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x
1
,
2
=
−
b
±
b
2
−
4
a
c
2
a
{\displaystyle x_{1,2}={-b\pm {\sqrt {b^{2}-4ac}} \over 2a}}
f
0
(
2
)
{\displaystyle f_{0}^{(2)}}
f
0
(
2
)
{\displaystyle f_{0}^{(2)}}
f
0
(
1
)
=
−
f
0
+
f
1
h
−
1
2
h
f
(
2
)
(
ξ
)
{\displaystyle f_{0}^{(1)}={\frac {-f_{0}+f_{1}}{h}}-{\frac {1}{2}}h\,f^{(2)}(\xi )}
f
1
(
1
)
=
−
f
0
+
f
1
h
+
1
2
h
f
(
2
)
(
ξ
)
{\displaystyle f_{1}^{(1)}={\frac {-f_{0}+f_{1}}{h}}+{\frac {1}{2}}h\,f^{(2)}(\xi )}
(n+1)-punts eerste afgeleide:
h
f
′
(
x
i
)
=
∑
j
=
0
,
i
≠
j
n
(
−
1
)
i
−
j
+
1
i
−
j
i
!
(
n
−
i
)
!
j
!
(
n
−
j
)
!
(
f
i
−
f
j
)
+
(
−
1
)
n
−
i
i
!
(
n
−
i
)
!
(
n
+
1
)
!
h
n
+
1
f
(
n
+
1
)
(
ξ
)
=
∑
j
=
0
,
i
≠
j
n
(
−
1
)
i
−
j
i
−
j
(
n
j
)
(
n
i
)
(
f
j
−
f
i
)
+
(
−
1
)
n
−
i
(
n
i
)
(
n
+
1
)
h
n
+
1
f
(
n
+
1
)
(
ξ
)
{\displaystyle h\,f'(x_{i})\;=\sum _{j=0,i\neq j}^{n}{\frac {(-1)^{i-j+1}}{i-j}}{\frac {i!(n-i)!}{j!(n-j)!}}(f_{i}-f_{j})\;+\;{\frac {(-1)^{n-i}i!(n-i)!}{(n+1)!}}h^{n+1}f^{(n+1)}(\xi )\;=\sum _{j=0,i\neq j}^{n}{\frac {(-1)^{i-j}}{i-j}}{\frac {\binom {n}{j}}{\binom {n}{i}}}(f_{j}-f_{i})\;+\;{\frac {(-1)^{n-i}}{{\binom {n}{i}}(n+1)}}h^{n+1}f^{(n+1)}(\xi )}
2-punts eerste afgeleide:
h
f
(
1
)
(
x
0
)
=
−
f
0
+
f
1
−
1
2
h
2
f
(
2
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{0})=-f_{0}+f_{1}-{\frac {1}{2}}h^{2}f^{(2)}(\xi )}
h
f
(
1
)
(
x
1
)
=
−
f
0
+
f
1
+
1
2
h
2
f
(
2
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{1})=-f_{0}+f_{1}+{\frac {1}{2}}h^{2}f^{(2)}(\xi )}
3-punts eerste afgeleide:
h
f
(
1
)
(
x
0
)
=
1
2
(
−
3
f
0
+
4
f
1
−
f
2
)
+
1
3
h
3
f
(
3
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{0})={\frac {1}{2}}(-3f_{0}+4f_{1}-f_{2})+{\frac {1}{3}}h^{3}f^{(3)}(\xi )}
h
f
(
1
)
(
x
1
)
=
1
2
(
−
f
0
+
f
2
)
−
1
6
h
3
f
(
3
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{1})={\frac {1}{2}}(-f_{0}+f_{2})-{\frac {1}{6}}h^{3}f^{(3)}(\xi )}
h
f
(
1
)
(
x
2
)
=
1
2
(
f
0
−
4
f
1
+
3
f
2
)
+
1
3
h
3
f
(
3
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{2})={\frac {1}{2}}(f_{0}-4f_{1}+3f_{2})+{\frac {1}{3}}h^{3}f^{(3)}(\xi )}
3-punts tweede afgeleide:
h
2
f
(
2
)
(
x
0
)
=
f
0
−
2
f
1
+
f
2
−
h
3
f
(
3
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{0})=f_{0}-2f_{1}+f_{2}-h^{3}f^{(3)}(\xi )}
h
2
f
(
2
)
(
x
1
)
=
f
0
−
2
f
1
+
f
2
−
1
12
h
4
f
(
4
