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In mathematics , Thiele's interpolation formula is a formula that defines a rational function
f
(
x
)
{\displaystyle f(x)}
finite set of inputs
x
i
{\displaystyle x_{i}}
f
(
x
i
)
{\displaystyle f(x_{i})}
interpolation . This interpolation formula is named after the Danish mathematician Thorvald N. Thiele . It is expressed as a continued fraction , where ρ represents the reciprocal difference :
f
(
x
)
=
f
(
x
1
)
+
x
−
x
1
ρ
(
x
1
,
x
2
)
+
x
−
x
2
ρ
2
(
x
1
,
x
2
,
x
3
)
−
f
(
x
1
)
+
x
−
x
3
ρ
3
(
x
1
,
x
2
,
x
3
,
x
4
)
−
ρ
(
x
1
,
x
2
)
+
⋯
{\displaystyle 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 }}}}}}}
Note that the
n
{\displaystyle n}
ρ
n
(
x
1
,
x
2
,
⋯
,
x
n
+
1
)
−
ρ
n
−
2
(
x
1
,
x
2
,
⋯
,
x
n
−
1
)
+
x
−
x
n
+
1
ρ
n
+
1
(
x
1
,
x
2
,
⋯
,
x
n
+
2
)
−
ρ
n
−
1
(
x
1
,
x
2
,
⋯
,
x
n
)
+
⋯
,
{\displaystyle \rho _{n}(x_{1},x_{2},\cdots ,x_{n+1})-\rho _{n-2}(x_{1},x_{2},\cdots ,x_{n-1})+{\cfrac {x-x_{n+1}}{\rho _{n+1}(x_{1},x_{2},\cdots ,x_{n+2})-\rho _{n-1}(x_{1},x_{2},\cdots ,x_{n})+\cdots }},}
while the
n
{\displaystyle n}
reciprocal difference is defined to be
ρ
n
(
x
1
,
x
2
,
…
,
x
n
+
1
)
=
x
1
−
x
n
+
1
ρ
n
−
1
(
x
1
,
x
2
,
…
,
x
n
)
−
ρ
n
−
1
(
x
2
,
x
3
,
…
,
x
n
+
1
)
+
ρ
n
−
2
(
x
2
,
…
,
x
n
)
{\displaystyle \rho _{n}(x_{1},x_{2},\ldots ,x_{n+1})={\frac {x_{1}-x_{n+1}}{\rho _{n-1}(x_{1},x_{2},\ldots ,x_{n})-\rho _{n-1}(x_{2},x_{3},\ldots ,x_{n+1})}}+\rho _{n-2}(x_{2},\ldots ,x_{n})}
The two
ρ
n
−
2
{\displaystyle \rho _{n-2}}
References
