[Axiom-developer] [RationalInterpolationAlgorithms] Move back RationalInterpolation; example, question

[wyscc]

Changes 
http://page.axiom-developer.org/zope/mathaction/RationalInterpolationAlgorithms/diff
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??changed:
-Next RationalInterpolation
-
The package below implements rational interpolation. 
\begin{axiom}
)abbrev package RINTERP RationalInterpolation
++ Description:
++ This package exports interpolation algorithms
RationalInterpolation(xx, F): Cat == Body   where
    xx: Symbol
    F:  IntegralDomain
    UP  ==> UnivariatePolynomial
    SUP ==> SparseUnivariatePolynomial
 
    Cat == with
        interpolate: (Fraction UP(xx, F), List F, List F, _
                      NonNegativeInteger, NonNegativeInteger) _
                      -> Fraction UP(xx, F)
        interpolate: (List F, List F, NonNegativeInteger, NonNegativeInteger) _
                      -> Fraction SUP F

    Body == add
        RIA ==> RationalInterpolationAlgorithms

        interpolate(qx, lx, ly, m, k) ==
            px := RationalInterpolation(lx, ly, m, k)$RIA(F, UP(xx, F))
            elt(px, qx)
 
        interpolate(lx, ly, m, k) ==
            RationalInterpolation(lx, ly, m, k)$RIA(F, SUP F)
\end{axiom}

Comments: Packages compiled on MathAction seems to be local to the page. 
Dependent packages therefore needs to be on the same page to load the packages 
in correct sequence.

Example:

\begin{axiom}
interpolate(xlist, ylist, 3, 2)$RINTERP('x, FRAC INT)
interpolate(1/6::FRAC UP(x,FRAC INT), xlist, ylist, 3,2)$RINTERP('x,FRAC INT)
interpolate(xxlist, yylist, 3, 2)$RINTERP('x, FRAC dom)
interpolate(4*z::FRAC UP(x,dom), xxlist, yylist, 3, 2)$RINTERP('x, FRAC dom)
\end{axiom}

Question: If <code>p(xx) = interpolate(lx, ly, m, k)</code>, what is the 
purpose of
<code>elt(px, qx) = p(qx)</code>, the composition of <code>p(xx)</code> and 
<code>qx</code>, especially when <code>qx</code> is from <code>FRAC UP(xx, 
F)</code> instead of from just <code>F</code>? and why is this function (the 
composition) also called <code>interpolate</code>?

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