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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