In mathematics, Thiele's interpolation formula is a formula that defines a rational function
f
(
x
)
from a finite set 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. 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
)
+
⋯
