Учебное пособие 800365
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F !/ 3 2 3 / 3 5
3 x 22 = (x, x) = [x]Te [x]e# R
sup Ax 22 = |
sup [x]eT [A]T [A][x]e = |
sup (x, A Ax). |
x E, x 2=1 |
x E, x 2=1 |
x E, x 2=1 |
H 3 3 3 3 5
! 3 / 33 3 ! [A]T [A] 3 3 # 5
/ 3 3 3 3 3 G 3 ! λmax# ? G 3 ! / 5
3 3 (Ax, Ax)
! 3/ 3 # H 3 3/
A 2 = λmax. (7)
F / 3 / 3 /
3 A 2 3 7
n |
(8) |
A 2 = ( aij2 )1/2. |
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i,j=1
3 / / λ1 + λ2 + ... + λn J
3 ! [A]T [A]/ ! λk, k = 1, 2, ..., n/
3 / 3 λmax# " /
A 2 A 2. |
(9) |
F 3 3/ 3 3 ! [A] 3 n × n
33 G 3 # 9 3 / 5
3 G ! λn−1 3 3 5
pA(λ) 3 ! / 3 3 3#
M 3 ! 3 3/ 5
3 3 3 /
3 3 3 3 3 : 3 / 3
A G / ! # " 3 3 @
3 / 3 A G 3 3 5
3 #
F 3 / 3 3
3 / 3 G
3 3 #
% 4 /- ' F 5
F / @ 3 5
G 3 A 4F : A → A6/ 3
# H x A/ F x = x#
2 G 3 3 B#O# % 5A @
#
# F /
@ E 3 3 5
#
H 3 @ 5
! 4 3# T(U/ T.U6# 23 G / 3 3 3 % !
3 3 C[a, b]#
R 3 C[a, b] 3 3 A C[a, b] 3 3
/ ! G 3 7
ε > 0 δ > 0 / |t − t | < δ/
! x(t) C[a, b] |x(t ) − x(t )| < ε#
# 1 D !
C[a, b] 3 / @ 5
/ 3
C[a, b] 4 3 6 !
C[a, b])#
$ 6-
&-.*
? A J / @ E E# H 5
n An / 4B#B6/ An A n# F 3 3/ A& 0 E E&
λ& 0
pA(λ) = det([A] − λI)& & &
A − λI # "
Sp(A)#
' r(A) = = sup{|λ|, λ Sp(A)}#
R 3 3 3 2#$# / 5
@ 3 4 3# TBU/ T(U/ T.U67
r(A) = lim |
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(10) |
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→∞ |
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? f (x) J ! / @
H 7
∞
f (x) = f (0) + 1!1 f (0)x + 2!1 f (0)x2 + ... + n1! f (n)(0)xn + ... = cnxn,
n=0
3 Rf # F 3 ! f A
∞ |
N |
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f (A) = |
cnAn : f (A)x = lim cnAnx, |
n=0 |
N →∞ n=0 |
@ # " @ 3 @ 5
/ 3 3 4 3# TBU/ T(U/ T.U6#
,
Rf 9 7 f (x)& 7 f (A)
A #
"3 G 3 3 @ /
# 1 3 3 !
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G 3 3 ! f (A) 3
f (λ1) 0 . . .
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cnAn = |
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R / λi @
N
f (λi) = lim cnλni / / / |λi|, i = 1, 2, ..., n
N →∞ n=0
3 3 ! f (x)#
H 3 q(x) / 5
3 / / / q(A)
@ A# G 3
3 3 @ / 3 3 3 5 *G 4 3#/ 3 / TBU6#
! E0 A& 0
E&
# 9 pA(A) = O#
3 3 3/ 3 ! A 3
n×n @ 3 R(x)/ R(A) =
O#
) / 3 3 ! 3 n × n
n2# ? G 3 3 ! I, A, A2, ..., An2 3 / / / @ 5
c0, c1, ..., cn2 / c0I + c1A + c2A2 + ... + cn2 An2 = O# 3 R(x) = c0 +c1x+c2x2 +...+cn2 xn2
3 ! A#
3 3 3 !
