41C/CV root finders

[Dieter]

(05-28-2015 01:54 PM)Ángel Martin Wrote:  With the new initial estimation there's a marginal improvement - two iteration less if compared against e-8, or just one less if using the e-9 criteria.

This sounds like there is something wrong with f(i) and/or f'(i). Could you post these two formulas here?

A Newton-style solver should converge roughly quadratically, and the version I use does so. Here are the iteration results for the sample case. "err_abs" is the applied correction (delta_i) in that iteration step, and "err_rel" is the quotient of this and the current approximation, i.e. delta_i : i.

Code:

Initial estimate: 0,08571428572

     err_abs =     err_rel =   new i =
it#  f(i)/f'(i)    err_abs:i   i - delta_i
-------------------------------------------
(1)  -6,23 E-02   -4,21 E-01   0,1480341636
(2)   3,67 E-03    2,54 E-02   0,1443681501
(3)   9,44 E-06    6,54 E-05   0,1443587134
(4)   2,49 E-10    1,72 E-09   0,1443587132
-------------------------------------------

This nicely shows the quadratic convergence: the error is order of 10^–1, 10^–2, 10^–5 and finally 10^–9,
Here the program quits since 1,72 E-9 is below the error limit of 5 E–8.

(05-28-2015 01:54 PM)Ángel Martin Wrote:  It occured to me the reason for the larger number of loops could be that I check for f(i)/f '(i) to meet that criteria, and not for f(i) alone.

Sure, this is what I do as well. f(i)/f'(i) is the correction term, the mentioned "delta i". My 10-digit program quits if delta_i : i drops below 5E–8. Due to the quadratic convergence one can expect the result to be exact to 12–14 digits, plus/minus a slight unavoidable error in the last digit(s) due to roundoff.

(05-28-2015 01:54 PM)Ángel Martin Wrote:  This is more demanding when f '(i)<1, as the quotient would be larger than the numerator and thus a bit counterproductive...

Are there really cases where f'(i) is less than 1? Usually f'(i) is of similar magnitude as the payments, i.e. around 100 or 1000. For the sample case I get values around 4000 or 5000. You seem to use a different definition of the TVM function and its derivative. Sounds interesting, so again I would like to ask for the formulas you use.

BTW, thank you for the MOD files. I will use them with V41 and see what's in there. ;-)

Dieter