How can I optimize bisection-method for polynomial root finding in Java?

Question

double findaroot(double x1, double x2){ //finds the root between two values
    double gap = Math.abs(x1 - x2); //find the initial interval
    while(gap > INTERVAL) { //check for precision           
        gap = gap / 2; //halve the interval
        double x3 = x1 + gap;
        if (f(x3) == 0) { //check for symmetry
            return x3;
        } else if (Math.signum(f(x1)) == Math.signum(f(x3))){
            x1 = x3; //redefine the interval
        } else {
            x2 = x3; //redefine the interval
        }
        findaroot(x1, x2); //call again
    }
    return (x1 + x2) / 2; //return the mean
}

I am trying to find a solution for f(x)=-21x^2+10x-1 in the intervals (-143, 0.222222). The guidelines state that I should implement a bisection method to solve this. Currently this method works fine for 8 of the 10 test cases that I must pass, but it gives a "Time-limit exceeded" error for the aforementioned values. It takes 15 seconds to approximate the root given a precision level of at least "0.000001" between the intervals.

I'm not sure how I can make this more efficient without changing the method. I have already implemented Horner's method to calculate the function because Math.pow(x1, x2) was taking too long.

Can you please tell what's the point for making the recursive call `findaroot(x1, x2);` ? You aren't using its result. — Rafi Kamal, Sep 23 '13 at 10:27
I am so foolish, yes you are right, thank you so much. This is the first time we're supposed to use recursion and I thought that the recursive call was needed for it to work. — user2806648, Sep 23 '13 at 10:32
This looks allot like your university's homework assignment which is due tommorow. :) — NoCode, Sep 23 '13 at 17:28

Rafi Kamal · Accepted Answer · 2013-09-23T10:57:19.963

Just remove the line findaroot(x1, x2);. You are not using the result of this recursive function call anyway.

EDIT: This is the recursive version of your code (not tested)

double findaroot(double x1, double x2){ //finds the root between two values
    double gap = Math.abs(x1 - x2); //find the initial interval
    if (gap > INTERVAL) { //check for precision           
        gap = gap / 2; //halve the interval
        double x3 = x1 + gap;
        if (f(x3) == 0) { //check for symmetry
            return x3;
        } else if (Math.signum(f(x1)) == Math.signum(f(x3))){
            x1 = x3; //redefine the interval
        } else {
            x2 = x3; //redefine the interval
        }
        return findaroot(x1, x2);
    }
    else
         return (x1 + x2) / 2; //return the mean
}

isnot2bad · Answer 2 · 2013-12-12T21:43:24.960

0

As others already said: The recursive invocation of findaroot is wrong/not required. This code works for me:

private final int NMAX = 100;

public double solve(Function f, double a, double b, double tolerance) {
    if (a >= b) throw new IllegalArgumentException("illegal interval!");

    double fa = f.value(a);
    if (Math.signum(fa) == Math.signum(f.value(b))) throw new IllegalArgumentException("solution not within interval!");

    for (int n = 0; n < NMAX; n++) {
        double c = (a + b) / 2;

        double fc = f.value(c);
        if ((fc == 0.0) || ((b - a) / 2 < tolerance)) {
            // solution found
            return c;
        }

        // adapt interval
        if (Math.signum(fc) == Math.signum(fa)) {
            a = c;
            fa = fc;
        } else {
            b = c;
        }
    }

    return Double.NaN;
}

edited Dec 12 '13 at 21:43

answered Sep 23 '13 at 10:56

isnot2bad

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For further speedup, define variables fa and fb outside the loop, using them for the check for different sign, and pass the value of fc to fa or fb when shrinking the interval. -- This is not really noticeable, for more visible improvements use Regula falsi with the Illinois modification. – Lutz Lehmann Dec 12 '13 at 18:17
@LutzL You're right! I've modified the code to make use of this optimization. Regula falsi is of course an improvement, but I think out of question as the QA wants to implement bisection method. – isnot2bad Dec 12 '13 at 21:47

How can I optimize bisection-method for polynomial root finding in Java?

2 Answers2