3 !#
" 4 0 A
exp(A) = I + 1!1 A + 2!1 A2 + ... + n1! An + ...
, ?&
I + A + A2 + ... + An + ...
(I − A)−1 ! & I − A#
? /
3 ! ex = exp(x) 5
#
) 3 3/
(I + A + A2 + ... + An)(I − A) = I + A + A2 + ... + An − A − A2 − ... − An − An+1 = I − An+1.
H A < 1/ An+1 → 0 / /
(I + A + A2 + ... + An + ...)(I − A) = nlim (I + A + A2 + ... + An)(I − A) = nlim (I − An+1) = I. |
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→∞ |
→∞ |
N #
" 3 @ 3 4,'6 3 5
3 / 3 A < 1 3 r(A) < 1# ?
@ 3 #
$! 0 & 7 # - &
" 3 @ 3 3 3
3 # ? G 3 3 @
3 ! 3 ! 3
3 / 3 ! / 3 43 5
] 5 3 3 3 3 @ G 3 5 6# R 3 3 3
3 / 3
3 3 # ? G 3
5 #
G 3 3 3 5
3 # M 3 3 3 &
#
? 3
a21x1 |
+ a22x2 |
+ ... + a2nxn |
= b2, |
a11x1 |
+ a12x2 |
+ ... + a1nxn |
= b1, |
. . . . . . . . . |
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an1x1 |
+ an2x2 |
+ ... + annxn |
= bn, |
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3 3 3 3 @ 5
Ax = b# ? 3/ / 5
3 ] 5 -77
A 0 0 & &
- &
=
n
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|aii| > |
|aij |, i = 1, 2, ..., n. |
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=1,j=i |
H / i5 aii xi
/ 3 3
x2 |
= c21x1 |
+ c22x2 |
+ ... |
+ c2nxn |
+ d2 |
, |
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x1 |
= c11x1 |
+ c12x2 |
+ ... |
+ c1nxn |
+ d1 |
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xn |
= cn1x1 |
+ cn2x2 |
+ ... |
+ cnnxn |
+ dn, |
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di = bi/aii, cii = 0, cij = −aij /aii, |
j=1 |cij | < 1, i = 1, 2, ..., n# 3 |
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3 G 3 x = Cx |
+ |
d/ 3 |
C |
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∞ |
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) / 3 3 x0 = d/
3 3 x1 = Cx0 + d/ @ 3 /
k5 xk/ k + 15 3 3
xk+1 = Cxk + d, k = 1, 2, ... |
(11) |
, C = q < 1&
0
5 6 x = Cx + d & & 0
Ax = b# -
0 x0#
? 3 x2
x1/ 3 x2 = C(Cx0 + d) + d = C2x0 + (Cd + d) = C2x0 + (C + I)d# % 3/ x3 = C3x0 + (C2 + C + I)d# @ 3
xk+1 = Ck+1x0 + (Ck + Ck−1 + ... + I)d, k = 1, 2, ... . |
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(12) |
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" 3 3 B#-/ k → ∞ |
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3 ! 4 6 (Ck + Ck−1 + ... + I) 3 (I − C)−1 |
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Ck+1 |
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# ? G 3 3 3 4,&6
3 # R 3 G x # H /
4,&6 / 3 3
x = θ + (I − C)−1d = (I − C)−1d. ? x
x = Cx + d / 3 / (I − C)x = d/ 3
(I − C)x = (I − C)(I − C)−1d = d.
H x 3 # H 3 #
R! 3 3 x − xk+1 E # ? 3 5
3 E 33 @ 3 5
/ 3 3
x − xk+1 E = (I + C + C2 + ... + Ck + Ck+1 + ...)d − Ck+1x0 − (I + C + C2 + ... + Ck)d E =
= Ck+1x0 E + Ck+1d + Ck+2d + ... E Ck+1x0 E + Ck+1d E + Ck+2d E + ...
