1
LU
2
x x b
p
±m · b ,
m b =2
p
10 m N N
m = 0, a1 a2 . . . aN =
b−1
ak b−k ,
k=1
∆x ∆m = x m
−N
≤ bb−1
≤ m < 1.
= b1−N .
p
L xmin
≤ p ≤ U
xmax
xmin = bL−1
U
≤ |x| ≤ b
= xmax .
n N n
−1
10−2
10
+3
3
x10 = 0, 3
·
425,, 33 425 6
− N − 1
x = 4 102 +2 10+5+3 y = 4 62 +2 6+5+3 6−1 +3 6−2
10
·
·
x = 1/3
·
·
·
·
·
x3 = 0, 1
10
N = 3 x = 8, 22 = 0, 0, 822 10 10,,
·
y = 0, 00317 = 0, 0, 317 10−2 ,
·
z = 0, 00432 = 0, 0, 432 10−2 .
·
x+y+z (x + y) + z x + y = 8, 8 , 22317 0, 822 10 (x + y ) + z 8, 22432 0, 822 10
·
·
x + (y (y + z ) y + z = 0, 00749 0, 749 10−2 x + (y (y + z ) = 8, 22749 0, 823 10
x2 N = 10
·
·
− 1634 1634x x+2 = 0
√ 816, 9987760 9987760,, √ ∆ = 816, x1 = 817 + ∆ 1633 998776,, √ 1633,, 998776 x2 = 817 − ∆ 0, 0012240 0012240.. ∆ = 667487, 667487,
5 x1 x2 = 2
·
x2 =
2 x1
1, 223991125 · 10−3.
x2
n n Ax = b, A = (aij ) 1
≤ i, j ≤ n b = (b ) 1 ≤ i ≤ n 1≤i≤n
n
i
x = (xi )
×n
n
aij xj = bi ,
i = 1, . . . , n .
j =1
A
detA = 0 x
b
A xi =
det( det(Ai ) , det( det(A)
i = 1, . . . , n ,
Ai
i
A
b
(n + 1)! n
det(A det(A) =
− (σ)
( 1)
ai,σ(i)
i=1
σ
σ
n!
n
109 9, 6 1047
50
·
xn
x
Ax = b
A = LU 1
L U Ax = b Ly = b
U x = y
A = (a ( aij ) aij = 0
∀i, j
: 1
≤ j < i ≤ n
aij = 0
∀i, j
: 1
≤ i < j ≤ n.
A aii = 0
aii A
det(A det(A) = i = 1, . . . , n
x1 = b1 /a11 , i = 2, 3, . . . , n 1 xi = aii
− i−1
bi
aij xj
.
j =1
A xn = bn /ann , i=n
− 1, n − 2, . . . , 2, 1 1 xi = aii
− n
bi
aij xj
.
j =i+1
n(n + 1)/ 1)/2
n(n 2
n
1)/2 − 1)/
LU A = (aij )
n (1)
A
×n
(1)
= A
aij = aij
i, j = 1, . . . , n
k = 1, . . . , n (k )
lik =
aik
(k+1)
aij
(k )
akk
, (k )
= aij
i = k + 1, 1, . . . , n; n; (k ) ik akj ,
−l
i = k + 1, 1, . . . , n .
U (i)
uij = aij , L 2n3 /3 A(k)
A(k) k
lij A(k+1) k
A(k+1) 2(n 2(n
− k)
2
A(n)
n−1
n−1
2(n 2(n
k =1
2
− k)
=2
j 2 = 2
(n
j =1
3 1)n(2n (2n − 1) − 1)n ∼ 2n .
6
3
(k )
(k )
akk
ukk
U
A
A=
1 2 2 4 7 8
3 5 9
A(2) =
1 0 0
2 0 6
3 1 12
− − −
LU
A
n
Ai = (ahk ) h, k = 1, . . . , i A A i = 1, . . . , n
n
|a | > ii
|
aij .
|
j =1,j = i
A j = 1, . . . , n
n
|a | > jj
A
|
aij .
|
i=1,j = i
LU LU O(n3 ) n
det(A det(A) = det(L det(L)det(U )det(U )) = det(U det(U )) =
ukk ,
k =1
k r>k
k k
r
1
prr = pkk = 0
0
prk = pkr = 1
P (k,r ) =
1
0 ... 0
0
... 0 ... 1 ... ... 1 ... 0 ... 0
0 ... 0
1
1 k r n
LU = P A, P
Ax = b ˜ LU x = b
x
˜ Ly = b
P A = LU ˜ ˜ = P b P Ax = b b
U x = y
x + y
= 1,
x+y
= 2.
x + y 1 1 y
= 1,
−
= 2
− 1 .
y y=
1 2 1
− 1. −
x x=
1
− y 0.
x+y
= 2,
x + y
= 1.
x+y (1 )y
−
x+y y
x =1
= 2, = 1 2,
−
= 2, = 1.
y = 1.
