Hi Tor,

Thank you so much for your help and response.  I feel like this group
provides kinda of like an "underground" educational curriculum.  It's kinda
nice knowing that science and mathematics have something to do with the
visual arts.

I did read your answer to my question on "Not a sphere. . ." and understood
it.  Thank you for that response as well.  I have visited the sites you have
given me and will say that (for me) it will take a while to absorb the
material.  Thank you for answering my questions, it is greatly appreciated.

Arie

in article 3BD8B1DF.82914778@hotmail.com, Tor Olav Kristensen at
tor### [at] hotmailcom wrote on 10/25/01 7.44 PM:

> 
> Arie, I have thought about how to explain
> what numerical methods are, but I find it
> difficult to give a good explanation.
> 
> But here's my try at it anyway:
> 
> When one are solving a mathematical problem
> analytically, one often manipulate SYMBOLS
> that are describing the problem. The mani-
> pulations are done according to a set of
> rules that preserves the exact description
> of the problem. (And by doing it this way,
> any found solutions will be exact too.)
> 
> If the manipulations are done in an intelli-
> gent way, each step of them will simplify the
> problem, or parts of it, until simple
> expressions containing the symbols are found.
> 
> But if one are solving a problem numerically,
> one may start with a symbolic description of
> the problem. And, sometimes, one thereafter
> rewrites (and maybe simplifies) that
> description a bit. Then one substitute
> NUMBERS for some or all of the symbols in
> the (modified) description and evaluate it.
> (These numbers can be results of analysis,
> guessing or measurements obtained from "the
> real world".)
> 
> Often one can use the result of this
> evaluation to determine how "far off" those
> initial numbers are from a solution (i.e.
> estimate the errors).
> 
> Some knowledge of similar problems can then
> often be used to try to guess or estimate how
> to alter those numbers in order to get closer
> to a solution. If so, one may repeat the
> process all over again and hope that a good-
> enough solution appears (i.e. has an
> acceptable error.)
> 
> If one find a solution to a problem, one
> sometimes have enough information
> to get closer to other solutions analytically.
> If one succeeds doing this, one can then go
> back and apply further numerical methods to
> seek out other solutions (or more accurate
> ones).
> 
> It is important to notice that with numerical
> methods one does not get exact results if one
> are using computers. This is because the
> numbers involved are always either truncated
> or rounded off. And also because some of the
> calculations the computers perform are done
> by numerical methods themselves or by
> approximations of the mathematical operations.
> 
> 
> I'll try to explain more about the mentioned
> methods later.
> 
> 
> Tor Olav
> 
> Btw:
> Arie, did you read my reply to your question in
> the "Not a sphere" thread (by Zebu 2. Oct.) ?
> 
> 
> Here's a web page that you hopefully will
> find interesting:
> 
> http://www.damtp.cam.ac.uk/user/fdl/people/sd/lectures/index.html
> 
> It contains links to some "Numerical Methods"
> lecture notes by Stuart Dalziel.
> 
> There's a html version here:
> http://www.damtp.cam.ac.uk/user/fdl/people/sd/lectures/nummeth98/index.htm
> 
> And if you look at this page;
> http://www.damtp.cam.ac.uk/user/fdl/people/sd/lectures/nummeth98/roots.htm
> 
> - you'll find these sub-chapters(?):
> 
> "3.3 Linear interpolation (regula falsi)"
> (Read the part "3.2 Bisection" first.)
> 
> "3.5 Secant (chord)"
> (Read the part "3.4 Newton-Raphson" first.
> 
> - where he talks about the methods I and
> Anton has mentioned.
> 
> It is often quite useful to study the graph-
> images supplied within, while you are
> figuring out what is _really_ going on.

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