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Dear GAP-Forum!

In ftp.math.rwth-aachen.de:/pub/incoming/ there is now a

file called 'approx.g' which implements GAP v3.4 functions

to compute rational approximations of cyclotomic field

elements.

The module is not meant to introduce numerical computation

into GAP --- rather it allows to find accurate rational

approximations cyclotomic field elements, embedded into

the complex numbers. Let E(n) mean exp(2 pi i/n) and

consider a cyclotomic field element z then

ApproxReCycSimple(z, absError)

computes a pair [a, b] of rationals such that

a <= Re(z) <= b and b - a < absError

for some positive rational absError. In fact, there

are three alternative methods of approximation:

Simple

finds *some* a, b (not necessarily with small numerator

or denominator)

Binary

finds a, b such that the denominators are powers of 2

(and are minimal under this condition)

SternBrocot

finds a, b such that the denominators are small (and

in case they represent principal terms of the associated

continued fraction series, are minimal)

For details refer to 'approx.g' and print out all lines

which contain the string #F (under UNIX: grep \#F approx.g).

In addition there is a simple function

StringDecimalRat(<rat>, <mantissaDigits>)

which returns a string for the 'scientific notation'

of the rational <rat> expressed with <mantissaDigits>

many digits after the decimal dot (e.g. "1.96883e5").

Again, the function is not meant to be numerical, but

it proved useful to me in that it often gives me an

impression of a number. (StringDecimalRat(6283/2000, 3)

yields "3.141e0".)

Have fun with it -- and don't forget to send me

the bug reports...

Sebastian.

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