A(k) (k)
akk
k
r
r
≥k
|a( ) | = max |a( ) | ≥ k rk
r
k sk
s k
k
A
L
L=
1 β 2
c1
b2
a2
0
1 β n β i
α1 = a1 , β 2 α1 = b2
0
bn
cn−1 an
U
0
0 αi
a1
1
,
U =
α1
.
LU c1
A 0
α2 0
cn−1 αn
.
⇒ β 2 = αb21 , β 2c1 + α2 = a2 ⇒ α2 = a2 − β 2c1, . . .
α1 = a1 , β i =
bi , αi = ai αi−1
− β c −1,
i = 2, . . . , n .
i i
n Ax = f 2 Ly = f U x = y
Ly = f
y1 = f 1 , yi = f i β i yi−1 , yn yi ci xi+1 xn = , xi = , αn αi
↔ ↔
−
3(n 3(n
A
U x = y
×n −1
−
− 1)
AA−1
i=n
− 1, . . . , 1. 5n
8n
−7
−4
v(1) , . . . , v(n)
n
A = I
i = 2, . . . , n ,
A−1 = ( v(1) , . . . , v(n) ) n
Av(k) = e(k) ,
1
≤ k ≤ n,
e(k)
0
k
1
e(k) =
L n
U
0 1 0
1 k . n A
(1) A
A
3
n
A−1 A(j ) vj = 1j , A(j ) (0, (0, 0, . . . , 0, 1)T
j vj = (v1j , . . . , vjj )T j
M
K 2 (M ) M ) =
λmax (M ) . λmin (M )
1j =
Ax = b. ˆ x ˆ x
x
x − xˆ ≤ K 2(A) r x b ˆ · r = b − Ax
r
A
x − xˆ
x(k)
Ax = b
x
lim x(k) . x = lim k→∞
x(k+1) = B x(k) + g,
B
A b x = B x + g. x = A−1 b
g = (I
− B)A−1b
(2)
B
k e(k) = x
e(k) = B e(k−1) ,
− x( ) , k
0 , 1, . . . e(k) = B (k) e(0) , k = 0,
limk→∞ e(k) = 0 ρ(B ) < 1 ρ(B )
e(0)
x(0)
B
ρ(B ) = max λi (B ) ,
|
|
g
λi (B )
B
A P
N
A = P
− N
P P
(0)
x x(k)
k
≥1
P x(k+1) = N x(k) + b, (3) g = P
≥ 0.
P x = N x + b (2)
x −1
k
Ax = b B = P −1 N
b
(3) P ( P (x(k+1) r(k) = b
− Ax( ) k
− x( )) = r( ), k
k
k
≥ 0,
k P ( P (x(k+1)
− x( )) = α r( ) , k
k
k
k
≥ 0,
αk (3) (4)
k
(5)
P
P P
A Ax = b 1 xi = aii
− − n
bi
aij xj
,
j =1,j = i
x(0)
(k+1)
xi
1 = aii
A = D D : E F
i = 1, . . . , n . x(k+1)
n
(k )
bi
− E − F
aij xj
,
i = 1, . . . , n .
j =1,j = i
A
A
: :
0 0 BJ = D−1 (E + E + F ) F ) = I
− D−1A.
(k )
xj
(k+1)
xj
(k+1)
xi
=
1 aii
− i−1
bi
n
(k+1)
aij xj
j =1
−
(k )
aij xj
j =i+1
x(k+1) = (D
− E )−1 b + F x( ) k
A P = D
i = 1, . . . , n .
