Newton Divided Differences and Interpolation Polynomials
Mth 351 May 8 2001 - Maple 5.00
Bent E. Petersen
Filename: 351s2001_newton_divided_diff.mws
> restart;
The procedure divdiff() computes symbolic Newton divided differences recursively. Not much error
checking is done so be careful. Note the use of a limit in the code. This allows us to compute the
divided differences even when the nodes are not all distinct. The purpose of this procedure is to give
you something to experiment with when you study Newton divided differences.
We use the divdiff() procedure to compute interpolation polynomials of functions in the procedure
Ninterp() below. One nice feature is that Ninterp() does the right thing for the case of repeated nodes.
Of course, Maple does have built-in interpolation, and one should use it. The Maple routine however
works with lists of points, rather than functions. Thus Maple has no sensible way of handling repeated
nodes, and returns an error in this case. We show below how the Maple routine can be used to
interpolate functions and how to handle repeated nodes in that case simply by blowing up the repeated
points and then passing to a limit.
> divdiff:=proc(f)
> local x;
> if nargs < 2 then ERROR(FAIL); fi;
> if nargs = 2 then RETURN(f(args[2]));
> else
>
limit((divdiff(f,args[3..nargs])-divdiff(f,x,args[3..nargs-1]))/(a
rgs[nargs]-x),x=args[2]);
> fi;
> end:
Here's some examples:
> divdiff(f,a);
( )f a
> divdiff(f,a,a);
( )( )Dfa
> divdiff(f,a,b);
โ โ ( )f b( )f a
โ + b a
> divdiff(f,a,a,a);
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