Практикум
.pdf
1 '0
. ( ( 5Z f 0 ( 3 "
( 35 " 3"
( 4 3 . ( 0
S
b ( ("
( 0 ( ( ( " ( 3
( ( 0 ( 3" ( ( 3 3 ( " 0 ] 0
3 " f 0 ( 00 (
0 3( " 3 ( ( e 4 ( (e
00 ( ' S %
/ 3 0 3 ( 4 (
" 0 % 4 " 00 ( / 3 0 3 (
c / ( 3d %g f 3
) 0 0 0 ( ( 0 3 % ( (
( 4 %
[\
(
(x1, y1), (x2, y2), . . . , (xm, ym)
cxi = xj 0 i = jd" 0 ( 3 yi 5 3 Z ( 4 ) 3 / 5 F (x)" 3
yi = F (xi) − ηi c[ Sd
0 i = 1, . . . , m" 0 ( 0 4 ηi ( ) 3 / 3 Z 3
/ 0 " ( " % ( /
% fj
|
m |
|
|
j |
|
F (x) = |
cj fj (x) |
c[ d |
|
=1 |
|
cj ( f / 2 3 c[ d ( (
( A m×n( / % / %
%
|
f1(x1) f2(x1) . . . fn(x1) |
|
|
A = |
|
f1(x2) f2(x2) . . . fn(x2) |
|
|
|
. . . |
|
|
|
|
|
|
|
|
|
|
f1(xm) f2(xm) . . . fn(xm) |
|
|
|
|
|
|
|
|
|
|
c n ( % f / )
|
f1(x1) |
f2(x1) |
. . . fn(x1) |
||
Ac = |
|
f1(x2) |
f2(x2) |
. . . |
fn(x2) |
|
|
|
|
|
|
|
|
|
|
. . . |
|
|
|
|
|
|
|
f1(xm) f2(xm) . . . fn(xm)
c1
c2
. . .
cn
=
F(x1)
F(x2)
. . .
F (xn)
m " 0 % ( 3 3
c[ Sd c[ Wd 0 (
η = Ac − y
c[ Wd
c[ Rd
η ' % ( 4 c[ Rd ( ("
3 ( ( 4 c d ! (
X
4 3 5Z 3
n |
n |
|
min ||η||22 = ||Ac − y||22 = |
( aij c − yi)2 |
c[ d |
i |
|
|
=1 |
j=1 |
|
( ( 3 (( ( 0 3(
% " f 0 3 ( ( ( 4 %
/ ( c[ d 0 ( 0 ( ( ck 0 3 ( 0
5 |
|
n |
|
n |
|
|
|
δ||η||22 |
= |
2( |
a c y )a |
||
|
|
j |
||||
|
δck |
|
|
ij − i ik |
||
|
|
i=1 |
|
=1 |
||
|
|
|
|
|
||
( n ck = 1, 2, . . . , nd ( 0 (
(Ac − y)T A = 0,
f 5
AT (Ac − y) = 0.
. " 0 % ( 5
AT Ac = AT y, |
c[ [d |
' ( ( ( ( |
$ |
/ AT A ( (( " A ( 0 % / "
0 0 ! f ( Z 3 ( / (AT A)−1" 4 ( c[ [d
c = (AT A)−1AT y = A+y.