,
− E,
BGS = (D (D
N = F,
− E )−1U. x(k)
x(k+1)
A
x (n+1)
(n+1)
max xi i
− xi
≤ ≤
A R
1 aii
a x ik
(n) k
− xk
i k=
−x
(n+1)
2
−x
k
k= i
k
∞
(n) ik | max xk − xk
|a K max K max x
(n) k
0≤K<1
k
maxi |yi | = y∞
N
x A
−1 aii
− xi =
xi
≤ K x(n) − x
∞
(5)
P
r( ), z( ) , Az( ), z( ) k
αk =
αk = 1
A αk
k
k
k
k
≥ 0,
z(k) = P −1 r(k)
P = I x
x, y
y
(k)
r
= b − Ax(k) = A x − x(k) = Ae(k)
e(k)
k e
(k+1)
=
e
(k+1)
=
r
r
(k+1)
(k)
(α) = I − αP −1 A e(k) ,
(α) − αAz(k) .
· A xA = Ax, x1/2
Ax, y 2 e(k+1) A
x, yA =
=
Ae(k+1) , e(k+1) = r(k+1) , e(k+1)
=
r(k) , e(k) − α r(k) , P −1 Ae(k) + Az(k) , e(k) + α2 Az(k) , P −1 Ae(k) .
α
e(k+1) A
αk d 2 e(k+1) A dα
αk =
= 0. α=αk
r(k) , z(k) . Az(k) , z(k)
2
x(k)
k
x (k+1)
x
− x
A
→∞
≤
K 2 (P −1 A) 1 K 2 (P −1 A) + 1
−
k
x(0)
P K 2 (P −1 A) << K 2 (A). K 2 (P −1 A) 1 K 2 (A) 1 << . K 2 (P −1 A) + 1 K 2 (A) + 1
−
A,
− x
−
k
≥ 0,
P x(k+1) = x(k) + αk p(k) ,
p(0)
p(k) = z(k) x(0) = z(0)
− β p( −1) k≥0 k
A k
≥ 0,
k
r(0) = b
z(0) = P −1 r(0)
r( ) , p( ) k
αk =
− Ax(0)
k
Ap( ), p( ) k
k
x(k+1) = x(k) + αk p(k) r(k+1) = r(k)
− α Ap( ) k
k
z(k+1) = P −1 r(k+1) Ap(k) , z(k+1)
β k+1 =
Ap(k) , p(k)
p(k+1) = z(k+1)
− β p( ). k
k
(6)
x( +1) − x ≤ 2 k
A
K 2 (P −1 A)
k
− 1 x(0) − x K (P −1 A) + 1 2
A,
k
≥ 0.
Ax = b A=
1 1 1 2 2 5 4 6 8
b=
1 2 5
P LU Ax = b
PA PA PA A
ρ(A)
A∞
A
A= aA
1 a a a 1 a a a 1
LU
≤ A ρ(A) = A2
J a G ρ(G)
a
A
A=D
A=
−
D
−F
E
Ax = b
D ω
− E
(k+1)
x
=
− 1
ω
ω
− E − F
D + F x(k) + b ω
∈ R∗ . Ax = b
Lω det(L det(Lω ) ρ(Lω )
≥ |1 − ω| A=
−
3 1 0
−1 0 3 −1 −1 3
b
R3 .
x
Ax = b Ax = b xn
n
x0 = 0 M
xn+1
M (x − x) − x = M ( n
A.
s M. 1
u
R3
|x| ≤ |b| |u| |x − x| ≤ |b| s
n
Sn s
M. A, S S<1
p
f : Ω
p
⊂R →R
x
∈Ω
f ( f (x) = 0 x
Ω
Ω.
f f ( f (x) = 0 xn x
f : a, b a < b f ( f (a)f ( f (b) < 0 f (a, b) I 0 = (a0 , b0 ) x0 = (a0 + b0 )/2 n 1 I n = (an , bn ) I n−1 = (an−1 , bn−1 ) xn−1 = (a ( an−1 + bn−1 )/2 f ( f (xn−1 ) = 0 x = xn−1 f ( f (xn−1 ) = 0 f ( f (an−1 ) f ( f (xn−1 ) < 0 R
→R
x
a0 = a b0 = b
≥
·
an = an−1 ,
bn = xn−1 ;
an = xn−1 ,
bn = bn−1 ;
f ( f (xn−1 ) f ( f (bn−1 ) < 0
·
I n = (an , bn )
f : R y (x)
k
1
x0 (xn , f ( f (xn ))
→R
y(x) = f (xn )(x )(x
−x
f (xn )
n)
+ f ( f (xn ).
xn+1 y (xn+1 ) = 0
0x
xn+1 = xn
f (x ) − f f ( , (x ) n
n
n
≥ 0.
f f xn+1 = xn
f f (x − f (
x0 , x1
n)
x
xn f ( f (xn )
− x −1 , f (x −1 ) − f ( n
n
n
≥ 0.
f (xn ) f (x − f ( q
n)
xn+1 = xn
,
q k
≥ 0.
q = f (x0 )
f ( f (b) b
q=
[a, b]
− f ( f (a) −a f ( f (x) = 0
x = φ(x) φ : [a, b]
→R α = φ(α)
f ( f (α) = 0.
