Thiele's Interpolation Formula
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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Thiele's interpolation formula is a formula that defines a
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
f(x) from a
finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, is a finite set with five elements. Th ...
of inputs x_i and their function values f(x_i). The problem of generating a function whose graph passes through a given set of function values is called
interpolation In the mathematics, mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one ...
. This interpolation formula is named after the Danish mathematician Thorvald N. Thiele. It is expressed as a
continued fraction A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...
, where ρ represents the reciprocal difference: : f(x) = f(x_1) + \cfrac Note that the n-th level in Thiele's interpolation formula is :\rho_n(x_1,x_2,\cdots,x_)-\rho_(x_1,x_2,\cdots,x_)+\cfrac, while the n-th reciprocal difference is defined to be :\rho_n(x_1,x_2,\ldots,x_)=\frac+\rho_(x_2,\ldots,x_). The two \rho_ terms are different and can not be cancelled.


References

* Finite differences Articles with example ALGOL 68 code Interpolation {{mathanalysis-stub