b A+ 0 3 ( / ( / A" 3 5
Z ( (
AA+A = A, |
|
A+AA+ = A+, |
c[ Xd |
(AA+)T = AA+. |
|
! 0 5 ( / ( 0 ( Z 5 ( -b# % c " A 3 ( / 0 " A+ = A−1d
XS
! /#" A+ = A−1 )
"5 /
n : $#"! * ># A %% R - b#n%
‰ R n m n R
! /#% *P L+ ! m×n 2 ! A /
: $#"! * ># A1%%
- ' (
. ( ( 0 ( 4 5Z V
(−1, 2)" (1, 1)" (2, 1)" (3, 0)" (5, 3) 0 ( 3 ( ( /
F (x) = c1 + c2x + c3x2
b ( x" y ( / A
$ E "A"A A A(F , E A"A"A!A F/.*= #(A
b 0 ( ( / A 3( % / c 3 f1 = 1" f2 = x" f3 = x2d
#OA"% " #OA % $ #OA % $ P
' ( 4
@ - b# % ,
( /
$ O! "O3 , " ! 4(4" $ ! "& $ P
-5=+#$A,A;-;A$A, A;-*;A$ A" ! 4(4" $ ! "& $ P %
X
' ( " 0 5Z 3 /
:=* "O( + E$# %A$# %F +" E,# %A, # %F 5 .#+A+"%
. >
! /#& *- -+ *1-+ *4 4+ *7 N+ 2 :
f (x) = c1 + c2x log x + c3ex
- 2 (
! 3 / -=5,: + -=5,b 5 0 3 0
( 00 ( / 0 ( ( 4 %
/ V 3 0 ( ( 0
% 0 ( Z 5 ( -5=+
$ E "A"A A A(A4A3A"!F , E A"A"A!A A "A A GF -5=+#$A,A;*=;%
- ( 5Z 0 ( 5 ==5 VS @ l ++ 6" 0 3 3 0 5Z ( 5" 0 ( %"
0 3 00 ( / 0 ( ( 4 %
! /#+ E :
J
!( ( ( %
/ :1 . *@Z" :1 @= " :1 \ @" 5 ) = 5 " 5 )@\*b.: +
% % 0 o-+ 1 / + = ==57=$"
V 3 " 0 " 0 0 0 0 (( " (
( ( 0 ( 00 ( / % 5Z
y = β1 exp λ1t + β2 exp(λ2t).
XW
] ( 5 00 ( / 5 3 % ( %" ( %
( 0 ( ( 4 % % 0 ( β1,2
% 0 ( λ1,2
" 5Z 5 0 (( " 3 5 (
0 0 (( :1 . *@Z :\ @+ = .$-: +>.1=
+ #!O "O %;
, E( G9(( (3 9 ("4 " 949! " G99! " 9 G " " (9
" !!93 " ! & ! G& ( ! 3G(3 ! 3"!! ! ( 9 ! 9&3
! 9! ! (&4& ! &(9 ! " 4! ! "" ! "4!& ! 3 3F; @5:
Z6.+#6@:A;>=\75.7\::.*;A;= ;% Z -5=+#+A,A;=;A+A! +A; ;% Z# % + +5.#;;%
$ #E! ! 3 (F% 5 17> ! E 3F;
5 17> :1 . *@Z#_.$-: +:\ A5 17> !AEFA+A,AZ%.+#Z# %A;@=5=*;A;75 @H;%
. > u
:\ @+ = *. .$-: +:\ #5 17> A+A,AZ% 1 5. 6+Z#+%
5. 6+Z#5 17> % R /.*= #1A % :=* 2 "O
R#OA2% .$-# 5 17> #2% +%
. >
7.+ RQ, / R 7.+ *. =*1#/ ,%.+#Z# %A;,> + ;A/%
.+#Z# %A; +* 6;A -* +:#;uG &: uG &:;A5 17> %% - \ .# "%
. >
% f 0 (( 9 0 ( .$-: +>.1=
XR
(( 0 3 3 % 0 0 ( 5 5
4 3
! /#. ! " 5 " Z[\G]^_[A`Z
+"O (
,E ( ! 9" 3 (!99 ( 333 & " 4 & 9&G
3 " 3" " ("&! "! !(! 9 "3"& 4 (344
4 9 ! "! ! (4 "" !4!G " &!&( " G&"(
"" 9333 "" !43( "" 444& "& (4!" "4 !&&!