(E, d) α φ
∈ E
φ K<1
∀x,y, ∈ E, φ : E
α
d(φ(x)
φ : E E φ(α) = α
→
K <1
− φ(y)) ≤ K d(x − y).
→ E φ x0 ∈ E
x1
φ
α
{x }
∈ E
xn+1 = φ(xn )
n
x2 d(x1 − x2 ) ≤ Kd( Kd (x1 − x2 )
K < 1 ⇒ x1 = x2 K −
φ
d(xn+1 − xn ) ≤ Kd( Kd (xn − xn−1 ) d(xn+1 − xn ) ≤ K n d(x1 − x0 ) p > n p
xk − xk−1 = xp − xn
k=n+1
(8) p
d(xp − xn ) ≤
∞
k−1
K
k=n+1
n
k
K d(x − x ) ≤ K d(x − x ) 1
0
1
0
k=0
n
=
K d(x1 − x0 ) → 0 1 − K
(n, p) → ∞ .
{xn }
E
E
α
φ φ(α) = α 2
φ K
K <1 φ [a, b]
x
[a, b] φ x
K
y
φ(x) ≤ K
⊂R
[a, b]
[a, b]
[x, y ]
[a, b] 1
φ(y ) − φ(x) =
φ (x + t(x − y ))(x ))(x − y )dt
0
1
φ(y ) − φ(x)
≤
φ (x + t(x − y ))(x ))(x − y ) dt
0
1
≤
φ (x + t(x − y )) (x − y ) dt
0
≤
K x − y .
2
[a, b]
⊂R
φ : [a, b] φ
C1
→ [a, b]
α
∈ [a, b]
|φ(α)| < 1 K φ (α) < K < 1 E = [α [ α h, α + h] φ K K E φ(E ) E
|
−
|
| |≤
∀x0 ∈ E,
φ φ
⊂
lim xn = α.
n→+∞
α
{x }
n
|x − α| ≤ K |x0 − α|
n
n
|φ(α)| = 0 2 φ ∈ C
|φ | ≤ M
E 2
− α)φ(α) + (x −2!α) φ(c) 1 = α + φ (c)(x )(x − α)2 , c ∈]α, x[, 2 2 1 1 |φ(x) − α| ≤ 12 M |x − α|2 2 M |φ(x) − α| ≤ 2 M |x − α| φ(x) = φ(α) + (x (x
| − α| ≤
|x − α| ≤
2 1 M x0 M 2
1 M xn 2 n
1 M x0 2
|
− |
2n
α
|
,
− | α
2n
.
α
|φ (α)| > 1 [α
− h, α + h] ∀x ∈ [α − h, α + h] \ {α}, |φ(x) − α| > |x − α|. φ
α α
h > 0 φ−1 φ(α) = α α φ−1
φ [α−h,α+h] h, α + h])
|
1/φ (α)
x = φ−1 (x) α
([α φ([α φ(x) = x (φ−1 ) (α) =
−
|φ (α)| = 1 α = 0 φ (α) = 1
x
∈]0, ]0, π/2] π/2]
φ(x) = sin x x [0, [0, π/2] π/ 2] x0 ]0, ]0, π/2] π/2] xn l = sin l
l =0
φ(x) = sinh x x
sinh x > x
∈
x>0
∈
sin x < x
{ }
[0, +∞] ∈ [0,
0
φ 2
C
xn+1
)(x − x) + O(x − x2 ) − x = φ(x)(x x =x x +1 − x → φ(x). x −x n
n
n
n
n
p xn+1 x (xn x) p
− →l −
p = 1
0< l <+
||
∞
p = 2
ψ (x) = x α
f (x) . − f f ( (x) φ (α) = 0
f f
|x0 − α| ≤ δ
∈ C2
f (α) = 0 (7)
f ( f (α) = 0
f (α) = 0
δ>0
α
xn+1 α f (α) = . n→∞ (xn α)2 2f (α)
− −
lim
α
f
p p
f ( f (α) = . . . = f ( p−1) (α) = 0
∈N
f ( p) (α) = 0.
