Антоник В.Г. Численные методы: математический анализ и дифференциальные уравнения
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N
n
(x) = c
0
+ c
1
(x − x
0
) + c
2
(x − x
0
)(x − x
1
) + . . . +
+c
n
(x − x
0
)(x − x
1
) . . . (x − x
n−1
) .
c
i
, i = 0, n
g(x) = N
n
(x)
N
n
(x
0
) = c
0
, N
n
(x
1
) = c
0
+ c
1
(x
1
− x
0
) ,
N
n
(x
2
) = c
0
+ c
1
(x
2
− x
0
) + c
2
(x
2
− x
0
)(x
2
− x
1
) ,
. . .
N
n
(x
n
) = c
0
+ c
1
(x
n
−x
0
) + . . . + c
n
(x
n
−x
0
) ···(x
n
−x
n−1
) .
c
0
, c
1
, . . . , c
n
N
n
(x
i
) = f
i
, i = 0, n .
f(x)
x
i
f
i
N
2
(x) = c
0
+ c
1
(x − x
0
) + c
2
(x − x
0
)(x − x
1
) =
= c
0
+ c
1
(x − 2) + c
2
(x − 2)(x − 4) .
N
2
(2) = c
0
, N
2
(4) = c
0
+ c
1
(4 − 2) = c
0
+ 2 c
1
,
N
2
(5) = c
0
+ c
1
(5 − 2) + c
2
(5 − 2)(5 − 4) = c
0
+ 3 c
1
+ 3 c
2
.
c
0
= −2 ,
c
0
+ 2c
1
= 1 ,
c
0
+ 3c
1
+ 3c
2
= 4 .
c
0
= −2 , c
1
=
3
2
, c
2
=
1
2
.
N
2
(x) = −2 +
3
2
(x − 2) +
1
2
(x − 2)(x − 4) =
=
1
2
x
2
−
3
2
x − 1 . ¤
g(x) x
i
, f(x
i
) i = 0, n
g(x)
P
m
(x, α) = a
0
+ a
1
x + . . . + a
m
x
m
, α = (a
0
, a
1
, . . . , a
m
)
n m ≤ n
α P
m
ϕ(α) =
n
X
i=0
³
P
m
(x
i
, α) − f(x
i
)
´
2
→ min
{α}
.
f(x)
α
∗
= (a
∗
0
, a
∗
1
, . . . , a
∗
m
)
P
m
(x, α
∗
)
m = n
n P
n
(x, α
∗
)
ϕ(α
∗
) = 0
m < n
P
m
(x, α
∗
)
α
∗
∂ϕ(α)
∂a
k
= 0 , k = 0, m .
n
X
i=0
³
P
m
(x
i
, α) − f(x
i
)
´
x
k
i
= 0 , k = 0, m .
a
0
, a
1
, . . . , a
m
m
X
j=0
s
kj
a
j
=
n
X
i=0
f(x
i
)x
k
i
,
s
kj
=
n
X
i=0
x
k+j
i
, k, j = 0, m .
f(x)
x
i
f
i
P
1
(x, α
∗
) f(x)
ϕ(α
∗
)
n = 2, m = 1, P
1
(x, α) = a
0
+ a
1
x
x
i
f
i
x
0
i
x
1
i
x
2
i
f
i
x
0
i
f
i
x
1
i
Σ
½
3a
0
+ 7a
1
= −2 ,
7a
0
+ 21a
1
= −7 .
a
∗
0
=
1
2
, a
∗
1
= −
1
2
.
P
1
(x, α
∗
) =
1 − x
2
,
ϕ(α
∗
) =
2
X
i=0
³
P
1
(x
i
, α
∗
) − f(x
i
)
´
2
=
= (0 − (1))
2
+ (− 0.5 − 1)
2
+ ( −1.5 − (−2))
2
= 3.5 . ¤
x
0
, x
1
, . . . , x
n
f(x) , x ∈ [a, b] ;
P
m
(x, α) = a
0
+ a
1
x + . . . + a
m
x
m
;
ϕ(α) =
b
Z
a
³
P
m
(x, α) − f(x)
´
2
dx → min
{α}
;
α
∗
∂ϕ(α)
∂a
k
= 0 , k = 0, m ⇔
⇔
b
Z
a
³
P
m
(x, α) − f(x)
´
x
k
dx = 0 , k = 0, m .
m
X
j=0
s
kj
a
j
=
b
Z
a
f(x)x
k
dx ,
s
kj
=
b
Z
a
x
k+j
dx , k, j = 0, m .
g(x)
m m ≤ n
g(x) = P
m
(x, α)
α = (a
0
,
a
1
, . . . , a
m
) P
m
ϕ(α) = max
0≤i≤n
|P
m
(x
i
, α) − f(x
i
)| → min
{α}
.
ϕ(α)
P
m
(x, α)
f(x) x
i
f(x)
α
∗
= (a
∗
0
, a
∗
1
, . . . , a
∗
m
)
P
m
(x, α
∗
)
f(x)
m
n
m = n P
n
(x, α
∗
)
ϕ(α
∗
) = 0
m = n − 1
P
n−1
(x, α
∗
)
α
∗
a
0
+ a
1
x
0
+ . . . + a
m
x
m
0
+ h = f(x
0
) ,
a
0
+ a
1
x
1
+ . . . + a
m
x
m
1
− h = f(x
1
) ,
. . .
a
0
+ a
1
x
n
+ . . . + a
m
x
m
n
+ (−1)
n
h = f(x
n
) .
h
P (x, α
∗
) f(x) x
i
, i = 0, n
|h| ϕ(α
∗
) = |h|
m < n −1
P
m
(x, α
∗
)
α
∗
P
m
m + 2 {x
i
0
, . . . , x
i
m+1
} {x
0
, . . . , x
n
}
m = n
m = n −1
m < n − 1
m + 2 {x
0
, x
1
, . . . , x
n
}
P
m
(x, α
∗
k
), k = 1, 2, . . .
φ(α
∗
k
)
f(x)
x
i
f
i
P
1
(x, α
∗
)
f(x) ϕ(α
∗
)
n = 2, m = 1, P
1
(x, α) = a
0
+ a
1
x
m = n − 1
a
0
− a
1
+ h = 0 ,
a
0
− h = 1 ,
a
0
+ a
1
+ h = 4 .
h =
1
2
, a
∗
0
=
3
2
, a
∗
1
= 2 .
P
1
(x, α
∗
) =
3
2
+ 2 x , ϕ(α
∗
) = |h| =
1
2
. ¤
f(x)
{x
i
, f(x
i
)}, i = 0, n
Q(x, α) = a
0
p
0
(x) + a
1
p
1
(x) + . . . + a
m
p
m
(x) .
α = (a
0
, a
1
, . . . , a
m
)
p
0
(x), p
1
(x), . . . , p
m
(x)
p
i
(x) = x
i
, i = 0, m
Q
m
(x, α)
a
0
, a
1
, . . . , a
m
ϕ(α) =
n
X
i=0
³
Q
m
(x
i
, α) − f(x
i
)
´
2
ϕ(α) → min
{α}
.
α
∗
= (a
∗
0
, a
∗
1
,
. . . , a
∗
1
)
∂ϕ(α)
∂a
k
= 0 , k = 0, m .
m
X
j=0
s
kj
a
j
=
n
X
i=0
f(x
i
)p
k
(x
i
) ,
s
kj
=
n
X
i=0
p
k
(x
i
)p
j
(x
i
) , k, j = 0, m .