Численные методы. Ч.2. Решение уравнений. Буслов В.А - 4 стр.

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СПбГУ | Санкт-Петербург

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Рубрика: 

x
∗
x
∗
= lim
n→∞
x
n
, x
n+1
= F (x
n
) ,
x
0
[a, b] F
q = sup
x∈[a,b]
|F
0
(x)| = ||F
0
||
C
< 1
|F (x) − F (y)| = |F
0
(ξ)||x − y| ≤ ||F
0
||
C
|x − y| = q|x − y| .
F
F ([a, b]) ⊆ [a, b] F (x) ≡ 2
q q
x
2
= a F F (x) =
a
x
x
n+1
=
a
x
n
x
0
x
∗
=
√
a F
F (x) =
1
2
[x +
a
x
]
x
n+1
=
1
2
[x
n
+
a
x
n
]
x
0
∈ (0, ∞)
F (x) =
a
x
, x
n+1
=
a
x
n
,
F (x) =
1
2
[x +
a
x
] , x
n+1
=
1
2
[x
n
+
a
x
n
] , x
n
→ x
∗
.
F
0
(x
n
) = −
a
x
2
n
F
0
(x
n
) < 1
x
2
n
> a |F
0
(x
n+1
)| = |
a
x
2
n+1
| =
a
a
2
x
2
n
=
x
2
n
a
> 1 F (x) =
a
x
F (x) =
1
2
[x +
a
x
]
x
F
0
(x
n+1
) =
1
2
·
1 −
a
x
2
n+1
¸
=
1
2
"
1 −
a
1
2
(x
n
+
a
x
n
)
2
#
=
=
1
2
"
1 −
a
x
2
n
1
2
(1 +
a
x
2
n
)
2
#
=
1
2
(1 +
a
x
2
n
)
2
−
2a
x
2
n
(1 +
a
x
2
n
)
2
=
1
2
1 +
¡
a
x
2
n
¢
2
(1 +
a
x
2
n
)
2
<
1
2
,
x
j
x
∗
f(x) = 0 , f ∈ C
1
f(x) x
j
f
0
(x
j
) =
y −f (x
j
)
x−x
j
y = 0 x = x
j+1
x
j+1
= x
j
−
f(x
j
)
f
0
(x
j
)
.

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