f (α) = 0
xn+1 = xn p
f (x ) , − p f f ( (x ) n
n
≥0
n
α
|x +1 − x | < n
n
en = α
n en+1 = α ξn
xn xn+1
n
− x +1 = φ(α) − φ(x n
n)
−x
n
= φ (ξn )en ,
α
−x
n
=α
− x − α + x +1 = e − e +1 = (1 − φ(ξ φ (ξ ) ≈ φ (α) = 0 e ≈ x +1 − x . n
n
n
))en . n ))e
n
n
n
n
n
(9)
f 1 (x1 , x2 ) = 0, f 2 (x1 , x2 ) = 0. f = 0
f
Df (x = (x1 , x2 )) =
(0)
(0)
( x1 , x2 ) x(0) = (x
f = (f 1 , f 2 ) ∂f 1 ∂x 1 ∂f 2 ∂x 1
∂f 1 ∂x 2 ∂f 2 ∂x 2
n
.
≥0 [Df (x( ) )](x( +1) − x( ) ) = −f (x( ) ). n
An = Df (x(n) )
n
n
n
P
n
R
f
∈ C0([a, ([a, b])
n
x0 , x1 , . . . , xn
∈ [a, b] n + 1 p
p( p(xi ) = f ( f (xi ),
0
n
n
≤ i ≤ n.
p = pn f
P = n + 1 n≥0
pn
x0 , x1 , . . . , xn
φi i = 0, . . . , n φi (xj ) = δij , δij = 1
i=j
δij = 0
n
i, j = 0, 0, . . . , n ,
i=j
φi (x) =
i= j
(x (xi
−x ) , −x ) j
0
j
≤ i ≤ n.
f
x0 , x1,ldots,xn
n
pn (x) =
f ( f (xi )φ(x).
i=0
pn qn (x)
xi qn (xi )
− p
n (xi )
qn pn
−
n
= 0, 0,
i = 0, . . . , n .
n
n+1
qn = pn p( p(xi ) = f ( f (xi ) 0 (10)
≤i≤n
p( p(t) = a0 + a1 x + p
f
i
1 1
x0 x1
1 xN
− x ) = 0 j
··· ···
xn0 xn1 xnn
a0 a1 an xi
=
f ( f (x0 ) f ( f (x1 ) f ( f (xn )
.
n nx
··· +a
f LU
Ax = b A−1
cond( cond(A) = A
· xi = 2i 0, i = 0, . . . , 20 cond( cond (A) = 1 A
≥1
cond( cond(A) = 35655 3565516259 1625901350 01350880 880 A−1 = AT
pk f [ f [x0 , x1 , . . . , xk ] pk pk−1 f [ f [x0 , x1 , . . . , xk ]
f
x0 , x1 , . . . , xk pk
≤k
−
pk (x)
cond( cond(A)
x0 , x1 , . . . , xk−1
f [x0 , x1 , . . . , x ](x ](x − x0 ) . . . (x − x −1 ). − p −1(x) = f [ k
k
k
p0 (x) = f ( f (x0 ) n
pn (x) = f ( f (x0 ) +
f [ f [x0 , x1 , . . . , x k ](x ](x
k=1
k
xi
− x0) . . . (x − x −1). k
f [ f [x0 , x1 , . . . , xk ] f [ f [x0 ] = f ( f (x0 )
f [ f [x0 , x1 , . . . , xk ] =
− −
f
x1 , x2 , . . . , xk (x − x0 )qk−1 (x) − (x − xk ) pk−1 (x) . xk − x0 xk
(12) 2
f ( f (xi ) n
≥1
f [ f [x1 , . . . , x k ] f [ f [x0 , . . . , xk−1 ] . xk x0
qk−1 ∈ P k−1
pk =
k
[n] [n 1] [n 2]
... n f [ f [xn−2 , xn−1 , xn ] . . . f [ x0 , . . . , xn ]
f ( f (xn ) f [ f [xn−1 , xn ] f ( f (xn−1 ) f [ f [xn−2 , xn−1 ] f ( f (xn−2 )
− −
[2] [1] [0]
f ( f (x2 ) f ( f (x1 ) f ( f (x0 )
f [ f [x1 , x2 ] f [ f [x0 , x1 ]
f [ f [x0 , x1 , x2 ]
n
[k ]
f [ f [x0 , . . . , xk ]
(11) un uk
= =
pn
[n] [k ] + (x (x
− x )u +1, k
k
0
≤ k < n,
u0 = pn (x)
C
n+1
x0 , x1 , . . . , xn n + 1 x [a, b]
([a, ([a, b])
[a, b]
f
∈
f ( f (x) πn+1 (x) =
i (x
− p
n (x)
=
1 πn+1 (x)f (n+1) (ξ ) (n + 1)!
