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TN³

x+y =2 =⇒ x = y = 1. x−y =0


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ipt

x + y = 1.9999 =⇒ x = 1.00005, y = 0.99985. x − y = 0.0002 x + 0.99y = 1.99 =⇒ x = y = 1. 0.99x + 0.98y = 1.97

x + 0.99y = 1.9899 =⇒ x = 2.97, y = −0.99. 0.99x + 0.98y = 1.9701



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f (x + δx)

δx

|f (x + δx) − f (x)| ≈ |f 0 (x)| · |δx|

UFV!MAK`C`XK

|f 0 (x)|

T—8˜šV Z HUnRTÎV—^PSHU Z `NO™V˜SU ™¹UFVXPNtO ¸"Ê«T f

x

|f 0 (x)| · |x| |δx| |f (x + δx) − f (x)| ≈ · , |f (x)| |f (x)| |x|

AM K Z TUnO™XRT|V—^PSHU Z `NO™V˜SU ™´UFVXPNtO ¸ À Cs?Cs:CÐÏ Á A ÃCÄ Á |f 0 (x)|·|x| |f (x)|

f

x



^ HKL a "O beL4c>Xb'FIHKL™rIcL6b'F u _Tm L4d _gfG_DaKu _ cF a } _ crMFIc~dH _ „MF a cYO m_ b'F aKu Lob _%a } _ crIcLob~P a u _šh F*} _ crIc _ r h FIc _ E•cF a } _ crIcF*ETH aKu F • FM{ F›NKFIb _ _ u h _ _ _ L u FIHO u LoETc _ b'F u _Tm_ S • d } crIcFIb{ FIETL X~} HO"} E•dH FI}TLocFIb § u _ d h™_ _ b'FIc_ FL  m OcYO"b'F aKu _ aKu f_ HOgrIXcYO"b _ u ETH _ F m c _DaKu U©Xc}G_ „IL NKF g k}TLN _¡h O"|} _ LoJMHOgu rI_TXm cYO"b _a } _ c rIcLob { FIETL b dsFIHOg„IL NMS<nO"b'F y = f (x) O"} LoJMHOgrIXcYO"b ye = g(x) S nO"dYO"}DOqb'F FBNKF D = y − yeS m

Ñ C Ã :<; ÃcÒ =VA!É Á ÅBCÓÅBCs?Cs:C

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D = D n + Dm + Dz

iw

S<ÀBF h N O

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x = 0.3142 · 100

x = π/10

—³×RT8]T!TLJLKSUFVXWK ¯ GµT!JLK8˜SUFV

O bSM^T PSHRT!J#V

f (x) = sin x

bXT

¸¿GµT8]T!Tž`XK

Dn = y − y = sin(π/10) − sin(0.3142) = −3.9 · 10 −5 . sin(x)

g(x)

g(x) = x − x3 /6

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x − x3 /6

Ø=KNM `XK

a1 a2 a3 yb

= = = =

g(x)

f l(x ∗ x) = f l(0.09872164) = 0.9872 · 10 −1 f l(a1 ∗ x) = f l(0.03101154) = 0.3101 · 10−1 f l(a2 /6) = f l(0.0051683 . . .) = 0.5168 · 10 −2 f l(x − a3 ) = f l(0.309032) = 0.3090 · 100

ye = g(x) = 0.3090302767 . . .

¬+`XKµRT8]T!T

Dz = ye − yb = 3.0 · 10−5 .

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alg

f

x

yb = alg(x)



O hofG_ HKL u b‰X

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f (x + δx)

ipy

S `^ _gu IF bEGF h N O

|alg(x) − f (x)| = |f (x + δx) − f (x)| ≈ |f 0 (x)| · |δx|.

dHKLkb%O(Nz|cLo|

f

δx

cYO"dYO"}DO _ WHO u c _aKu O"WL h cF€b'F u _Tm F•b%O(Nz|cYOÊLoc

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p0 = x 0 i = 1, . . . , n pi = pi−1 xi p = pn