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{1})=f_{0}-2f_{1}+f_{2}-{\frac {1}{12}}h^{4}f^{(4)}(\xi )}
h
2
f
(
2
)
(
x
2
)
=
f
0
−
2
f
1
+
f
2
+
h
3
f
(
3
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{2})=f_{0}-2f_{1}+f_{2}+h^{3}f^{(3)}(\xi )}
4-punts eerste afgeleide:
h
f
(
1
)
(
x
0
)
=
1
6
(
−
11
f
0
+
18
f
1
−
9
f
2
+
2
f
3
)
−
1
4
h
4
f
(
4
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{0})={\frac {1}{6}}(-11f_{0}+18f_{1}-9f_{2}+2f_{3})-{\frac {1}{4}}h^{4}f^{(4)}(\xi )}
h
f
(
1
)
(
x
1
)
=
1
6
(
−
2
f
0
−
3
f
1
+
6
f
2
−
f
3
)
+
1
12
h
4
f
(
4
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{1})={\frac {1}{6}}(-2f_{0}-3f_{1}+6f_{2}-f_{3})+{\frac {1}{12}}h^{4}f^{(4)}(\xi )}
h
f
(
1
)
(
x
2
)
=
1
6
(
f
0
−
6
f
1
+
3
f
2
+
2
f
3
)
−
1
12
h
4
f
(
4
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{2})={\frac {1}{6}}(f_{0}-6f_{1}+3f_{2}+2f_{3})-{\frac {1}{12}}h^{4}f^{(4)}(\xi )}
h
f
(
1
)
(
x
3
)
=
1
6
(
−
2
f
0
+
9
f
1
−
18
f
2
+
11
f
3
)
+
1
4
h
4
f
(
4
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{3})={\frac {1}{6}}(-2f_{0}+9f_{1}-18f_{2}+11f_{3})+{\frac {1}{4}}h^{4}f^{(4)}(\xi )}
4-punts tweede afgeleide:
h
2
f
(
2
)
(
x
0
)
=
2
f
0
−
5
f
1
+
4
f
2
−
f
3
+
11
12
h
4
f
(
4
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{0})=2f_{0}-5f_{1}+4f_{2}-f_{3}+{\frac {11}{12}}h^{4}f^{(4)}(\xi )}
h
2
f
(
2
)
(
x
1
)
=
f
0
−
2
f
1
+
f
2
−
1
12
h
4
f
(
4
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{1})=f_{0}-2f_{1}+f_{2}-{\frac {1}{12}}h^{4}f^{(4)}(\xi )}
h
2
f
(
2
)
(
x
2
)
=
f
1
−
2
f
2
+
f
3
−
1
12
h
4
f
(
4
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{2})=f_{1}-2f_{2}+f_{3}-{\frac {1}{12}}h^{4}f^{(4)}(\xi )}
h
2
f
(
2
)
(
x
3
)
=
−
f
0
+
4
f
1
−
5
f
2
+
2
f
3
+
11
12
h
4
f
(
4
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{3})=-f_{0}+4f_{1}-5f_{2}+2f_{3}+{\frac {11}{12}}h^{4}f^{(4)}(\xi )}
4-punts derde afgeleide:
h
3
f
(
3
)
(
x
0
)
=
−
f
0
+
3
f
1
−
3
f
2
+
f
3
−
3
2
h
4
f
(
4
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{0})=-f_{0}+3f_{1}-3f_{2}+f_{3}-{\frac {3}{2}}h^{4}f^{(4)}(\xi )}
h
3
f
(
3
)
(
x
1
)
=
−
f
0
+
3
f
1
−
3
f
2
+
f
3
−
1
2
h
4
f
(
4
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{1})=-f_{0}+3f_{1}-3f_{2}+f_{3}-{\frac {1}{2}}h^{4}f^{(4)}(\xi )}
h
3
f
(
3
)
(
x
2
)
=
−
f
0
+
3
f
1
−
3
f
2
+
f
3
+
1
2
h
4
f
(
4
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{2})=-f_{0}+3f_{1}-3f_{2}+f_{3}+{\frac {1}{2}}h^{4}f^{(4)}(\xi )}
