a guess, call it x n and we wish to create a new
(improved) guess x n+1 . This new guess will only be an approximation to x. Let δx be
the error in x n , that is
x n = x + δx
Our game now is to compute (or estimate) δx.
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