"4 ! 9G "( 9!39 "( &G(! "( ("" "4 3(4 F
0 ! 5 "
! /#/ ( z = ax2 + bxy + cy2 + dx + ey + f
" x, y 2 (x, y) z = 0
)
b2 − 4ac
5 "
ERA[F 1. Z6* >#$1 O>.5+ $O$1 $A,1 O>.5+ ,O,1 $% 8 R P 7 R [ @ [ P > R . [ : @= +=\*#RA[A8AE! !F%
3 : ! a ;
) ! )
; E "
"
"5 (x, y) /
$ E" ! 9( G4 44 34 (3 && ! "3 !"F;
, E! 9 4 "G "( " " " "(F;
& : ) !
; * > "
: H 66C 66CI ( ) !
" 5 " "
& 9
X
3 (
(( 3 0 3 3
typK_o" ( / 5 0 (
!" #" $% &' %( )*+,#$-
5Z ( ( 3(
•2 5 ( 3 f "
5Z 5 0 ( Z 5 e( 4 e ! 0 Z c
d / 0 ( ( ( 0 e e c e ed 0 5Z ( ( 5" 0 3 35Z ( 3 0 Z 0
0 e( 4 e
•] 3 / 0 ( 0 ( (
0 e ( e f ( 5
•b 0 / Z 3 3 0 ( Z 5
/ @ . ' 0 ( " / 5
f (x) = |
|
sin(x)/x, |
x = 0 |
|
|
1, |
x = 0 |
(
:#$% @ .#$UV! V #$% $%.5 .#"%
X[
:#$% @ .#$UV! V #$% $Y$ ! V"%
c f ( †( 0 % "
0 0 % e( 4 % f0 ed
/ 5
f (x) = |
|
1, |
x ≥ 1 |
x < 1 |
|
x, |
1 |
||
|
|
|
− ≤ |
|
|
|
|
|
|
|
|
−1, |
x < −1 |
|
|
|
|
|
|
(
:#$% @ .#$V " V " Y $ E " "% V $ Y $U " V "%
• b 0 4 ( Z 3 3 0 ( Z 5
/ = .# %" |
: |
† (0 4 ( |
|
||
f ( 0 |
|
0 ( |
|||
! |
) " |
:#$% #$% = .#! "% |
|
4 ( |
|
(0 ! "" 3 ( |
/ #$% |
/ 3 |
|||
:#$% #" = .#! "%% #$% |
5 ( 3/ 5 |
(0 |
|||
e / 3e :#$% #$ = .#! "%% ( / 5 (
35Z 0 e ( e ( $ e / 3e :#$% ##" = .#! !"%% $% ( / 5
0 4 e ( 3e ( $" 5 S…
4 % !
(
. ( ( 0 / 3 / " 0 3 (
(0 5 0 ( Z 5 ( 4 3 /
/ df / ( dx 4 ( 0 Z 3 /
∆f f ∆x ! 0 0 / 5 5Z
•! 3 ( 0 ( T %
/ ( ( 0 % 3
XX
T 0 0 5 " % / ( ( ( (
3 % 0 3/ 0 ( ( 0 c
" d - / 3 3 3 3
/ 5Z " 0 ( 3( Z (
( / 0 5
5Z 0 4 3 3 4 ( $
( % 4 ( / %
•( / 3 3 0 %
0 3 ( ( 3 ) # (
5Z 0 4 (
•(0 5 ( 0 4 ( ( 0
( ( / % ! Z % 3 3 ( 0 5 5 c ( e( 4 ( f0 ed"
3Z ( % 3 ! ( 3 (
( 3 0 0 4 " 3
3( ( # ( 3 f ( 0 4
0 4 3( (0 5 0 3 Z %
•! 3 0 ( ( / 3 %
5 3 ( ( 4 0
∆x 0 5 0 ( ( ( ∆xopt" 0 ( 4 ∆x 0 5 f ( (
•] " % ( 0 / " 0
( " 0 5Z (
/ 3 ( 5 0 4 " 0 3 0 4 5Z e( 4 f0 e
' ( ( 0 0
2 0 0 3 0 4 5 " 0 (
% / % % ( 3 ( 0 3 3
3 / / c (
0 0 ( Z 0 (( % d $ ( (
/ " 0 % ( ( 0 XY