»eV—NOJ#V ° K Z `šT UFVI]T|VX_8KNR+OJ#V

pb0 = x0 i = 1, . . . , n pbi = pbi−1 xi (1 + δi ), pb = pbn

x 0 , x1 , . . . , x n

|δi | ≤ u

pb = p(1 + γ) = p(1 + δ1 ) · · · (1 + δn ). (1 − u)n ≤ (1 + γ) ≤ (1 + u)n ,

(1 + u)n = 1 +

    n n 2 u+ u + · · · = 1 + nu + O(u2 ), 1 2

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(1 − u)n+1 ≥ (1 − nu)(1 − u) = 1 − (n + 1)u + nu2

nu  1

(x1 · · · xn

)T

s0 = 0 i = 0, . . . , n pi = x i yi si = si−1 + pi s = sn

y = (y1 · · · yn

|γ| < nu

)T

u

s=

yT x

=



Pn

n

i=1 xi yi

sb0 = 0 i = 0, . . . , n pbi = xi yi (1 + αi ), |αi | ≤ u sbi = (b si−1 + pbi )(1 + βi ), |βi | ≤ u sb = sbn

x=

¬


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i]

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xi yi (1 + γi ),

i=1

1 + γ1 = (1 + α1 )(1 + β2 ) · · · (1 + βn )

1 + γi = (1 + αi )(1 + βi ) · · · (1 + βn ), i = 2, . . . , n.

|γ1 | ≤ nu

|γi | ≤ (n − i + 2)u x

sb − s =

UFV!MAK`

n X

n X

i = 2, . . . , n

sb

y

xi yi γi ,

i=1

n X

|xi | · |yi | = nu|y|T |x|.

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i=1

|xi | · |yi | · |γi | ≤ nu

i=1

sb − s |y|T |x|

s ≤ |y T x| nu.

sb−s

s ≤ nu

xi yi



p(x) = a0 xn + a1 xn−1 + · · · + an ,

x

p0 = a 0 i = 1, . . . , n pi = pi−1 x + ai p = pn

»eV—NOJ#V `XKNMC`XK

pb0 = a0 i = 1, . . . , n pbi = (b pi−1 x(1 + αi ) + ai )(1 + βi ) pb = pbn

pb = a0 xn (1 + γ0 ) + a1 xn−1 (1 + γ1 ) + · · · + an (1 + γn ),

1 + γ0 = (1 + α1 ) · · · (1 + αn )(1 + β1 ) · · · (1 + βn ), 1 + γi = (1 + αi+1 ) · · · (1 + αn )(1 + βi ) · · · (1 + βn ), i = 1, . . . , n − 1, 1 + γn = (1 + αn ),

UFV!MAK` Z T® !V¹VX_8KNR+OJ#V

|γ0 | ≤ 2nu

OR

|γi | ≤ (2(n − i) + 1)u

bXT

i = 1, . . . , n

¸


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ai

pb − p = a0 xn γ0 + a1 xn−1 γ1 + · · · + an γn

|b p − p| ≤ 2nu(|a0 ||xn | + |a1 ||xn−1 | + · · · + |an |),

|b p − p| 2nu(|a0 ||xn | + |a1 ||xn−1 | + · · · + |an |) ≤ . |p| |a0 xn + · · · + an |

(x − 2)9 = x9 − 18x8 + 144x7 − 672x6 + 2016x5 − 4032x4 + 5376x3 − 4608x2 + 2304x − 512 2

−11

4

x 10

3

2

1

0

−1

−2

−3

−4

−5

1.95

2

2.05

2.1

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−11

1

x 10

0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1

0

200

400

600

800

1000

1200

−52

x=2+k*2

1400

1600

1800

2000


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iv

p0 = a 0 e0 = |a0 | i = 1, . . . , n pi = pi−1 x + ai ei = ei−1 |x| + |ai | p = pn e = 2nu e|p|n

`XK VX_8KNRT´bXTLMAK Z TUnO™XRVLRT8]T!V¸ e

20

15

10

5

0

−5

−10

−15

−20 −2

−1

0

1

2

3

4

5

6

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π

π

π

r=

an

a2n

1 2

an

Sn

n


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ÀBF h N O _Tm•u _Tm dYO­LoJ

a2n

v u u a 2 n =t + 2

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aKh F m L

1 − 2

r

1 − 4

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 a 2 n

2

!2

=

v q u u t1 − 1 − 2

s

1−

 Sn 2 n

p 1 − a2n , 2

È i SZ É

.

