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Asked by David on 30 Sep 2013

Hello. I'm a student whose been working on a program to find the maxs or mins of a given function with in a specified range using an iterative function that cannot use the min or max function in as few iterations as possible. To this end I have used Halley's method for finding zeros of the derivative of the function, and from there determine the minimum. The function works to my satisfaction, but if an equation takes more than 4 cycles of the loop to complete, the function takes substantially longer. Any Help would be greatly appreciated. The code is:

function [X_final,Y_final]=Lab5_Real_1_4(equation) x=0; syms x;

Derivative=diff(equation);

Derivative_2=diff(diff(equation));

Derivative_3=diff(Derivative_2);

Guess1=10-(2*(subs(Derivative,x,10))*(subs(Derivative_2,x,10)))/(2*((subs(Derivative_2,x,10))^2)-((subs(Derivative,x,10))*(subs(Derivative_3,x,10))));

Guess2=4-(2*(subs(Derivative,x,4))*(subs(Derivative_2,x,4)))/(2*((subs(Derivative_2,x,4))^2)-((subs(Derivative,x,4))*(subs(Derivative_3,x,4))));

Guess3=-10-(2*(subs(Derivative,x,-10))*(subs(Derivative_2,x,-10)))/(2*((subs(Derivative_2,x,-10))^2)-((subs(Derivative,x,-10))*(subs(Derivative_3,x,-10))));

iterations=0;

Iterations_required=1;

Guess1_old=10;

Guess2_old=4;

Guess3_old=-10;

while ((iterations<Iterations_required && iterations<=5));

Guess1=Guess1-(2*(subs(Derivative,x,Guess1))*(subs(Derivative_2,x,Guess1)))/(2*((subs(Derivative_2,x,Guess1))^2)-((subs(Derivative,x,Guess1))*(subs(Derivative_3,x,Guess1))));

Guess2=Guess2-(2*(subs(Derivative,x,Guess2))*(subs(Derivative_2,x,Guess2)))/(2*((subs(Derivative_2,x,Guess2))^2)-((subs(Derivative,x,Guess2))*(subs(Derivative_3,x,Guess2))));

Guess3=Guess3-(2*(subs(Derivative,x,Guess3))*(subs(Derivative_2,x,Guess3)))/(2*((subs(Derivative_2,x,Guess3))^2)-((subs(Derivative,x,Guess3))*(subs(Derivative_3,x,Guess3))));

if (abs(Guess1_old-Guess1)>.001 || abs(Guess2_old-Guess2)>.001 ||abs(Guess3_old-Guess3)>.001)

Iterations_required=Iterations_required+1;

end Guess1_old=Guess1;

Guess2_old=Guess2;

Guess3_old=Guess3;

iterations=iterations+1;

end cycles=iterations

X_array=[Guess1,Guess2,Guess3];

Trial1=subs(equation,x,Guess1);

Trial2=subs(equation,x,Guess2);

Trial3=subs(equation,x,Guess3);

Y_Array=[Trial1, Trial2, Trial3];

Y_Length=length(Y_Array);

Y_sorted=sort(Y_Array);

Y_min=Y_sorted(end-(Y_Length-1));

Y_final=vpa(Y_min)

X_index=find(Y_Array==Y_min);

X_final=vpa(X_array(X_index));

X_final=X_final(1)

*No products are associated with this question.*

Answer by Jan Simon on 30 Sep 2013

The main strategy to increase the speed is to avoid repeated calculations. In your case, e.g.

subs(Derivative_2,x,Guess3)

Is evaluated twice. So better use a temporary variable:

c1 = subs(Derivative_2,x,Guess3);

The same happens for other `subs()` calls also and because this function is the bottleneck of your program, obtaining a speedup of a factor or almost 2 seems possible.

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