In mathematics, 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. ...

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 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 often has ...

. This interpolation formula is named after the

Danish Danish may refer to: * Something of, from, or related to the country of Denmark People * A national or citizen of Denmark, also called a "Dane," see Demographics of Denmark * Culture of Denmark * Danish people or Danes, people with a Danish ance ...

mathematician

Thorvald N. Thiele Thorvald Nicolai Thiele (24 December 1838 – 26 September 1910) was a Danish astronomer and director of the Copenhagen Observatory. He was also an actuary and mathematician, most notable for his work in statistics, interpolation and the three ...

. It is expressed as a

continued fraction In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integ ...

, where ρ represents the reciprocal difference: :

f(x) = f(x_1) + \cfrac

References

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