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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] hotmail com 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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