C k+1 x0 E + C k+1 d E + C k+2 d E + ... = C k+1 x0 E + C k+1 d E. 1 − C
F ! 3 3 3
3 3 xm − xm−1 E 3 (1 − q)ε/q/
q = C / ε J / /
3 / x − xm E ε#
) 3 3 3 3 3 3 5
3 ! 5
3 3 T,/ PU# ) 3
! 3 3 T(/ .U#
" @ @ G 3 3 5
! / G 3 3 !
4 ! 6#
* 8 * " ( &- $ $* +) " *. ,)& * +
( " #
R J 3 5
/ 3 / 3 3 3 5
3 # ? G 3 5
/
3 3 / 3 3 5
3 4 3 3 6# ! 5
3 5
3 3 3 3 # F 3
3 /
L L / G 3 3 5
# 3 3 3 3 5
3 / 3
#
! ! #
-
7 & 7 # ! 5
3 3 3 G 3 3 !
3 # ! 3 5
/ / ! #
% ) * ) &
,# ( ) ) ,-, F
A(x1, y1) B(x2, y2)/ x1 < x2# H / 5
@ A B/ / 3 3 #
( ) % % ? x1 < x2/ y1 > y2
! y(x) / A B7 y(x1) = = y1, y(x2) = y2# H ! y(x) / 3 5
/ @ 3 /
A B 3 3 #
O# ( ) ! ! ) 3
% # ? /
3 3
" 3 3 / 3 5 # 3 / # H / 5
/ ) 3 /
/ 5
3 * # M 3/
3 @ 3 3
@ / 3 @ 3 4 3 / OX6#
G 3 G 3 4 5 / 3 @ 6 @ 3
! # H 3 3 3 /
3 / 3/ 3 5
/
3 # * 3 / 3 J / 3
@ J ! 3 5
# H G @ G 3 3 ! / /
! ! # 9 7 7 0 7
# F 3 / @ 3 3 3 !
y(x), x1 x x2
x |
2 1 + y (x)2dx, |
J1(y(x)) = x1 |
@ G 3 3 ! @
J3(y(x)) = x2 |
y(x)dx, |
x1 |
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/ J1(y(x)) # M !
y = y(x), x1 x x2, G 4 6/
) / 3 @ [x1, x2] OX# ? G 3 y(x1) = y(x2) = 0#
* ! 3 3 / 3 3 3
#
( 3 - -
? 3 3 3/ 3
G 3 3 77 ! f (M) 5
3 # ? M0 G 3 3 # H M0
M = M0 + |
M |
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f (M0, |
M) = f (M) − f (M0) = df (M0, M) + α(ΔM) |
(1) |
/ !
#
M df (M0, M) ! / ! 5 3 3 f (M0)ΔM/ M = M − M0/ ! 3
3 3
df (M0) = grad f (M0) · M,
M M0M#
77 0 4 f (M0) = 0
gradf (M0) = 06/ ! 0 0
@ |
f (M0) 3 M |
f (M0)# |
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H df (M0, |
M)/ / 7 M/ 3 5 |
df(M0, − M) = −df (M0, M)/ G 3 f (M0)
M0 3 #
9 & - 7
M0& df (M0, M) ≡ 0#
? 3 / 3 5
G 3 3/ M/ f (M) = 0
gradf (M) # H 5
! / !
! - #
R / G 3 3 ! J(y) 3
3 3 # % 3 / 3 3 ! y(x)
! y(x) + δy(x)/ 3 !
δy 3 @ 3 ! ! y/ 5
@ 3 4 @ 6 ! J
3 / 3 4,67
J = J(y + δy) − J(y) = δJ(y, δy) + α(δy). |
(2) |
M δJ(y, δy) α(δy) 5
@ J# ? / δy, δ1y, δ2y
γ
6 δJ(y, δ1y + δ2y) = δJ(y, δ1y) + δJ(y, δ2y):