‚ _m F›N OšNKh F m i O~HOgrIXcYO"cgNKFíJ(Og_ rIcFIb _g_ ušdh HKL Su _ `J(O }DO _ u FIf H F f O~EGFIb _  m OeNKF ŸS a = 3m <d u _gu FI_ b¢dYO Jí_ Xd _ H_gO"u W _ Um HK_ b‰X F¿È u S Z É LoJM_DHaKOgu rIXcY_ O"b h π _T} m _ LobeL a S  m } m_•H F_Tm nu → ∞ S¿a ‚ƒJM_ }DO F F _ O hO"} E¡u FIc hNzcL } E E a NzcL`cYO _gŸ O"crIc L<U a HKb‰_TX m O u d EGh F a _ O(NldHKL F m { FIh EDO"cgN m O } HO(NFIcYO"} EGF Lo}TLo|¿{ FIETL cYO"dYO"}DO dYO Fbec L4J 2n S<À d cgNzL O"WsF L cYO"dYOgrIcF FM„ILob%O }GFqH FMrMFDS 6

6

n

n 6 12 24 48 96 192 384

Sn n Sn 3.0000000 768 3.1417003 3.1058285 1536 3.1430793 3.1326280 3072 3.1374769 3.1393509 6144 3.1819811 3.1410384 12288 3.3541021 3.1414828 24576 3.0000000 3.1414297 49152 0.0000000

†^*O*NKmF _ a _gU h _ HKb‰Xh h™_ È i S Z É c_ FI}DO(Nlu cYO"H _ _ WsFsd _ }D_ O Ÿ m F Ÿ F­_ }THO u FI} HO"_DJMaKbeu L h™aK_ h FI}sS*À m_h Lobeh L u L<cY_ O(N[_ WL S { F h dH _gu L π S O"} b%O(Nz|c  OšW 1 − O 1 Loc WL L4W b S = 0 S HKL E NkEGF Lo}GFIb n W ž O aKu O"WL h c _ HOgrIXcYO"cgNKF NKFd _gu H FIWc _ U _ HKb‰X h™_ dH FIXH F m L u L@S ¥ u O"WL h cYO _ W h Lo}DOlNKF Sn n

S2n

v  q q u  u Sn 2 1 − 1 − 1 + 1− u n u   = 2nu q  t Sn 2 2 1+ 1− n

n

Sn n

 Sn 2 n

2n



v u = Sn u t

2 q 1+ 1−

 Sn 2 n

.


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ZD\

¥TF m O(N m_ WLob _ dHOMETL h cF*H FIJMX hou O u FDP n 6 12 24 48 96 192 384

S2n

Sn n Sn 3.0000000 768 3.1415837 3.1058285 1536 3.1375901 3.1326284 3072 3.1415918 3.1393499 6144 3.1415923 3.1410317 12288 3.1415925 3.1414523 24576 3.1415925 3.1415575 49152 3.1415925

v  q q u  u Sn 2 1+ 1− u 1− 1− n u   = 2nu q  t Sn 2 2 1+ 1− n

 Sn 2 n



v u = Sn u t

2 q 1+ 1−

 . Sn 2

È i St É

n

Å %_ f H F a F m O(NkE h LobeL u L n → ∞  f H F a dsF u dH _gu L 0 3O a F m O(NlLoJ‰U _ HKb‰X h F€È i S t É aKh F m LF m O'W _ S }DO"H FIcYO"} S  u _ H F›N _DaKu O"cF π S ‚ƒJ u F _ f O'du HKLob'FIH_DO'aKu ET_L m Lob ^ _  m O'}DO mu O"H[Lob%O"b _ cm F aKu u O"W_DL h aKFIu cÊ_ d _DaKu u _ dsFI_ }?m cYO"a bcF‰m d _ b%O f O'cL u L`HOgrIXcYf O"cgNKF­h J EGFMr›u N cYO _DO"aKucrIc N S HOMEDO‰H FM{KL FIENKFkdH FIXH F L Ld dsFI} O"}  O Fkb'F HOgrIXcYO"cgNKFIbÂcFkLoJ XW N O cYO O"crIc S Sn n

2n

n

ôÆõÔ8EõÔ1 ú Á\ù A Á ÉsCÆÅ\È Á(û Cü Ç à ;8È Á É Á É ;>9(A Á×ý C ÀBFIb _  m OlNKF e−x =

e−x ∞ X

(−1)n

xn n!