h
3
f
(
3
)
(
x
3
)
=
−
f
0
+
3
f
1
−
3
f
2
+
f
3
+
3
2
h
4
f
(
4
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{3})=-f_{0}+3f_{1}-3f_{2}+f_{3}+{\frac {3}{2}}h^{4}f^{(4)}(\xi )}
5-punts eerste afgeleide:
h
f
(
1
)
(
x
0
)
=
1
12
(
−
25
f
0
+
48
f
1
−
36
f
2
+
16
f
3
−
3
f
4
)
+
1
5
h
5
f
(
5
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{0})={\frac {1}{12}}(-25f_{0}+48f_{1}-36f_{2}+16f_{3}-3f_{4})+{\frac {1}{5}}h^{5}f^{(5)}(\xi )}
h
f
(
1
)
(
x
1
)
=
1
12
(
−
3
f
0
−
10
f
1
+
18
f
2
−
6
f
3
+
f
4
)
−
1
20
h
5
f
(
5
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{1})={\frac {1}{12}}(-3f_{0}-10f_{1}+18f_{2}-6f_{3}+f_{4})-{\frac {1}{20}}h^{5}f^{(5)}(\xi )}
h
f
(
1
)
(
x
2
)
=
1
12
(
f
0
−
8
f
1
+
8
f
3
−
f
4
)
+
1
30
h
5
f
(
5
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{2})={\frac {1}{12}}(f_{0}-8f_{1}+8f_{3}-f_{4})+{\frac {1}{30}}h^{5}f^{(5)}(\xi )}
h
f
(
1
)
(
x
3
)
=
1
12
(
−
f
0
+
6
f
1
−
18
f
2
+
10
f
3
+
3
f
4
)
−
1
20
h
5
f
(
5
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{3})={\frac {1}{12}}(-f_{0}+6f_{1}-18f_{2}+10f_{3}+3f_{4})-{\frac {1}{20}}h^{5}f^{(5)}(\xi )}
h
f
(
1
)
(
x
4
)
=
1
12
(
3
f
0
−
16
f
1
+
36
f
2
−
48
f
3
+
25
f
4
)
+
1
5
h
5
f
(
5
)
(
ξ
)
{\displaystyle h\,f^{(1)}(x_{4})={\frac {1}{12}}(3f_{0}-16f_{1}+36f_{2}-48f_{3}+25f_{4})+{\frac {1}{5}}h^{5}f^{(5)}(\xi )}
5-punts tweede afgeleide:
h
2
f
(
2
)
(
x
0
)
=
1
12
(
35
f
0
−
104
f
1
+
114
f
2
−
56
f
3
+
11
f
4
)
−
5
6
h
5
f
(
5
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{0})={\frac {1}{12}}(35f_{0}-104f_{1}+114f_{2}-56f_{3}+11f_{4})-{\frac {5}{6}}h^{5}f^{(5)}(\xi )}
h
2
f
(
2
)
(
x
1
)
=
1
12
(
11
f
0
−
20
f
1
+
6
f
2
+
4
f
3
−
f
4
)
+
1
12
h
5
f
(
5
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{1})={\frac {1}{12}}(11f_{0}-20f_{1}+6f_{2}+4f_{3}-f_{4})+{\frac {1}{12}}h^{5}f^{(5)}(\xi )}
h
2
f
(
2
)
(
x
2
)
=
1
12
(
−
f
0
+
16
f
1
−
30
f
2
+
16
f
3
−
f
4
)
+
1
90
h
6
f
(
6
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{2})={\frac {1}{12}}(-f_{0}+16f_{1}-30f_{2}+16f_{3}-f_{4})+{\frac {1}{90}}h^{6}f^{(6)}(\xi )}
h
2
f
(
2
)
(
x
3
)
=
1
12
(
−
f
0
+
4
f
1
+
6
f
2
−
20
f
3
+
11
f
4
)
−
1
12
h
5
f
(
5
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{3})={\frac {1}{12}}(-f_{0}+4f_{1}+6f_{2}-20f_{3}+11f_{4})-{\frac {1}{12}}h^{5}f^{(5)}(\xi )}
h
2
f
(
2
)
(
x
4
)
=
1
12
(
11
f
0
