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| In [[mathematics]], '''Thiele's interpolation formula''' is a formula that defines a [[rational function]] <math>f(x)</math> from a [[finite set]] of inputs <math>x_i</math> and their function values <math>f(x_i)</math>. The problem of generating a function whose graph passes through a given set of function values is called [[interpolation]]. This interpolation formula is named after the [[Danish people|Danish]] mathematician [[Thorvald N. Thiele]]. It is expressed as a [[continued fraction]], where ρ represents the [[reciprocal difference]]:
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| :<math> f(x) = f(x_1) + \cfrac{x-x_1}{\rho(x_1,x_2) + \cfrac{x-x_2}{\rho_2(x_1,x_2,x_3) - f(x_1) + \cfrac{x-x_3}{\rho_3(x_1,x_2,x_3,x_4) - \rho(x_1,x_2) + \cdots}}} </math>
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| ==Implementation==
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| Here is a version of Thiele interpolation in [[Algol 68]]:
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| <source lang="algol68">
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| ¢ The MODE of lx and ly here should really be a UNION of "something REAL", "something COMPLex",
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| and "something SYMBOLIC" ... ¢
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| PROC thiele:=([]REAL lx,ly, REAL x) REAL:
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| BEGIN
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| []REAL xx=lx[@1],yy=ly[@1];
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| INT n=UPB xx;
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| IF UPB yy=n THEN
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| ¢ Assuming that the values of xx are distinct ... ¢
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| [0:n-1,1:n]REAL p;
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| p[0,]:=yy[];
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| FOR i TO n-1 DO p[1,i]:=(xx[i]-xx[1+i])/(p[0,i]-p[0,1+i]) OD;
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| FOR i FROM 2 TO n-1 DO
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| FOR j TO n-i DO
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| p[i,j]:=(xx[j]-xx[j+i])/(p[i-1,j]-p[i-1,j+1])+p[i-2,j+1]
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| OD
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| OD;
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| REAL a:=0;
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| FOR i FROM n-1 BY -1 TO 2 DO a:=(x-xx[i])/(p[i,1]-p[i-2,1]+a) OD;
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| yy[1]+(x-xx[1])/(p[1,1]+a)
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| ELSE
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| error; ~ ¢ Unequal length arrays supplied ¢
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| FI
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| END;
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| </source>
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| 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 may reduce to the ratio of two polynomials, which may have factors in common - thus yielding a 0/0 form.)
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| ==References==
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| * {{mathworld|urlname=ThielesInterpolationFormula|title=Thiele's Interpolation Formula}}
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| [[Category:Finite differences]]
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| [[Category:Articles with example ALGOL 68 code]]
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| {{mathanalysis-stub}}
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