The following warnings occurred:
Warning [2] Undefined array key 191098 - Line: 275 - File: inc/plugins/threaded_mode.php PHP 8.1.2-1ubuntu2.14 (Linux)
File Line Function
/inc/class_error.php 153 errorHandler->error
/inc/plugins/threaded_mode.php 275 errorHandler->error_callback
/inc/plugins/threaded_mode.php 23 ThreadedMode::showthread_threaded
/inc/class_plugins.php 142 threaded_mode_showthread_threaded
/showthread.php 918 pluginSystem->run_hooks
Warning [2] Undefined array key 191193 - Line: 275 - File: inc/plugins/threaded_mode.php PHP 8.1.2-1ubuntu2.14 (Linux)
File Line Function
/inc/class_error.php 153 errorHandler->error
/inc/plugins/threaded_mode.php 275 errorHandler->error_callback
/inc/plugins/threaded_mode.php 23 ThreadedMode::showthread_threaded
/inc/class_plugins.php 142 threaded_mode_showthread_threaded
/showthread.php 918 pluginSystem->run_hooks
Warning [2] Undefined variable $thread - Line: 295 - File: inc/plugins/threaded_mode.php PHP 8.1.2-1ubuntu2.14 (Linux)
File Line Function
/inc/class_error.php 153 errorHandler->error
/inc/plugins/threaded_mode.php 295 errorHandler->error_callback
/inc/plugins/threaded_mode.php 23 ThreadedMode::showthread_threaded
/inc/class_plugins.php 142 threaded_mode_showthread_threaded
/showthread.php 918 pluginSystem->run_hooks
Warning [2] Trying to access array offset on value of type null - Line: 295 - File: inc/plugins/threaded_mode.php PHP 8.1.2-1ubuntu2.14 (Linux)
File Line Function
/inc/class_error.php 153 errorHandler->error
/inc/plugins/threaded_mode.php 295 errorHandler->error_callback
/inc/plugins/threaded_mode.php 23 ThreadedMode::showthread_threaded
/inc/class_plugins.php 142 threaded_mode_showthread_threaded
/showthread.php 918 pluginSystem->run_hooks
Warning [2] Undefined variable $fid - Line: 295 - File: inc/plugins/threaded_mode.php PHP 8.1.2-1ubuntu2.14 (Linux)
File Line Function
/inc/class_error.php 153 errorHandler->error
/inc/plugins/threaded_mode.php 295 errorHandler->error_callback
/inc/plugins/threaded_mode.php 23 ThreadedMode::showthread_threaded
/inc/class_plugins.php 142 threaded_mode_showthread_threaded
/showthread.php 918 pluginSystem->run_hooks
Warning [2] Undefined array key 191193 - Line: 331 - File: inc/plugins/threaded_mode.php PHP 8.1.2-1ubuntu2.14 (Linux)
File Line Function
/inc/class_error.php 153 errorHandler->error
/inc/plugins/threaded_mode.php 331 errorHandler->error_callback
/inc/plugins/threaded_mode.php 332 ThreadedMode::buildtree
/inc/plugins/threaded_mode.php 304 ThreadedMode::buildtree
/inc/plugins/threaded_mode.php 23 ThreadedMode::showthread_threaded
/inc/class_plugins.php 142 threaded_mode_showthread_threaded
/showthread.php 918 pluginSystem->run_hooks
Warning [2] Undefined variable $theme - Line: 3 - File: inc/plugins/threaded_mode.php(305) : eval()'d code PHP 8.1.2-1ubuntu2.14 (Linux)
File Line Function
/inc/class_error.php 153 errorHandler->error
/inc/plugins/threaded_mode.php(305) : eval()'d code 3 errorHandler->error_callback
/inc/plugins/threaded_mode.php 305 eval
/inc/plugins/threaded_mode.php 23 ThreadedMode::showthread_threaded
/inc/class_plugins.php 142 threaded_mode_showthread_threaded
/showthread.php 918 pluginSystem->run_hooks
Warning [2] Trying to access array offset on value of type null - Line: 3 - File: inc/plugins/threaded_mode.php(305) : eval()'d code PHP 8.1.2-1ubuntu2.14 (Linux)
File Line Function
/inc/class_error.php 153 errorHandler->error
/inc/plugins/threaded_mode.php(305) : eval()'d code 3 errorHandler->error_callback
/inc/plugins/threaded_mode.php 305 eval
/inc/plugins/threaded_mode.php 23 ThreadedMode::showthread_threaded
/inc/class_plugins.php 142 threaded_mode_showthread_threaded
/showthread.php 918 pluginSystem->run_hooks
Warning [2] Undefined variable $theme - Line: 3 - File: inc/plugins/threaded_mode.php(305) : eval()'d code PHP 8.1.2-1ubuntu2.14 (Linux)
File Line Function
/inc/class_error.php 153 errorHandler->error
/inc/plugins/threaded_mode.php(305) : eval()'d code 3 errorHandler->error_callback
/inc/plugins/threaded_mode.php 305 eval
/inc/plugins/threaded_mode.php 23 ThreadedMode::showthread_threaded
/inc/class_plugins.php 142 threaded_mode_showthread_threaded
/showthread.php 918 pluginSystem->run_hooks
Warning [2] Trying to access array offset on value of type null - Line: 3 - File: inc/plugins/threaded_mode.php(305) : eval()'d code PHP 8.1.2-1ubuntu2.14 (Linux)
File Line Function
/inc/class_error.php 153 errorHandler->error
/inc/plugins/threaded_mode.php(305) : eval()'d code 3 errorHandler->error_callback
/inc/plugins/threaded_mode.php 305 eval
/inc/plugins/threaded_mode.php 23 ThreadedMode::showthread_threaded
/inc/class_plugins.php 142 threaded_mode_showthread_threaded
/showthread.php 918 pluginSystem->run_hooks
Warning [2] Undefined variable $lang - Line: 5 - File: inc/plugins/threaded_mode.php(305) : eval()'d code PHP 8.1.2-1ubuntu2.14 (Linux)
File Line Function
/inc/class_error.php 153 errorHandler->error
/inc/plugins/threaded_mode.php(305) : eval()'d code 5 errorHandler->error_callback
/inc/plugins/threaded_mode.php 305 eval
/inc/plugins/threaded_mode.php 23 ThreadedMode::showthread_threaded
/inc/class_plugins.php 142 threaded_mode_showthread_threaded
/showthread.php 918 pluginSystem->run_hooks
Warning [2] Attempt to read property "messages_in_thread" on null - Line: 5 - File: inc/plugins/threaded_mode.php(305) : eval()'d code PHP 8.1.2-1ubuntu2.14 (Linux)
File Line Function
/inc/class_error.php 153 errorHandler->error
/inc/plugins/threaded_mode.php(305) : eval()'d code 5 errorHandler->error_callback
/inc/plugins/threaded_mode.php 305 eval
/inc/plugins/threaded_mode.php 23 ThreadedMode::showthread_threaded
/inc/class_plugins.php 142 threaded_mode_showthread_threaded
/showthread.php 918 pluginSystem->run_hooks





