20120906, 02:23  #1 
∂^{2}ω=0
Sep 2002
República de California
11,743 Posts 
Webpage on NewtonRaphson
I just uploaded a first iteration (pun!) of Iterative Methods for Solving Nonlinear Algebraic Equations, a web tutorial on the NewtonRaphson method, its higherorderconvergent analogs, and applications in areas such as solution of nonlinear differential equations and number theory. (The latter are tobeadded 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 smallscreen mobile devices like smartphones, although I'll be interested to hear if it's at all readable on such.) Comments, questions, suggestions welcome. 
20120906, 02:52  #2  
Undefined
"The unspeakable one"
Jun 2006
My evil lair
2^{2}·17·97 Posts 
Quote:
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 nterm 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. Last fiddled with by retina on 20120906 at 03:27 Reason: The 〈 and 〉 characters render as empty square boxes on the iSlave 

20120906, 03:11  #3 
Mar 2006
2×3^{2}×29 Posts 
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. 
20120906, 04:01  #4  
"Ben"
Feb 2007
111001001110_{2} Posts 
Quote:


20120906, 18:20  #5  
∂^{2}ω=0
Sep 2002
República de California
11743_{10} Posts 
Quote:
Quote:
Quote:
Quote:
Thanks for the notes! 

20120906, 18:37  #6  
∂^{2}ω=0
Sep 2002
República de California
11,743 Posts 
Quote:
The 3rdderivative formula is correct with the 2s ... a quick way to derive it is to use the 2ndorderaccurate centraldiff approx for f'' at [j+1] and [j1], subtract the resulting approximations f''[j+1]  f''[j1] and divide the result by 2.delta. 

20120906, 19:09  #7 
∂^{2}ω=0
Sep 2002
República de California
11,743 Posts 
Found one more bug ... formula [E1] has the k1 and k subscripts reversed. Revised version uploaded.

20120906, 21:36  #8 
Aug 2006
5987_{10} Posts 
This is very nice, thank you for posting.

20120906, 23:09  #9 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
14704_{8} Posts 

20120907, 00:09  #10 
∂^{2}ω=0
Sep 2002
República de California
26737_{8} Posts 
I'm not sure about the ASCII equivalent thing ... does not this note in effect say <> are not equivalents of ⟨ and ⟩ ?
In any case, I use <> all over the page already for lessthan/greaterthan ... and IMO the result of replacing anglebrackets with <> is rather fugly. I wonder if there's a simple way to to implement the analog of a C typedef here, where ⟨ and ⟩ would get aliased to < and > on browsers known to have problems with the former. 
20120907, 02:13  #11 
Mar 2006
2·3^{2}·29 Posts 
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 recolor the background to white and trim off the excess whitespace to the left and right of the actual bracket. Then you could put in <img> tags like so: Code:
<img src="http://www.escapecodes.info/images/9001.gif" height=12/>ƒ, <i>g</i><img src="http://www.escapecodes.info/images/9002.gif" height=12/> 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 imagebracket 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. 
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