− x ) ξ ∈ [a, b] i
x = xi πn+1 (xi ) = 0 ξ pn+1 (t) f f ( f (x) − pn (x) = pn+1 (x) − pn (x) n+1 x0 , . . . , xn+1
xi
≤ n+1
pn+1 (t) − pn (t) = cπn+1 (t),
x x, x0 , . . . , xn+1 pn+1 − pn
c ∈ R.
g (t) = f ( f (t) − pn+1 (t) = f ( f (t) − pn (t) − cπn+1(t). ξ
∈
g
n +2 (ξ ) = 0
(n+1)
(n+1) pn = 0,
(n+1)
πn+1
= (n ( n + 1)!
g (n+1)(ξ ) = f (n+1) (ξ ) − c(n + 1)! = 0
f − p ≤ (n +1 1)! π +1 f ( +1). n
n
n
c
2
(n+1)
π +1 n
f
f ( f (x) f xi
f − p ≤ n
1 4(n 4(n + 1)
a
[a, b]
f ( +1). n
n
|πn+1 (x)| [xn−1 , xn ]
[x0 , x1 ]
|(x − x0 )(x )(x − x1 )| ≤
n (x)
n+1
− b
− p
x ∈ [x0 , x1 ]
(x1 − x0 )2 h2 = 4 4
h = ( b − a)/n
i> 1
|(x − xi )| ≤ ih. 2
h n! b − a |(x − x )| ≤ max 2h3h . . . n h =
x∈[a,b] a,b]
i
i
4
4
n+1
n
2
1 lim n→∞ 4(n 4(n + 1)
− b
a
n+1
=0
n
f − p
n
n
tn (x) = cos(n cos(n arccos x), θ = arccos x
x = cos θ
x
→∞
∈ [−1, 1]. 1]. θ ∈ [0, [0, π]
tn (x) = cos(nθ) nθ), tn+1 (x) + tn−1 (x) = cos((n + 1)θ 1)θ) + cos((n cos((n
1)θ ) − 1)θ
= 2 c os os(nθ)cos nθ)cos θ = 2xt 2 xtn (x). tn t0 (x) = 1,
t1 (x) = x,
tn+1 (x) = 2xt 2 xtn (x)
− t −1(x). n
2
tn
n−1
n
≥1
n n
tn 2i + 1 π ] 2n
ˆi = cos x
1[, ∈ − 1, 1[,
0
≤ i ≤ n − 1.
x ˆi
0 x ˆn−i = 1
−1
−xˆ
i
[a, b]
xi =
a+b b a + x ˆi . 2 2
−
πn+1 n
πn+1 (x) =
(x
i=0
n x i=0 (ˆ
n
− xˆ ) = 1/2 i
−x ) = i
a
2
− b
(ˆ x
i=0
− xˆ ) i
πn+1 (ˆ x)
n
a
n+1
n
− b
a
π +1 n →∞
[ 1, 1]
−
(n+1)
.
4
√2π e ∼ 4√n
− b
a
n+1
e
xi
n
0
b
tn+1
π +1 = 2 n! 4
(n+1) n
−
e/4) +1 ) O(√n (e/4)
n
n
(13)
√ n/e) n! ∼ 2πn (n/e)
n
x0 = a < x1 < .. . < xN = b I j = [xj , xj +1 ] H = I j
b
I = [a, b] H
− a.
N
f |I j
n pH n
f
∈ C ( +1)(I ) n
H n (x)
f (x) − p f (
(n+1)
H f ( +1) . ≤ 4(n +1 4(n + 1)n 1)n n
n
f
I (f ) f ) = a < b
[a, b]
b a
f ( f (t)dt
I (f ) f ) J (f ) f ) n
J (f ) f ) = (b (b
− a)
ξi
ωi f ( f (ξi )
i=0
[a, b]
ωi
n i=0
ωi = 1
ξi N
f
f
∈ P 0
n i=0
f
∈ P
N N
∈ P +1 N N
f ( f (x) = 1 ωi = 1
ξ 0 p0 (x) = f ( f (ξ )
∈ [a, b]
f
[a, b]
b
f ( f (x)dx
a
(b − a)f ( f (ξ ).