oL c m OÇETH aKu O } m_ _ c>EGFIH f _ LoHOÊJ(OÇ_ghE a O"} x ∈hou Cu Sò_ d Fí^ dYO u _ ETH aKu _Êa FM{ _ u FIEDO"b _ u c>Xb'_DFIaKHKuL™rIc u _ d _•_ mET_ H aKu LF _ d _ga u _gFIu b _ J(O x > 0 cF WLob cYO(NzW _ NKu {KLo|ÇH FIaJMX a O h EsS a HKL_Tm x = u 10 EÇh FIc NzcL<cYO O"crIc L O"} _•a Wu Lob u _ E a _gu _ cgNzL O"WsF LFY}TL6dHKLo}DO"JMXgNKF‰c>Xb'FIHKL™rIc FM{ F E −7.265709 · 10 Y}DO"HBNKF rIL cL6cF beL F SkÀ d J(O x = 1, 2, . . . , 10  a _ cYO"dYOgrIcF m FM„ILob%O h }GFqH m FMrMFDS n=0

−5

x 1 2 3 4 5 6 7 8 9 10

e−x vrsta relativna napaka 3.678795 · 10−1 3.678794 · 10−1 1.6 · 10−7 1.353353 · 10−1 1.353353 · 10−1 2.2 · 10−7 −2 −2 4.978707 · 10 4.978701 · 10 1.1 · 10−6 1.831564 · 10−2 1.831532 · 10−2 1.7 · 10−5 −3 −3 6.737947 · 10 6.737461 · 10 7.2 · 10−5 −3 −3 2.478752 · 10 2.477056 · 10 6.8 · 10−4 9.118820 · 10−4 9.139091 · 10−4 2.2 · 10−3 −4 −4 3.354626 · 10 3.486091 · 10 3.9 · 10−2 −4 −4 1.234098 · 10 1.799157 · 10 4.6 · 10−1 4.539992 · 10−5 −7.265709 · 10−5 2.3 · 10−0

£ O"J h™_gf NKF m O%J(O"d _ H F m NKFer h FIc _ EÇETH aKu FšO hou FIHKcLoHO3d _gh F f€u F f O%dYO%d _ O"W a _gh X u cL ETH F m c _DaKu L<cFI}DO(NrpO a O cYO"HOg{ rpO(N _ sdH F m FIc¿J(OgrIcF›N _ dYO m O u L dH _gu L 0 S Å _€a _ r h FIcL cYO(NzEGFMr›NzLF a F‰J(O"b'F fDh L N _ b%O(Nz|cF m FM„ILob%O h }GF


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Zi

}TL _DaKu O"cF›N _ cF u _ rIcF m_ } _ c„pOšHOgrIXcYO"cgN OQS n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

an sn n an sn 1.000000 1.000000 20 41.103188 13.396751 −10.000000 −9.000000 21 −19.572947 −6.176195 50.000000 41.000000 22 8.896794 2.720599 −166.666672 −125.666672 23 −3.868171 −1.147572 416.666687 291.000000 24 1.611738 0.464166 −833.333374 −542.333374 25 −0.644695 −0.180529 1388.888916 846.555542 26 0.247960 0.067430 −1984.127075 −1137.571533 27 −0.091837 −0.024407 2480.158936 1342.587402 28 0.032799 0.008392 −2755.732178 −1413.144775 29 −0.011310 −0.002918 2755.732178 1342.587402 30 0.000380 0.000852 −2505.211182 −1162.623779 31 −0.001216 −0.000364 2087.676025 925.052246 32 0.000380 0.000016 −1605.904663 −680.852417 33 −0.000115 −0.000099 1147.074707 466.222290 34 0.000034 −0.000065 −764.716492 −298.494202 35 −0.000010 −0.000075 477.947815 179.453613 36 0.000003 −0.000072 −281.145782 −101.692169 37 −0.000001 −0.000073 156.192108 54.499939 38 0.000000 −0.000073 −82.206375 −27.706436 39 −0.000000 −0.000073

