Антоник В.Г. Численные методы: математический анализ и дифференциальные уравнения
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p(x) q(x)
x
0
, x
1
, . . . , x
n
n
X
k=0
p(x
k
)q(x
k
) = 0 .
p
0
(x), p
1
(x), . . . ,
p
m
(x)
x
i
i = 0, n
n
X
k=0
p
i
(x
k
)p
j
(x
k
) = 0 , i, j = 0, m , i 6= j .
p
i
(x),
i = 0, m
s
kj
= 0, k 6= j
α
∗
a
∗
0
=
n
P
i=0
f
i
p
0
(x
i
)
n
P
i=0
p
2
0
(x
i
)
, . . . , a
∗
m
=
n
P
i=0
f
i
p
m
(x
i
)
n
P
i=0
p
2
m
(x
i
)
.
{x
i
}, i = 0, n
p
0
(x) = 1 , p
1
(x) = x − b ,
p
i
(x) = (x − c
i
)p
i−1
(x) − d
i
p
i−2
(x) , i = 2, m ,
b =
1
n + 1
n
X
j=0
x
j
, c
i
=
n
P
j=0
x
j
p
i−1
(x
j
)
n
P
j=0
p
2
i−1
(x
j
)
,
d
i
=
n
P
j=0
x
j
p
i−1
(x
j
)p
i−2
(x
j
)
n
P
j=0
p
2
i−2
(x
j
)
.
f(x)
x
i
f
i
Q
1
(x, α
∗
) ϕ(α
∗
)
n = 2 , Q
1
(x, α) = a
0
p
0
(x) + a
1
p
1
(x) ,
p
0
(x) = 1 , b =
1
3
3
X
j=0
x
j
= 2 , p
1
(x) = x − 2 .
x
i
f
i
p
0
(x
i
) p
1
(x
i
) p
2
0
(x
i
) p
2
1
(x
i
) f
i
p
0
(x
i
) f
i
p
1
(x
i
)
Σ
a
∗
0
=
7
3
, a
∗
1
=
5
2
.
Q
1
(x, α
∗
) =
7
3
x +
5
2
(x − 2) ,
ϕ(α
∗
) =
2
X
i=0
³
Q
1
(x
i
, α
∗
) − f(x
i
)
´
2
=
=
³
−
1
6
− 0
´
2
+
³
7
3
− 2
´
2
+
³
29
6
− 5
´
2
=
=
1
36
+
1
9
+
1
36
=
1
6
.
g(x)
[a, b]
g(x) =
g
1
(x) , x ∈ [x
0
, x
1
) ,
g
2
(x) , x ∈ [x
1
, x
2
) ,
. . .
g
n
(x) , x ∈ [x
n−1
, x
n
] ,
g
i
(x) = c
i
x + d
i
, i = 1, n .
c
i
, d
i
, i = 1, n
g(x)
g(x
i
) = f
i
, i = 0, n ;
g(x) x
i
, i = 1, n − 1
g
i
(x
i
) = g
i+1
(x
i
) , i = 1, n − 1 .
g
i
(x)
c
1
x
0
+ d
1
= f
0
,
c
1
x
1
+ d
1
= f
1
,
c
2
x
1
+ d
2
= f
1
,
c
2
x
2
+ d
2
= f
2
,
. . .
c
n
x
n−1
+ d
n
= f
n−1
,
c
n
x
n
+ d
n
= f
n
.
c
i
, d
i
i = 1, n 2(n−1)+ 2 =
2n
g
i
(x), i = 1, n
g(x)
f(x)
x
i
f
i
g(x)
f(x)
n = 2
g
1
(x) = c
1
x + d
1
, g
2
(x) = c
2
x + d
2
.
c
1
, d
1
, c
2
, d
2
c
1
x
0
+ d
1
= f
0
,
c
1
x
1
+ d
1
= f
1
,
c
2
x
1
+ d
2
= f
1
,
c
2
x
2
+ d
2
= f
2
.
−c
1
+ d
1
= 0 ,
2c
1
+ d
1
= 2 ,
2c
2
+ d
2
= 2 ,
3c
2
+ d
2
= −1 .
c
1
= d
1
=
2
3
, c
2
= −3 , d
2
= 8 .
g(x) =
2
3
x +
2
3
, −1 ≤ x ≤ 2 ,
−3x + 8 , 2 ≤ x ≤ 3 . ¤
f(x)
x
i
f
i
f(1)
f(x)
x
i
f
i
f(2)
f(x) =
1
x + 2
, x ∈ [−1, 1]
{−1, 0, 1}
f(x) = −sin
x
2
, x ∈ [−π, π]
{−π, 0, π}
x
1
0x
2
−1, 3
1, 1 2, 3
x
2
= =
f(x
1
)
f(x) =
1
a
2
− x
, x ∈ [−4, −1]
{−4, −3, −2, −1}
a
10
−5
f(x)
x
i
f
i
P
1
(x, α
∗
) f(x)
f(x)
x
i
f
i
P
2
(x, α
∗
) f(x)
f(x)
x
i
f
i
P
1
(x, α
∗
) f(x)
x
i
f
i
f(x)
f(2)
f(x) = e
x
, x ∈ [0, 1]
P
1
(x, α
∗
)
x
i
f
i
f(1.5)
f(x)
x
i
f
i
P
2
(x, α
∗
)
f(x) ϕ(α
∗
)
f(x)
x
i
f
i
P
1
(x, α
∗
)
f(x) ϕ(α
∗
)
P
3
(x, α)
P
3
(−1, α) = 0 , P
3
(0, α) =
7
3
, P
3
(1, α) = 1 , P
3
(2, α) = 2 .
f(x)
x
i
f
i
g(x)
f(x)
f(x)
x
i
f
i
Q
1
(x, α
∗
) ϕ(α
∗
)
f(x)
x
i
f
i
Q
1
(x, α
∗
) ϕ(α
∗
)