[Octave-bug-tracker] [bug #60539] Slow performance of betaincinv.m

[Rik]

Follow-up Comment #2, bug #60539 (project octave):

I instrumented the code to see where some of the hots spots were.  I then ran
with a random 1 million point sample.


y = betaincinv (rand (1e6,1), 1, 3);


In the newton_method() function I put separate timings around the the range
checking code and around the actual objective function calls


    tic;
    x(x(todo) < 0) = eps;
    x(x(todo) > 1) = 1-eps;
    bm1(it) = toc;
    tic;
    res(todo) = -F(x(todo), a(todo), b(todo), y(todo)) ...
                ./ JF (x(todo), a(todo), b(todo));
    bm2(it) = toc;


The bisection() and newton_method() functions are called twice for the lower
tail and upper tail.  The results were


bm_bisect: 1.36663
bm1: 0.0612214
bm2: 1.36306
bm_bisect: 1.50331
bm1: 0.0524464
bm2: 1.32451


Lots to optimize, or not, here.  First, performance of the code to do range
checking is immaterial (25X smaller) compared to objective function
evaluations.

Second, the code performs 10 rounds of bisection followed by 20 rounds of
Newton's Methods.  The bisection starts with initial values of 0, 1 so that
the center of the first round will be 0.5.  Given that Newton's method will
converge faster than bisection, I think we could just skip this step entirely
and start with an initial guess of 0.5 for the solution.  Handily, the
bisection function accepts a parameter for number of iterations.  I set that
to 1 and then re-ran the function.  The calculated output differed slightly
from the original run, but the maximum deviation was 2 eps which I think is
acceptable.



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