ξ=a ξ=b b ξ = a+ 2
0 0 1
[a, b]
ξ0 = a f
p1 p1 (x) =
(x
a
f ( f (x)dx
f (a) + f ( f (b) . (b − a) f ( 2 n+1
ξi = a + i
b
− a. n
f a, b
f (b) − (x − b)f ( f (a) − a)f ( . b−a
b
ξ1 = b
[ 1, 1]
τ i =
−
f
∈ C ([−1, 1])
−1 + i 2
n
n
pn (x) =
f ( f (τ i )φi (x),
i=0
φi
φi (x) =
j= i
1
f ( f (x)dx
−1
1 ωi = 2
− τ − τ
j j
n
1
x τ i
pl (x)dx = 2
−1
ωi f ( f (τ i )
i=0
1
φi (x)dx
τ i
0
−1
τ n−i =
−τ ,
φn−i (x) = φi ( x),
ωn−i = ωi .
−
i
ωi n
b
f ( f (x)dx
a
f
≥n
pn = f f
∈ P
n
(b − a)
ωi f ( f (ξi ).
i=0
n
∈ C ([−1, 1]) n
1
f ( f (x)dx = 0 = 2
−1
ωi f ( f (τ i ).
i=0
f ( f (x) = xn+1
n f
∈ P +1 n
n n+1
n
n
n n = 1 n
n =1 ω0 = ω1 =
1 2
n =2 1 ω0 = ω2 = , 6
ω1 =
2 3
n =4 ω0 = ω4 =
7 , 90
ω1 = ω3 =
16 , 45
ω2 =
2 15
n=6 ω0 = ω6 = n
41 , 840
9 , 35
ω1 = ω5 =
≥8
9 , 280
ω2 = ω4 =
ω3 =
34 . 105
ωi < 0 4
N
C
N +1
[a, b] n
b
E (f ) f ) =
f ( f (x)dx
a
− (b −
K N N
1 a) ωi f ( f (xi ) = N ! N ! i=0
≥0
b
(N +1) K N (t)dt, N (t)f
a
[a, b]
→
K N N (t) = E x
(x
N
− t)+
,
t
∈ [a, b].
g (x, t) g : (x, t) → [a, b] × I
E x →
g(x, t)dt
I
=
b
1 (N +1) N +1) (x − t)N (t)dt. + f N ! N !
a
pN ∈ P N N
g(x, t)) dt. E ( E (x →
I
f ( f (x) = pN (x) + E ( pN ) = 0
b
E (f ) f )
=
1 E x → (x − t) f (t)dt N ! N ! 1 → − E x (x t) f (t) dt N ! N ! 1 (x − t) dt f (t) · E x → N ! N ! 1 N (N +1) +1) +
a
b
=
N (N +1) +1) +
a b
=
(N +1) N +1)
N +
a
b
=
N ! N !
(N +1) +1) K N (t)dt. N (t)f
a
2
E (f ) f )
1 f (N +1) N ! N !
≤
b
∞
· |
K N N (t) dt.
a
f
|
f : y:t
R+
∈ R+ → y(t) ∈ R y (t) = f ( f (t, y(t)) y(0) = y0 .
