Polynomial arithmetic and roots

The final goal of our attempt is to provide a representation of real numbers that will be used in symbolic computations, especially related to polynomials.

Up to now, we have implemented a polynomial arithmetic that can handle both univariate and multivariate polynomials with erna real or complex coefficients.

We have also implemented procedures for root isolation and extraction of a root as an erna real or complex, for square-free univariate polynomials. A preliminary implementation of Corless et al. algorithm [(Corless et al., 95)] for GCD computation is also available.

Figure 2: Excerpt from a Maple session: polynomials with erna coefficients.

Here is a list of the operations implemented:

• arithmetic operations for polynomials with erna real coefficients:

  Function	Input	Output
  rPolAdd	$p,q$	$r$ $(=p+q)$
  rPolProd	$p,q$	$r$ $(=pq)$
  rPolDiv	$p,q,var$	[quo,rem]
  		$p = q\cdot {\rm quo}+{\rm rem},$
  		deg(rem,$var$)$<$deg($q,var$)

  Similarly, for polynomials with erna complex coefficients: cPolAdd, cPolProd, cPolTail.

• evaluation:

  Function	Input	Output
  rPolEval	$var_1=val_1,...,p$	$r\ (=p\vert _{var_1=val1,...})$
  	$val_1,...$ are erna reals ,	an erna real
  	$p$ has erna real coefficients	or a polynomial
  cPolEval	$var_1=val_1,...,p$	$r\ (=p\vert _{var_1=val1,...})$

• root computation:

  Function	Input	Output
  sqFreePolRootsIsol	$p,var$	$L=[r_1,\varepsilon_1,\dots]$
  		$r_i$ approximates a root
  		of $p$ with accuracy $\varepsilon_i$
  sqFreePolRoot	$p,var,[r,\varepsilon]$	$o$ an erna object
  	$r$ - initial $\varepsilon$-approximation
  	of a root of $p$

In addition, there are functions that convert a polynomial with erna coefficients to a polynomial with rational coefficients, either by dump-ing (rPolDump), epsint-ing (rPolEpsint) or refine-ing (rPolRefine) the coefficients.