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I have been working on a pair of similar root-seeking algorithms in the past few weeks. I finished writing an article (click here) that describes the algorithms (which I name the Probing Slopes Algorithms and Probing Steps Algorithm) and compares them with Newton's method. My article also includes three test functions used to compare the new algorithms with Newton's method, using Excel VBA. I am excited to report that the new algorithms find the root in fewer loops and also make fewer function calls than Newton's method.


PS: Based on the feedback of kind folks on this web site, you will need to use FireFox to view the article's text AND figures. some how, Microsoft IE does not show the images.

Edited: 8 Nov 2007, 12:36 p.m. after one or more responses were posted

Hi Namir,

Following the link I see your article but none of the figures.



I checked the site on two different laptops (which do not have the source files) at home and was able to see the figures. I will ask a few friends to see if they have the same problem you are having.


FWIW, I can see the figures.


Me too.


I can't see the figures :(

See 'em... (with Firefox, but not with Internet Explorer engine)

Edited: 8 Nov 2007, 11:04 a.m.


I can the the figues in Firefox but cannot see them in Microsoft Intenet Explorer 7.0.


Yes, I can see them in Firefox.


I see them in Opera but not Internet Explorer.


Ok, I can see it with Firefox but not MS IE. This is very interesting, because I created the text using MS Word and then copied it to Microsoft Web Expressio--both MS products!!!

I will have to create a new copy of the article that I assemble with a simple text editor and insert fewer HTML tags. This will take a few days, so please be patient. In the mean time, use FireFox to view the article.


Hi Namir,

have not gone through you article in detail. For some days ago I have read some interesting articles about safeguarded root finding. A lot of thinking needs to be invested to make a root finder as bulletproof as possible. And Newtons method used to be quick, but has its pitfalls. How about yours?

Here are some articles which might be of interest. Would be cool to see how your algorithm copes with these tricky functions mentioned in the articles.

Have fun and let us know about the results of your further testing.

// Frank

First hand information about the very first HP-solver (page 20)

[link:http://www.cs.berkeley.edu/~wkahan/Math128/RealRoots.pdf ]Some serious stuff about root finding[/link]

Another article comparing root finding methods

Thanks Frank. Will do!


Netscape 8.1.3 shows the images with no problem. In the "about" dropdown, it says "based on Firefox" so Firefox seems to be the magic key! (Just another reason to dislike Micro$oft.)

PS Namir - Met your son at an NAU event a couple of weeks ago. He saw my name on my nametag and recognized it - from our discussion at HHC2007 which I guess you must have told him about!!

Edited: 8 Nov 2007, 5:21 p.m.