y0
×R → R
∈R
t>0
f
R+
L>0
×R
|f ( f (t, x1 ) − f ( f (t, x2 )| ≤ L|x1 − x2 | ∀x1 , x2 ∈ R, ∀t > 0, (14) t>0
y : [a, b] R [a, b] y (ti )
→
C 1 a = t0, t1, . . . , t h = (b − a)/n
= b n+1
n
y (ti + h) y (ti ) , h→0+ h y (ti ) y(ti h) = li m , h→0+ h y (ti + h) y (ti h) = li m . h→0+ 2h
y (ti ) =
−
li m
−
−
−
y (ti )
(Dy )i
(Dy Dy))iP =
y (ti+1 ) y (ti ) , h
−
(Dy Dy))R i =
(Dy Dy))C i = y
∈ C 2(R)
−
t
y (ti )
y (ti+1 )
i = 0, 0, . . . , n
− y(t −1) , i
i = 1, . . . , n; n;
h
− y(t −1) , i
2h
y(t) = y (ti ) + y (ti )(t )(t
i = 1, 1, . . . , n η
∈R
− 1;
ti
− 1. t
− t ) + y 2(η) (t − t )2. i
i
t = ti+1 (Dy Dy))iP = y (ti ) +
h y (η), 2
|y(t ) − (Dy Dy)) | ≤ C h, ] |y (t)| P i
i
C = 12 maxt∈[ti ,ti+1 t = ti−1
(Dy Dy))R i = y (ti )
− h2 y (η),
|y(t ) − (Dy Dy)) | ≤ Ch, ] |y (t)| R i
i
C = 12 maxt∈[ti−1 ,ti t = ti+1 t = ti−1
= y (ti ) + y (ti )h +
y (ti−1 )
= y (ti )
(Dy Dy))C i = y (ti ) +
−
y (η1 ) + y (η2 ) 2 h , 12
Dy)) | ≤ Ch 2 , |y(t ) − (Dy ] |y (t)| Dy)) | |y(t ) − (Dy C i
i
C =
1 6
maxt∈[ti−1 ,ti+1
y
y (ti ) 2 y (η1 ) 3 h + h , 2 6 y (ti ) 2 y (η2 ) 3 y (ti )h + h h , 2 6
y (ti+1 )
−
2
P i
i
ti
1 2
0 = t0 < t1 < . . . < tn < tn+1 < . . . h = tn+1 tn
−
yn
y (tn ). (14)
t = tn
y (tn ) = f ( f (tn , y (tn )). )). y (tn )
yn+1 −yn h
= f ( f (tn , yn )
n = 0, 1, 2, . . .
y0 .
yn+1 −yn h
y0 .
= f ( f (tn+1 , yn+1 )
n = 0 , 1, 2, . . .
∈ C3
yn+1 = yn + hn Φ(t Φ(tn , yn , hn ),
0
≤ n < N,
Φ
en en = y (tn+1 ) yn = y (tn ) en = y(tn+1 )
y
− y +1,
0
n
− y(t ) − h n
≤ n < N,
Φ(tn , y (tn ), hn ). n Φ(t y
|
y
n
0
en
0
|
yn
h
εn
S (yn ) (˜ yn ) yn+1 = yn + hn Φ(t Φ(tn , yn , hn ), y˜n+1 = y˜n + hn Φ(t Φ(tn , y˜n , hn ) + εn ,
max y˜n n
y
| − y | ≤ S |y˜0 − y0| + n
≤ n < N, ≤ n < N,
| | εn
n
(yn ) max yn n
y0
0 0
→ y(0)
h
| − y(t )| → 0 n
→0
y˜n = y (tn ) y˜n+1 = y˜n + hn Φ(t Φ(tn , y˜n , hn ) + en . S max yn n
| − y(t )| ≤ S |y0 − y(0)| + n
| | en
n
.
≥0
1 Φ
∀(t, y) ∈ R+ × R.
Φ(t,y, Φ(t,y, 0) = f ( f (t, y),
Φ
t
y
∈ [0, [0, T ] T ]
Λ
Φ(t, y1 , h) − Φ(t, Φ(t, y2 , h)| ≤ Λ|y1 − y2 |, |Φ(t, (y1 , y2 ) ∈ R2 h∈R S = eλT
1
≥ p
y y = f ( f (t, y) C
p
f
C
≥0 |e | ≤ C h +1,
y
p n
n
0
≤ n < N. 2
y (t) = f ( f (t, y(t))
tn
tn+1
tn+1
y(tn+1 )
− y(t
n) =
f ( f (t, y(t))dt. ))dt.
tn
yn+1
−y
n
=
h [f ( f (tn , yn ) 2
− f ( f (t +1 , y +1 )], )], ∀n ≥ 0.
yn+1
−y
n
=
h [f ( f (tn , yn ) 2
f (t +1 , y − f (
2 yn+1
n
n
n
n
+ hf (tn , yn ))]. ))].
h
−y
n
= hf (tn+ 12 , yn+ 12 ).
yn+ 12 1 yn+ 12 = yn + f ( f (tn , yn ), 2
yn+1
−
1 yn = hf tn+ 12 , yn + f ( f (tn , yn ) . 2 2 4
yn+1 = yn +
h (K 1 + 2K 2K 2 + 2K 2K 3 + K 4 ), 6
K i K 1 K 2 K 3 K 4
= f ( f (tn , yn ) h h = f ( f (tn + , yn + K 1 ) 2 2 h h = f ( f (tn + , yn + K 2 ) 2 2 = f ( f (tn+1 , yn + hK 3 )
