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thiele

Thiele's continued fraction interpolation formula

 Calling Sequence thiele (x, y, v)

Parameters

 x - list of independent values, $\left[{x}_{1},..,{x}_{n}\right]$ y - list of dependent values, $\left[{y}_{1},..,{y}_{n}\right]$ v - variable or value to be used in a rational function

Description

 • Important: The thiele function has been deprecated. Use the superseding function CurveFitting[ThieleInterpolation] instead.  A call to thiele automatically generates a call to CurveFitting[ThieleInterpolation].
 • The thiele function computes the rational function of variable v (or evaluated at numerical value v) in continued fraction form which interpolates the points {(x[1], y[1]), (x[2], y[2]), ..., (x[n], y[n])}.  If n is odd then the numerator and denominator polynomials will have degree $\frac{n}{2}-\frac{1}{2}$. Otherwise, n is even and the degree of the numerator is $\frac{n}{2}$ and the degree of the denominator is $\frac{n}{2}-1$.
 • If the same x-value is entered twice, it is an error, whether the same y-value is entered.  All independent values must be distinct.

Examples

Important: The thiele function has been deprecated. Use the superseding function CurveFitting[ThieleInterpolation] instead.  A call to thiele automatically generates a call to CurveFitting[ThieleInterpolation].

 > $\mathrm{thiele}\left(\left[1,2,a\right],\left[3,4,5\right],z\right)$
 ${3}{+}\frac{{z}{-}{1}}{{1}{+}\frac{{z}{-}{2}}{\frac{{1}{-}{a}}{{3}{-}{a}}{+}{1}}}$ (1)

References

 The function Thiele uses Thiele's interpolation formula involving reciprocal differences.  For more information, refer to:
 Abramowitz, M., and Stegun, I., eds. Handbook of Mathematical Functions. New York: Dover Publications, Inc., 1965. Chap. 25 p. 881, Formula 25.2.50.

 See Also