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Newton’s polynomials.
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Let us consider a divided difference:
f(x, x0) = 
.
The unknown value is:
f(x) = f(x0)+f(x, x0)(x–x0). (1)
We know f(x0), but we do not know f(x, x0), then
f(x, x0, x1) = 
,
so
f(x, x0) = f(x0, x1)+f(x, x0, x1)(x–x1). (2)
Now let us put (2) into (1):
f(x) = f(x0)+f(x0, x1)(x–x0)+f(x, x0, x1)(x–x0)(x–x1).
Continuing this process we shall obtain:
f(x)=f(x0)+f(x0, x1)(x–x0)+f(x0, x1, x2)(x–x0)(x–x1)+…+f(x0, x1, …, xn)(x–x0)(x–x1)…
(x––xn–1)+ f(x, x0, x1, …, xn)(x–x0)(x–x1)…(x–xn).
The member f(x, x0, x1, …, xn)(x–x0)(x–x1)…(x–xn) is unknown, so we can say that it is an error, then
f(x)»f(x0)+f(x0, x1)(x–x0)+f(x0, x1, x2)(x–x0)(x–x1)+…+f(x0, x1, …, xn)(x–x0)(x–x1)…(x––xn–1)
f(x)=Pn(x) + f(x, x0, x1,.., xn)(x-x0)(x-x1)...(x-xn)
f(x)@Pn(x)
Pn(x)= y0+y0, 1(x-x0)+y0, 1, 2(x-x0)(x-x1)+...+y0, 1,...n(x-x0)...(x-xn-1)
When we have proportional table:
(3)
x0 y0
Dy0
x1 y1 D2y0
Dy1 D3y0
x2 y2 D2y1 D4y0
Dy2 D3y1 D5y0
x3 y3 D2y2 D4y1
Dy3 D3y2
x4 y4 D2y3
Dy4
x5 y5
Formula (3) named “First interpolational formula of Newton”, or “formula of forward interpolation ”
It use divided differences, which form high incline in the table of differences.
Formula (3) is used when unknown point x is at the beginning of the table.
Consider difference
etc.
f(x) @ yn + yn, n-1(x-xn) + yn, n-1, n-2(x-xn)(x-xn-1)+..+yn, n-1,..., 1, 0(x-xn)(x-xn-1)...(x-x1)
(4)
Formula (4) named «Second interpolational formula of Newton» or «formula of back interpolation». It use divided differences, which form lower incline in the table of differences. Formula (4) is used when point x is at the ending of table.
Interpolation in the middle of the table.
Let’s rename table points in such way
x-3 y-3
Dy-3
x-2 y-2 D2y-3
Dy-2 D3y-3
x-1 y-1 D2y-2 D4y-3
D y-1D3y-2D5y-3
x0 y0 D2y-1D4y-2 D6y-3
~~ Dy0 ~~ D3y-1 ~~ D5y-2 ~~
x1 y1 ~~ D2y0 ~~~ D4y-1 ~~
Dy1 D3y0
x2 y2 D2y1
Dy2
x3 y3
Consider first interpolational formula of Newton, that use differences, formed lower braked line in the middle of the table(underline «~»):
(5)
(5)- first interpolational formula of Gauss. Analogously, consider differences, formed high braked line:
(6)
(6) - second interpolational formula of Gauss.
Numerical integration.
When F(x) is unknown then:
x0 x1 x2... xn
y0 y1 y2... yn yi=f(xi)
f(x) @ L(x)
.
Coefficients Ai are depended only of points and not depend of function.
When we have proportional table: x=x0+ht
dx=hdt
-coefficients of Kotes.
They depend only of number of points and may be calculated for various n.
- formula of Newton-Kotes.
Particular cases.
1)
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b
ò f(x)dx=SX0, Y0, Xn, Yn
a
- formula of left rectangles.
- formula of right rectangles.
2) n=1 (particular formula of trapezes): h=b-a
.
Total formula of trapezes: 
3) n=2 (particular formula of Simpson)
.
Total formula of Simpson: 
4) n=3 (particular formula of Newton);
Total formula of Newton (law of 
)
n=3m
Error:
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