Re: New Root Seeking Algorithms



#2

Namir,

Your comparison with Newton's method is wrong.

Newton's method requires that you use f '(x) explicitly. If you don't, what you are doing is really measuring the convergence of fixed point iteration using a function that approximates f '(x). Simply stated, you are comparing your method to the accuracy of the approximation of f '.

The last example you give is for a polynomial with 6 unique roots. If you use the derivative explicitly, an initial guess of 7 will give the root to 10 significant digits in 6 iterations: six calls of f and six calls of f '. If you do one more iteration and you are using sufficient significant digits, you will have the root to 20 significant digits. Your example fails because the inaccuracy of f ' slows the convergence rate.

One of shortcomings of Newton's method is that you have to know and call the derivative. If you don't know the derivative and only want to call the function itself, use the secant method. This is closer to what you are doing but the error in f ' is reduced with every iteration.

I suggest you look at the fixed point iteration theorem and calculate theoretical convergence. This will give you a better picture of how your method behaves.

Hans


#3

hi,

i understand secant dominates when f' is numerical. is this true?


Possibly Related Threads...
Thread Author Replies Views Last Post
  [HP Prime] Using n-root symbol and exponent problem uklo 7 1,118 11-11-2013, 01:39 AM
Last Post: Alberto Candel
  OT--TI-36X Algorithms Matt Agajanian 48 4,555 09-01-2013, 08:13 PM
Last Post: robert rozee
  Cubic root (-8) = 2 ? Gilles Carpentier 37 3,786 08-12-2013, 10:26 PM
Last Post: jep2276
  HELP WANTED ON ALGORITHMS Joerg Woerner 19 1,988 04-27-2013, 12:56 PM
Last Post: Eric Smith
  Square Root Simplifier for HP39gII Mic 4 806 03-11-2013, 08:25 AM
Last Post: Eddie W. Shore
  Seeking DIY4X update David Griffith 10 1,212 11-15-2012, 12:41 AM
Last Post: Eric Smith
  Cube root on standard calculator Thomas Klemm 22 2,521 11-09-2012, 06:11 AM
Last Post: Pierre
  Seeking new resources for TI NSpire Programming Namir 6 925 07-29-2012, 03:57 AM
Last Post: fhub
  ROOT bug? HP 48S/48G Eddie W. Shore 8 1,101 07-13-2012, 07:05 PM
Last Post: Eddie W. Shore
  OT: primitive mult/div algorithms Egan Ford 9 1,049 05-27-2012, 09:27 PM
Last Post: Egan Ford

Forum Jump: