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In progress
The following is the table of formulae for equidistant interpolation , i.e. interpolation where the interpolation poles are placed at equal distances from each other
x
i
=
0
,
1
,
2
,
3...
,
n
{\displaystyle x_{i}=0,1,2,3...,n}
Interpolation method
Interpolation formula
Newton series
L
n
(
x
)
=
∑
m
=
0
n
(
x
m
)
∑
k
=
0
m
(
m
k
)
(
−
1
)
m
−
k
f
(
k
)
{\displaystyle L_{n}(x)=\sum _{m=0}^{n}{\binom {x}{m}}\sum _{k=0}^{m}\,{\binom {m}{k}}\,(-1)^{m-k}f(k)}
Lagrange interpolation
L
n
(
x
)
=
(
x
n
)
∑
k
=
0
n
x
−
n
x
−
k
(
n
k
)
(
−
1
)
n
−
k
f
(
k
)
{\displaystyle L_{n}(x)={\binom {x}{n}}\sum _{k=0}^{n}{\frac {x-n}{x-k}}{\binom {n}{k}}(-1)^{n-k}f(k)}
Barycentric interpolation
L
n
(
x
)
=
∑
k
=
0
n
(
−
1
)
k
f
(
k
)
(
x
−
k
)
k
!
(
n
−
k
)
!
∑
k
=
0
n
(
−
1
)
k
(
x
−
k
)
k
!
(
n
−
k
)
!
{\displaystyle L_{n}(x)={\frac {\sum _{k=0}^{n}{\frac {(-1)^{k}f(k)}{(x-k)k!(n-k)!}}}{\sum _{k=0}^{n}{\frac {(-1)^{k}}{(x-k)k!(n-k)!}}}}}
Ramanjuan interpolation
L
n
(
x
)
=
sin
x
π
π
∫
0
∞
t
−
x
−
1
∑
k
=
0
n
(
−
1
)
k
t
k
f
(
k
)
d
t
{\displaystyle L_{n}(x)={\frac {\sin x\pi }{\pi }}\int _{0}^{\infty }t^{-x-1}\sum _{k=0}^{n}(-1)^{k}t^{k}f(k)dt}
In progress