^ KH L3HOgrIXcYO"cgNzX e u FI| u F Ÿ OME•cLF a O(N a _ E a L3r h FIcL3d _ JML u LoETcLFQH FIJMX hou O u dYO­EGF h Lo}íLocšNKFH F h O u LoETcYO­cYO"dYO"}DO d _gu IF bìb%O(Nz|cYOQS už O"f HO m LfDh u F m f O h O"m |} _ _ cYmO u O"hcrIc _ _‰Lo_TJMmHOgrIu XcYO"b _ m e u O"} _ h  Ÿ m _ O•LoJMHOgrI_ XcYO"h b _ e u Loc¿h cYO_Tu m _ { FIEDh O"cgNKF _ EGFI|%dHKLoW L h™_ c FIcYh™_šO"} u _ EGF Lo}TLo|íu { FIETL E L e = 1/e S<‚ƒJ F OqJ F O*ETL Lob  O O"|} h o h u k u u f T _ m u £ E%EGF Lo}GF*cYO"dYO"}GFDS FIJMX O O"}Q{KcF O { FIEDO"cgN O­Lob%O O"|} cYO"beH FMrqJpF b%O rIcLo| { FIET}sS 10

−10

−10

10

10

ôÆõÔ8Eõþ ö Á\ùÁ sÉ CÆÅ\È Á :«ÉsC Ä ;GCA Å ÁòÁ ÅBCÆ÷ÿ Á ÀBFIb _  m O aKu O­H FM{KL u ETL4}TEDO m HO u cFqFIcYOgrIWsF d _Tm O"cL a U _ HKb‰X h™_

ax2 + bx + c = 0

È i Sw É jld _ HO"uWYO•J fGh™_ _ HKcgNKh F'U _ HKb‰h X h F h™_ h O"|} _ EÊcFI}D_gO u u FIHKLo|Ëh dHKLob'_ FIHKLo|Ëm dHKmLods_ F h NKF m_ _ |>X u m Lo|ËcYh O"dYO"_T}sm S€_ nldHpS‚u `rMF _ a _ a } f _  FâáY„IL™FIc LJpF EGF Lo}TL O LBJpF b%O(Nz|cLF6d FIb O"|} dHKL F dH FI} HOgrIL EGFíO Ld } HOgrIL EGF W F O d_HKLsHOgrIXcYu O"_ cgNzX b − 4ac SB§6FIb‰X a F h O"_ |} _ a Lo_gJ h _gf u cFI_ b _ m rMF[_DcaKdu HpSQdH F m | _Tm c _ „MF h™_ FIcYOgrIW _šm F h Lob _'a u L aKu Lob } FâáY„IL™FIc b a, b, c }TL3Lob%OšcYO(NzEGFMr›N O"W X c ETH F c S †[HKX f O u _ F Ÿ OMEDi O NKF h O"|} _ d _ga u FIc„I_gLˆO f h c __T_~m { u u FIED_ O"cgNKm FBFIcYO"_} _ EGF h u Lo}TLo|š{ _Tu m FIETL ha "}T_~L _Ta F m h O"|} m _ d _ N OMETL>EqFIcFIbÞLoJMb'F m_ LoJMHOgrIXc EÈ S w É S%§6FIb‰X FšLoJ cFIb O"}  O€FIc H FM{KL FIEÈ ETL c dH F JMcYO"}DO bÉ LoJMHOgrIXcYO"b d _ U _ HKb‰X h L È i S w É  m HKX fG_ dYOšd _ U _ HKb‰X h L x = c/(ax ) S dd _ F UET_ J(HKO"b‰b'X FIh bL È_ i cS w dÉ HpmS _aW=Lob 1.2345678  b = 76543210.5  c = 0.1122334455 d _gu FIbÄE m E _ NzcLcYO u O"crIc _DaKu L _ x1,2 =

