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Rational expressions simplification

ALG48 provides two powerful commands for simplifying multivariate polynomials and rational functions. These commands will work on any algebraic expressions by treating them as the quotient of two polynomials in several ``variables'', which can actually be non-rational subexpressions (see the second example in Section 2).

The simplification consists of two main steps

  1. Multiplying out all products of polynomials and collecting the terms of same degree for the numerator and denominator polynomials;
  2. Simplification of the rational function by the multivariate polynomial greatest common divisor (GCD) of the the numerator and denominator.

Depending on the type of the simplified rational function, RSIM returns it in one of the following forms:

Similarly, FCTR returns the simplified rational function as either

FCTR computes the true factorization of a polynomial over the integers (or, equivalently, over the field of rational numbers), and not approximate roots over the real or complex field as computed by a root finding program (like the command ROOT or PROOT of the HP48). Polynomials of arbitrary degree can be irreducible over the integer, and a factorization might therefore entail high degree polynomials. For instance:

displaymath1658

ALG48 version 4.2 uses Berlekamp p-adic factorization algorithm and can compute the complete factorization of virtually any polynomial (up to degree 256). For more on RSIM and FCTR performances see Section 4.16.

ALG48 also provides the command RAT->, which operates like RSIM, but returns the numerator and denominator of the simplified rational function as two separated polynomials. In addition, the commands GCD and LCM respectively compute the greatest common divisor and the least common multiple of two polynomials. These two functions do not accept rational functions as input, since the GCD and LCM are not well-defined notions in this case.

All the simplification commands produce a ``Infinite Result'' error if the denominator of the simplified expression is zero.


next up previous
Next: Output format for polynomials Up: Commands Previous: Generalities

Claude-Nicolas Fiechter (fiechter@cs.pitt.edu), 12 May 1998