−
56
f
1
+
114
f
2
−
104
f
3
+
35
f
4
)
+
5
6
h
5
f
(
5
)
(
ξ
)
{\displaystyle h^{2}f^{(2)}(x_{4})={\frac {1}{12}}(11f_{0}-56f_{1}+114f_{2}-104f_{3}+35f_{4})+{\frac {5}{6}}h^{5}f^{(5)}(\xi )}
5-punts derde afgeleide:
h
3
f
(
3
)
(
x
0
)
=
1
2
(
−
5
f
0
+
18
f
1
−
24
f
2
+
14
f
3
−
3
f
4
)
+
7
4
h
5
f
(
5
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{0})={\frac {1}{2}}(-5f_{0}+18f_{1}-24f_{2}+14f_{3}-3f_{4})+{\frac {7}{4}}h^{5}f^{(5)}(\xi )}
h
3
f
(
3
)
(
x
1
)
=
1
2
(
−
3
f
0
+
10
f
1
−
12
f
2
+
6
f
3
−
f
4
)
+
1
4
h
5
f
(
5
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{1})={\frac {1}{2}}(-3f_{0}+10f_{1}-12f_{2}+6f_{3}-f_{4})+{\frac {1}{4}}h^{5}f^{(5)}(\xi )}
h
3
f
(
3
)
(
x
2
)
=
1
2
(
−
f
0
+
2
f
1
−
2
f
3
+
f
4
)
−
1
4
h
5
f
(
5
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{2})={\frac {1}{2}}(-f_{0}+2f_{1}-2f_{3}+f_{4})-{\frac {1}{4}}h^{5}f^{(5)}(\xi )}
h
3
f
(
3
)
(
x
3
)
=
1
2
(
f
0
−
6
f
1
+
12
f
2
−
10
f
3
+
3
f
4
)
+
1
4
h
5
f
(
5
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{3})={\frac {1}{2}}(f_{0}-6f_{1}+12f_{2}-10f_{3}+3f_{4})+{\frac {1}{4}}h^{5}f^{(5)}(\xi )}
h
3
f
(
3
)
(
x
4
)
=
1
2
(
3
f
0
−
14
f
1
+
24
f
2
−
18
f
3
+
5
f
4
)
+
7
4
h
5
f
(
5
)
(
ξ
)
{\displaystyle h^{3}f^{(3)}(x_{4})={\frac {1}{2}}(3f_{0}-14f_{1}+24f_{2}-18f_{3}+5f_{4})+{\frac {7}{4}}h^{5}f^{(5)}(\xi )}
5-punts vierde afgeleide:
h
4
f
(
4
)
(
x
0
)
=
f
0
−
4
f
1
+
6
f
2
−
4
f
3
+
f
4
−
2
h
5
f
(
5
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{0})=f_{0}-4f_{1}+6f_{2}-4f_{3}+f_{4}-2h^{5}f^{(5)}(\xi )}
h
4
f
(
4
)
(
x
1
)
=
f
0
−
4
f
1
+
6
f
2
−
4
f
3
+
f
4
−
h
5
f
(
5
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{1})=f_{0}-4f_{1}+6f_{2}-4f_{3}+f_{4}-h^{5}f^{(5)}(\xi )}
h
4
f
(
4
)
(
x
2
)
=
f
0
−
4
f
1
+
6
f
2
−
4
f
3
+
f
4
−
1
6
h
6
f
(
6
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{2})=f_{0}-4f_{1}+6f_{2}-4f_{3}+f_{4}-{\frac {1}{6}}h^{6}f^{(6)}(\xi )}
h
4
f
(
4
)
(
x
3
)
=
f
0
−
4
f
1
+
6
f
2
−
4
f
3
+
f
4
+
h
5
f
(
5
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{3})=f_{0}-4f_{1}+6f_{2}-4f_{3}+f_{4}+h^{5}f^{(5)}(\xi )}
h
4
f
(
4
)
(
x
4
)
=
f
0
−
4
f
1
+
6
f
2
−
4
f
3
+
f
4
+
2
h
5
f
(
5
)
(
ξ
)
{\displaystyle h^{4}f^{(4)}(x_{4})=f_{0}-4f_{1}+6f_{2}-4f_{3}+f_{4}+2h^{5}f^{(5)}(\xi )}
A backward difference uses the function values at x and x − h x + h x :
∇
h
[
f
]
(
x
)
=
f
(
x
)
−
f
(
x
−
h
)
.
