CoCoA:Interpolate

Interpolate

interpolating polynomial

Description

This function returns a multivariate polynomial which takes given

values at a given set of points.

NOTE:

* the current ring must have at least as many indeterminates as the
  dimension of the space in which the points lie;

* the base field for the space in which the points lie is taken to be
  the coefficient ring, which should be a field;

* in the polynomials returned, the first coordinate in the space is
  taken to correspond to the first indeterminate, the second to the
  second, and so on;

* if the number of points is large, say 100 or more, the returned
  value can be very large.  To avoid possible problems when printing
  such values as a single item we recommend printing out the elements
  one at a time as in this example:

Example

  X := Interpolate(Pts,Vals);
  Foreach Element In X Do
    PrintLn Element;
  EndForeach;

  Use Q[x,y];
  Points := [[1/2, 2], [3/4, 4], [5, 6/11], [-1/2, -2]];
  Values := [1/2,1/3,1/5,-1/2];
  F := Interpolate(Points, Values);
  F;
-46849/834000y^2 - 1547/52125x + 13418/52125y + 46849/208500
-------------------------------
  [Eval(F,P) | P In Points] = Values;  -- check
TRUE
-------------------------------

Syntax

Interpolate(Points:LIST,Values:LIST):POLY

where Points is a list of lists of coefficients representing a set of
*distinct* points and Values is a list of the same size containing
numbers from the coefficient ring.

   <type>list</type>
   <type>points</type>
   <type>polynomial</type>