Lyuka and the Ostrowski method's for Root Seeking



#3

A few threads back Hugh Steers pointed out a method for finding the roots of a nonlinear function by Russian mathematician Ostrowski. I checked Luyka's web page which discussed that method and presented the code for the algorithm. It did real well.

I further searched the Internet for articles about root-seeking method based on Ostrowski's method. I found several articles that further accelerated the convergence for that algorithm. A few articles were published by Chinese mathematician Zou (and colleagues) and offer somewhat elaborate enhancements to Ostrowski's method. These enhancements can solve for a root in TWO iterations (requiring 9 function calls) as compared to 6 iterations (and 12 function call) with Newton's method!. The fast Halley's method required 3 iterations and 9 function calls. The basic Ostrowski method took 3 iterations and 7 function call.

So thank you Lyuka for pointing me to Ostrowski's method. I really enjoyed learning about it and about its variants.

More gems in the root-seeking toolbox. Just when I thoght I learned about all the algorithms available, I get to learn about new and even more efficient ones!!

Namir


#4

Hi,

The Ostrowski's method is a kind of root finding algorithm that uses reverse-interpolation to approximate the root.

So, any other interpolation, such as Lagrange interpolation, can be used as a root finding algorithm.

If you use that of n=3, a recurrence equation


can be used instead of Ostrowski's


This can be tested replacing a line in _ost.c

Quote:
t = (h * a - b) / (h - 1.0);



by
Quote:
t  = a * e * f * (f - e) - b * d * f * (f - d) + c * d * e * (e - d);
t /= (f - e) * (f - d) * (e - d);



Though the convergence of the method shown above is almost quadratic (order of about 1.8) for a zero of multiplicity 1 in a neighborhood of the zero, it's NOT recommended as it tends to diverge when the guess is not near the zero.

IMHO, the most important thing as a root finding algorithm is not
the order of convergence, but the stability of convergence, i.e. ability to find a root with very few chance of divergence.

Regards,

Lyuka


Possibly Related Threads…
Thread Author Replies Views Last Post
  A fast Bernoulli Number method for the HP Prime Namir 16 5,360 11-22-2013, 04:46 PM
Last Post: Namir
  [HP Prime] Using n-root symbol and exponent problem uklo 7 2,835 11-11-2013, 01:39 AM
Last Post: Alberto Candel
  A brand new calculator benchmark: "middle square method seed test" Pier Aiello 25 6,761 09-13-2013, 01:58 PM
Last Post: Pier Aiello
  Cubic root (-8) = 2 ? Gilles Carpentier 37 10,032 08-12-2013, 10:26 PM
Last Post: jep2276
  Downhill Simplex Method (Nelder Mead) for WP-34S Marcel Samek 0 866 07-07-2013, 03:05 AM
Last Post: Marcel Samek
  Square Root Simplifier for HP39gII Mic 4 1,937 03-11-2013, 08:25 AM
Last Post: Eddie W. Shore
  Seeking DIY4X update David Griffith 10 2,789 11-15-2012, 12:41 AM
Last Post: Eric Smith
  Cube root on standard calculator Thomas Klemm 22 6,244 11-09-2012, 06:11 AM
Last Post: Pierre
  Seeking new resources for TI NSpire Programming Namir 6 2,263 07-29-2012, 03:57 AM
Last Post: fhub
  New Quadratic Lagrangian Interpolation Method Namir 2 1,316 07-20-2012, 04:32 PM
Last Post: Namir

Forum Jump: