Численные методы. Ч.1. Исследование функций. Буслов В.А - 33 стр.

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= (∆x)
k+1
f
(k+1)
(x + kΘ∆x + Θ
0
∆x) = (∆x)
k+1
f
(k+1)
(x + (kΘ + Θ
0
)∆x) .
Θ
0
< 1 Θ) Θ
00
=
kΘ+Θ
0
k+1
(∆x)
k+1
f
(k+1)
(x + (k + 1)Θ
00
∆x) .
Θ
00
< 1
f
(k)
(x) =
∆
k
f
(∆x)
k
+ o(1) .
∆
k
f
(∆x)
k
= f
(k)
(x + Θk∆x) ∆x → 0 f
(k)
(x) = lim
∆x→0
∆
k
f
(∆x)
k
∆
x
x
[n]
≡ x(x − h)(x − 2h) . . . (x − (n − 1)h) , x
[0]
≡ 1.
h = 0 x
[n]
= x
n
∆
k
x
[n]
= n(n − 1) . . . (n − (k −1))h
k
x
[n−k]
.
∆x
[n]
= (x + h)
[n]
− x
[n]
=
= (x + h)x(x − h) . . . (x − (n − 2)h) − x(x − h) . . . (x − (n −1)h) =
= x(x − h) . . . (x − (n − 2)h)[x + h − (x − (n − 1)h)] = nhx
[n−1]
,
∆
∆
2
x
[n]
= ∆(∆x
[n]
) = ∆(nhx
[n−1]
) = nh(n − 1)hx
[n−2]
=
= n(n − 1)h
2
x
[n−2]
,
∆
d
k
x
n
= n(n − 1) . . . (n − (k −1))x
n−k
(dx)
k
.
x
0
, x
1
, . . . , x
N
: x
i
= x
0
+ ih f
0
, f
1
, . . . , f
N
p(x) : p(x
i
) = f
i
, i = 0 , 1 , . . . , N , deg p(x) = N . (4)
{x
i
, f
i
}
N
i=0
p(x) =
N
X
k=0
f
012 ... k
N
k
(x) .

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