Webpage on Newton-Raphson

[ewmayer]

I just uploaded a first iteration (pun!) of Iterative Methods for Solving Nonlinear Algebraic Equations, a web tutorial on the Newton-Raphson method, its higher-order-convergent analogs, and applications in areas such as solution of nonlinear differential equations and number theory. (The latter are to-be-added at a later date).

It's a long page, roughly the equivalent of a book chapter, so I suggest interested readers save to their PC and peruse the individual sections at their leisure.

I have viewed the page in Firefox and Safari; let me know if you encounter any formatting issues using your browser of choice. (Note: the page is not intended for small-screen mobile devices like smartphones, although I'll be interested to hear if it's at all readable on such.)

Comments, questions, suggestions welcome.

[retina]

Originally Posted by ewmayer

(Note: the page is not intended for small-screen mobile devices like smartphones, although I'll be interested to hear if it's at all readable on such.)

It is perfectly readable on one of my minions iSlave devices. Although the right edge of text wraps at about 60% of the total page width. So there ends up being a 40% wide white vertical strip on the right side for the entire length of the page except for some of the tables. Table 4(a) and 4(b) end up slightly wider than the surrounding text. The wider tables appear to be because the browser won't wrap the long numbers and will render the full number all on one line.

The line with all the asterisks (in the Hilbert space section) is rendered to almost the entire page width without a break.

Also you have an errant '-->' on the page just before "Once one has computed the coefficients of one's n-term Chebyshev ...". The aqua box just above there is rendered across the entire iSlave screen and appears to be the element that is defining the maximum page width.

ETA: The 〈 and 〉 characters render as empty square boxes on the iSlave.

[WraithX]

Excellent web site. It will take quite a while to go through and digest all the tasty morsels it contains.

Here are some small things I noticed on first glance:
In table 3c you have index 4 listed twice, ie 0,1,2,3,4,4, instead of 0,1,2,3,4,5.

Then, right after Inv3 you have exponent 2 listed twice, ie ε^0, ε^1, ε^2, ε^2, instead of ε^0, ε^1, ε^2, ε^3

And this one I'm not sure about, but right after you say:
"The second and third derivatives might be approximated"
you give two equations, one for f'' and one for f'''. It looks like the formula for f''' should be:

instead of:

But again, this last one is just a guess.

[bsquared]

Originally Posted by Most Excellent NR tutorial

For a discussion of this x87 implementation including assembly-code samples, optimization ideas and the resulting timing data, we refer the reader to this math-forum thread. (As it happens, the foregoing thread is what prompted the present author to finally put together some long-simmering thoughts on the topic of iterative schemes in a coherent webpage form).

I thought some the material looked familiar and then saw the above quote. Glad you took the time to write it up - nice page!

[ewmayer]

Originally Posted by retina

It is perfectly readable on one of my minions iSlave devices. Although the right edge of text wraps at about 60% of the total page width. So there ends up being a 40% wide white vertical strip on the right side for the entire length of the page except for some of the tables. Table 4(a) and 4(b) end up slightly wider than the surrounding text. The wider tables appear to be because the browser won't wrap the long numbers and will render the full number all on one line.

If you or any other of the iPhone users thinks of a way to improve the formatting result - perhaps by hacking the page source to just one table and surrounding text - let me know.

The line with all the asterisks (in the Hilbert space section) is rendered to almost the entire page width without a break.

Those are unneeded - a misguided experiment in section-highlighting - so I snipped them.

Also you have an errant '-->' on the page just before "Once one has computed the coefficients of one's n-term Chebyshev ...". The aqua box just above there is rendered across the entire iSlave screen and appears to be the element that is defining the maximum page width.

There are actually about a half-dozen stray "-->" on the rendered page - those are thanks to my text editor (which is otherwise nice except for 2-3 such annoyances) screwing up the highlighting of about every 4th inline html-comment snip, in a fashion that causes the ensuing non-comment text to be highlighted as a comment, making it very hard to read. The only workaround I've found is to add a gratuitous html comment-closing "-->" to the comments fubared in this way. I may have to simply add an item to my pre-upload checklist, where I search for such doubled "-->-->", replace with single "-->', upload to the ftp server, then undo the edit to restore readability of my source file.

ETA: The 〈 and 〉 characters render as empty square boxes on the iSlave.

You mean the angle-brackets, e.g. in the Hilbert space section? Not sure if I can work around that.

Thanks for the notes!

[ewmayer]

Originally Posted by WraithX

Excellent web site. It will take quite a while to go through and digest all the tasty morsels it contains.

Here are some small things I noticed on first glance:
In table 3c you have index 4 listed twice, ie 0,1,2,3,4,4, instead of 0,1,2,3,4,5.

Then, right after Inv3 you have exponent 2 listed twice, ie ε^0, ε^1, ε^2, ε^2, instead of ε^0, ε^1, ε^2, ε^3

Thanks - fixed the repeated-index bugs in my local version (will re-upload later, after a few more edits).

The 3rd-derivative formula is correct with the 2s ... a quick way to derive it is to use the 2nd-order-accurate central-diff approx for f'' at [j+1] and [j-1], subtract the resulting approximations f''[j+1] - f''[j-1] and divide the result by 2.delta.

[ewmayer]

Found one more bug ... formula [E1] has the k-1 and k subscripts reversed. Revised version uploaded.

[CRGreathouse]

This is very nice, thank you for posting.

[retina]

Originally Posted by ewmayer

You mean the angle-brackets, e.g. in the Hilbert space section? Not sure if I can work around that.

Can you not use the ASCII equivalents? <>

[ewmayer]

Originally Posted by retina

Can you not use the ASCII equivalents? <>

I'm not sure about the ASCII equivalent thing ... does not this note in effect say <> are not equivalents of &lang; and &rang; ?

In any case, I use <> all over the page already for less-than/greater-than ... and IMO the result of replacing angle-brackets with <> is rather fugly.

I wonder if there's a simple way to to implement the analog of a C typedef here, where &lang; and &rang; would get aliased to < and > on browsers known to have problems with the former.

[WraithX]

I'm wondering if you could just use image files for the angle brackets? I found some images online that could be tailored for your use here:
http://www.escapecodes.info/
I would re-color the background to white and trim off the excess white-space to the left and right of the actual bracket. Then you could put in <img> tags like so:

<img src="http://www.escapecodes.info/images/9001.gif" height=12/>&fnof;, <i>g</i><img src="http://www.escapecodes.info/images/9002.gif" height=12/>

This doesn't look too bad in my testing here.

I was originally thinking that CSS could handle the C typedef / multiple browser issue, but after some searching it doesn't seem like this would catch "all" browsers.

The one issue I can think of with the image-bracket approach is that getting the height of the image to match the height of the font might be difficult across different browsers and screen resolutions. There may be an easy way around this, but this was all I had time to search for tonight.