{\displaystyle \nabla _{h}[f](x)=f(x)-f(x-h).}
∇
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{\displaystyle \nabla _{h}^{n}[f](x)=\sum _{i=0}^{n}(-1)^{i}{\binom {n}{i}}f(x-ih),}
The Newton series Newton forward difference equation , named after Isaac Newton ; in essence, it is the Newton interpolation formula , first published in his Principia Mathematica [ 1]
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{\displaystyle f(x)=\sum _{k=0}^{\infty }{\frac {\Delta ^{k}[f](a)}{k!}}\,(x-a)_{k}=\sum _{k=0}^{\infty }{\binom {x-a}{k}}\,\Delta ^{k}[f](a),}
which holds for any polynomial function f and for many (but not all) analytic functions (It does not hold when f is exponential type
π
{\displaystyle \pi }
π
{\displaystyle \pi }
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!
{\displaystyle {\binom {x}{k}}={\frac {(x)_{k}}{k!}}}
is the binomial coefficient , and
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⋯
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{\displaystyle (x)_{k}=x(x-1)(x-2)\cdots (x-k+1)}
is the "falling factorial " or "lower factorial", while the empty product (x )0 is defined to be 1. In this particular case, there is an assumption of unit steps for the changes in the values of x , h = 1
Note the formal correspondence of this result to Taylor's theorem . Historically, this, as well as the Chu–Vandermonde identity ,
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{\displaystyle (x+y)_{n}=\sum _{k=0}^{n}{\binom {n}{k}}(x)_{n-k}\,(y)_{k},}
(following from it, and corresponding to the binomial theorem ), are included in the observations that matured to the system of umbral calculus .
In a compressed and slightly more general form and equidistant nodes the formula reads
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{\displaystyle f(x)=\sum _{k=0}{\binom {\frac {x-a}{h}}{k}}\sum _{j=0}^{k}(-1)^{k-j}{\binom {k}{j}}f(a+jh).}
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⇒
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{\displaystyle (-1)^{k}ke_{k}(x_{1},\ldots ,x_{n})=\sum _{j=1}^{k}(-1)^{k-j+1}p_{j}(x_{1},\ldots ,x_{n})e_{k-j}(x_{1},\ldots ,x_{n})\Rightarrow ka_{k}=-\sum _{j=1}^{k}a_{k-j}p_{j}(x_{1},\ldots ,x_{n})}
=f(x)-f(x-h).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4916660138355c77211844935d43b20ba90144f6)
=\sum _{i=0}^{n}(-1)^{i}{\binom {n}{i}}f(x-ih),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2d02b454bc5bec308d6b40c4b1d6fd0e5177a6f)
}{k!}}\,(x-a)_{k}=\sum _{k=0}^{\infty }{\binom {x-a}{k}}\,\Delta ^{k}[f](a),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0dcfd8646a4b3c907637c9a16dfef5fabeefac80)
