Question

Comparing convergence efficiencies among various numerical analysis methods. I'm looking for a good upper-undergraduate-level text (or info) or beginning graduate-level text for comparing convergence speed among various methods for approximating irrational zeros for high-order polynomials (or other rational functions). Perhaps something that might contrast Newton-Raphson with Jenkins-Traub or Laguerre or Grobner-Buchberger. I created 3 algorithms, but I'm concerned about convergence speed and I'm looking for optimization. Can anyone recommend a good (introductory?) text covering at least J

Answer 1

The last part of that question is:

... covering at least Jenkins-Traub and Newton-Raphson?

Answer 2

@amistre64
@TuringTest
@Mertsj
@AravindG
@mathstudent55
@satellite73
@ajprincess
@precal
@jhonyy9
@Luis_Rivera

Answer 3

@kropot72

Answer 4

Now that some of you folks are here, yes this is a bit higher-level, but I don't need a full answer, just any suggestions will do.. Fire away, folks!

Answer 5

Even if all anyone has to write is just what numerical methods books they have studied and benefited from, that will be greatly appreciated.

Answer 6

sorry, it has been too long since I have seen graduate level math

Answer 7

http://link.springer.com/chapter/10.1007%2FBFb0106616?LI=true#

Answer 8

np, I appreciate it even if all anyone did was take time to read. Thanks to everyone to his/her time. I'm still going to keep the question around, though.

Answer 9

http://www4.ncsu.edu/~stsynkov/book_sample_material/frontmatter.pdf

Answer 10

For a little more background, I came up with some semi-efficient ways to compute the interest rate for a series of annuity payments given the number of payments, the compounding period and the future value. It's really a trivial problem, and yes I could just put the whole thing into a business calculator. But it sparked an interest in the underlying pure mathematical theory. So, there is no pressing reason to get this done except for the pure love of study.

Answer 11

A last note: I'll keep the problem open for quite a while, so don't worry, I will come back to it often and check it to see if anyone has some more input. Thanks to everyone for stopping by! It's nice talking to you all "from the other side".

Answer 12

for convergence technique there are a few good books like
- an introduction to numerical analysis by (David F mayers) sorry i dont remember actual name its been 2 years :(
- theory of matrices in numerical analysis by Alston S. Householder
- Analysis of numerical methods by Eugene Issacson and Herbet Bishop Keler
- Introductry method of numerical analysis by S.S. Sastry
- Numeerical analysis by Richard L burden, John Douglas Faires

these are the best books i know and i would recommend you the first one but i am not 100% sure about author's name

moreover, if you are talking about rate of convergence then Newton Raphson is the one that gives fastest convergence with a bit less accuracy that could be ignored due to less number of iteration. wish you all the best. I tried to derive a few algorithm for convergence but unfortunately failed lol :P finally gave up haha

Answer 13

by the way 3rd book is also very nice :)

Answer 14

Thanks again to everyone and most recently to you @ghazi for what will be a great help. It's funny to look back on one's career in education and industry and see the "holes". I used to program in financial applications and some of the interest routines were purchased software where I never had to roll up my shirt sleeves on certain number crunching tasks. I always wanted to learn certain finer points. Now that I'm semi-retired, I can do such things. Everyone has been very attentive and a huge help. Great community this Openstudy! Some people play golf when they retire. Some read math books and torture themselves!

Answer 15

hahaha you are absolutely right but i am hoping to keep my studies on till i want and for now i am studying and still student :( by the way if you are deriving some new technique for fast convergence i think you should focus on the background of it, like from where this has developed and then i believe you can find the gap and hole in them...so again wish you all the best :D and i would be very happy if my recommendations work out for you :D thanks :)