Looking for TVM contributions

[robve]

(06-19-2024 03:47 PM)robve Wrote:  I'm not completely happy about the iguess() (it's your guess_i() code by the way), which works fine in END mode, but is not as good in BEGIN mode. It also can return an overflow >100% causing NaN. I can replicate your symbolic derivation and understand it, but inserting a BEGIN mode adjustment that requires i% itself that we want to guess is something to figure out.

After some more experimentation with TVM problems for which the initial guess isn't that good, such as
n=10 pv=-100 pmt=10 fv=<small> for <small>=1e-1 to 1e-99
n=40 pv=900 pmt=-400 fv=-1000 BGN
n=480 pv=100000 pmt=-208.333333333333343 fv=0
and others, the following updated version is a nice improvement:

Code:

double iguess()
{
  double P = pv + b*pmt;
  double F = fv - b*pmt;
  double A = (F + P)/n;
  double B = (F - P) - A;
  double C = (pmt + A)/B;
  double D = (n*n - 1.0)*A*C/B;
  double I = 100.0*C*(D - 3.0)/(D - 1.5);
  if (fabs(I) < 1e-16)
    I = copysign(1e-8, I);
  else if (fabs(I) > 99)
    I = copysign(99, I);
  return I;
}

It is based on Albert's guess_i(). It may not start at a point to approach the root from a side that does not flip the npmt sign (it doesn't affect these root finders). It does appear to offer a fairly accurate starting point. In the Hybrid method the next point we pick after the guess is often quadratically closer to the root using the Newton derivative. That's all we need for some TVM problems as it hits the root exactly with only 1 to 3 npmt evaluations.

Results for Hybrid with the new guess compared to the previous implementation looks good, saving 5 iteration steps total over all tests: 98 instead of 103, not counting the NaN case that is now fixed. Final accuracy is the same.

  tvmperfcomp-hybrid-newguess.pdf (Size: 33.22 KB / Downloads: 2)

- Rob

EDIT: minor edit to correct typo and I=1e-8 that works for Newton when I=1e-9 fails with NaN in Newton.