mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 rat ...

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 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 a n ...

. 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 a ...

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 (mathematics), expression obtained through an iterative process of representing a number as the sum of its integer part and the multiplicative inverse, reciprocal of another number, then writ ...

, where ρ represents the

reciprocal difference In mathematics, the reciprocal difference of a finite sequence of numbers (x_0, x_1, ..., x_n) on a function f(x) is defined inductively by the following formulas: :\rho_1(x_1, x_2) = \frac :\rho_2(x_1, x_2, x_3) = \frac + f(x_2) :\rho_n(x_1,x_2 ...

: :

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

References

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