|Ilya Zakharevich on Fri, 30 Jul 1999 20:13:03 -0400 (EDT)|
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|Re: ebcdic packed numbers|
[A complimentary Cc of this posting was sent to Abigail <email@example.com>], who wrote in article <firstname.lastname@example.org>: > _Finite_ or _repeated_ fractions can be. But I challenge you to find 2 bignums > that form a ratio equal to the square root of 2. Say, BignumSQRT(2)/1. ;-) > [Not that's it's impossible to do exact arithmetic on algebraic numbers. > C-K Yap had an interesting talk on CCCG'92 explaining how to do > arithmetic on algebraic numbers, using just integers. It's all based on > the fact that an algebraic number is a root of a polynomial of bounded > degree, and integer coefficients. But I disgress... ] Well, Math::Pari module has no problem dealing with algebraic numbers: perl -MMath::Pari=:DEFAULT,Mod -wle \ ' $t = PARIvar("t"); $q = PARIvar("q"); \ $root2 = Mod($t, $t**2 - 2); \ $root3 = Mod($q, $q**2 - 3); \ print "OK" if 24 == (($root2 - $root3)**2 - 5)**2' OK However, not that these are "abstract" algebraic numbers, not complex algebraic numbers. Say, $root2 represents both -sqrt(2) and sqrt(2). To do calculations with complex algebraic numbers, one needs to keep a pair of an "abstract" algebraic number (an element of a finite-dimensional commutative algebra over Q) and a complex floating-point approximation to this number (a complex number which is significantly closer to one embedding of the above number into C than to other embeddings). PARI has no built-in datatype for such pairs. But since PARI supports arbitrary-precision (complex) floating-point arithmetic as well, it should be a quick exercise to write a Perl module which would use Math::Pari and will make calculations with complex algebraic numbers. Ilya P.S. BTW, how hard is it to add a new datatype to PARI nowadays?