What Secrets the Bisection Method Hides?

[Namir]

(06-01-2018 06:38 PM)Claudio L. Wrote:  

(05-31-2018 11:40 PM)Namir Wrote:  The paper shows different cases for w1+w2 (using manual selection):

1) Sum is under 2. I just chose to assign 1 to one weight and calculate teh other weight as sum - 1. I could have used different numbers.
2) Sum is equal to 2, but w1 <> 1 and w2 <> 1.
3) Sum > 2 and w1 and w2 are selected to enhance the advantageous root-bracketing end.

Not trying to be a pain, just observing that those 3 cases are one and the same, simply by multiplying numerator and denominator by the same value, you'll always arrive to:

((w1/w2)*A+B) / (1+w1/w2)

because there is one single independent parameter. For example, case 2) above is the same as case 1) if you multiply numerator and denominator of a generic case 1) by (2/Sum). You get a w1' and w2' such that their sum is exactly 2 (and it represents the exact same function). Just multiplying by a larger value would change it into case 3). They are all numerically equivalent, one and the same.
The magnitude of w1+w2 doesn't really define the problem as much as the relationship between w1 and w2, hence my suggestion to use that as a parameter.

(05-31-2018 11:40 PM)Namir Wrote:  You have total freedom to select w1 and w2 as long as it obeys:

C = (w1 * A + w2 * B) / (w1 + w2)

The relative values of w1 and w2 that you select will influence convergence rate, which of course greatly depends on the f(x) and the tolerance value for the refined guess for the root.

Namir

I agree with your last sentence, it's the relative values of w1 and w2 that matter. I disagree with the sentence above the formula, as you don't have total freedom to choose w1 and w2. If you choose a second set w1' and w2' such that:

w1' = k * w1
w2' = k * w2

They will produce the exact same function C for all k. You can simply replace it in the formula to see how the k gets canceled out.

I think you have a good point in trying to improve it by getting it closer to doing a linear interpolation, but without adding the extra computing time required by a true interpolation (is like doing a "guessed" interpolation without actually interpolating anything). I just thought the choice of parameter was a little confusing to me.
Other than that, it's a great paper, an eye opener.

Yes, your approach works very well too.

Namir