172:¢ The MODE of lx and ly here should really be a UNION of "something REAL" and "something SYMBOLIC" ... ¢ PROC thiele:=(REAL lx,ly, REAL x) REAL: BEGIN REAL xx=lx,yy=ly; INT n=UPB xx; IF UPB yy=n THEN ¢ Assuming that the values of xx are distinct ... ¢ REAL p; p:=yy; FOR i TO n-1 DO p:=(xx-xx)/(p-p) OD; FOR i FROM 2 TO n-1 DO FOR j TO n-i DO p:=(xx-xx)/(p-p)+p OD OD; REAL a:=0; FOR i FROM n-1 BY -1 TO 2 DO a:=(x-xx)/(p-p+a) OD; y+(x-xx)/(p+a) ELSE error ¢ Unequal length arrays supplied ¢ FI END;
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The article on Thiele's interpolation formula duplicates the form given in
Abramowitz and Stegun's "Handbook of Mathematical Functions" - as does Eric Weisstein's. However, it is not particularly easy to implement, from the description given, even by writing recursively. In the (antique) version of
158:"Maple" to which I have access, there is a good non-recursive working implementation of Thiele's algorithm. (Dognose (as Anthony Burgess wrote it in one novel), I've used it to good effect often enough.) If anyone is interested, I can pass on the code (copyright abuse permitting).
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Note that, although it works in most cases, it is sensitive to input values, especially those due to equally spaced abscissæ. (Essentially, in such a case, it reduces to the ratio of two polynomials, which may have factors in common - thus yielding a 0/0 form.)
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Here is a version of Thiele interpolation in Algol 68:
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