−b ±

√ b2 − 4ac . 2a

2

2

x1 = −6.200000558900046 · 107 ,

1

x2 = −6.034970778375905 · 10−9


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ZDZ

u _ rIcL3cL™r h L4dYO aKu O ^ HKETL`dHKLoW h L Ÿ FI}%NKF‰cYO u O"crMFIcc3dHKL m HKX f FIbìdYO a b _ J(O"HO m L _Tm { u FIEDO"cgN O%FIcYO"} _ EGF h Lo}TLo|Ê} _gh L™rILoc¿E~WL aKu ETX LoJ f XWL h L E a F u _ rIcF m FM„ILob%O h }GFDSLd F m HKX fG_ cL™r h™_ LoJMHOgrIXcYO"b _ dH FI} _ x = c/(ax )  m_ WLob _ dHOMETL h c _ x = −1.466275647008561 · 10 S x e1 = −6.200000558900046 · 107 ,

x e2 = −1.466275647008561 · 10−9 . 2

−9

2

ôÆõÔ8Eõ ö ÆC ÷øfÅBCÆÅ\È Á 

1

I10

‚ƒc u F f HO h F

Z

1

xn ex−1 dx,

 h O"|} _ c>Xb'FIHKL™rIc _ HOgrIXcYO"b _ H FI}TXHKJMLoETc _ dH FI} _ U _ HKb‰X h F In =

0

n = 0, 1, . . .

In =

1 xn ex−1 0

−n

Z

1

xn−1 ex−1 dx = 1 − nIn−1 ,

a O(N d _ JMcYO"b _ J(OgrMF u c _ ETH F m c _DaKu I = 1 − e S £ FIJMX hou O u LÈÔE€FIc _ NzcL3cYO u O"crIc _DaKu L É cL a _ cYO(NzW _gh NK{KL@P 0

n 0 1 2 3 4 5 6

0

−1

In 0.6321205 0.3678795 0.2642411 0.2072767 0.1708932 0.1455340 0.1267958

n In 7 0.1124296 8 0.1005630 9 0.0949326 10 0.0506744 11 0.4425812 12 −4.3109741 13 57.0426636

£ O"J h™_gf NKFE%U _ HKb‰X h L I − 1 − nI S nO"dYO"}DOšdHKL6r h FIc>X I a Fd _ bec _gŸ L6J n Loc u _ H F›Nd _ O"W a _gh X u cL ETH F m c _DaKu L3|L u H _ cYO"HOg{ rpO u _ rIcF*ETH F m c _DaKu L I dYO­dYO m O(N _ S n

n−1

n−1

n

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1−In n

n


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I12

Zt ÈÔEÊFIc _ NzcLBcYO u O"crIc _DaKu L É E a F

I0

n 0 1 2 3 4 5

In 0.6321205 0.3678795 0.2642411 0.2072766 0.1708934 0.1455329

n 8 9 10 11 12 13

SS

In 0.1009320 0.0916123 0.0838771 0.0773522 0.0717733 0.0669477

SS

6 0.1268024 7 0.1123835 26 0.0000000

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x>0 x≥1



0<x<1

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−12

1 2 8

12

p 1 1 1 − 10−12 = 1 − 10−12 − 10−24 + · · · = 0. 9| .{z . . 9} 4 9| .{z . . 9} 87 . . . , 2 8

Loc•{ u FIETL h™_­a FlJ(O _ }TH _gŸ LscYO 0. 9| .{z. . 9} S §`O"} _ša } _ H FIcgNKFIcgNKFIbÂcLo} _gh LscFldHKL m FIb _šm_ i >} _ dYOq}TEDO m HKLoHO"b _  NKF{ u FIETL h™_ EGF m c _ b%O"cgNK{ F m_ } h FIHkcF*dHKL m F m_ d _Tm } _ HOgrIL u EGFqLoc m_ WLob _ \QS ^ HKE _ dH F maKu OME h NzLoE _ { u FIETL h™_ }TLTNKFEGFMr›NKF _Tm i NKF 1 + 10 SB§4X m_ WLob _ 12

11

12

−11

p 1 1 1 + 10−11 = 1 + 10−11 − 10−22 + · · · = 1. 0| .{z . . 0} 4 9| .{z . . 9} 87 . . . , 2 8

}DO"H a FJ(O _ }TH _gŸ L4cYO i SB§`O"} _'a } _ H FIcgNKFIcgNKFIb-dHKL m FIb _%m_ i  a }TEDO m HKLoHO"cgNKFIb-dYO a F u _ cF a dH FIb'FIcL@S 11

10

ôÆõÔ8EõÔ8 û ;>=÷Ç Á Å9":C; Á :CøB; ý =QÉ ÅBC à ; Ï(øfÇ@C 

†*O"cYOlNKF u HKL™r h FIc a }DOšH FI}TXHKJMLoETcYO­U _ HKb‰X h O

xk+1 = 2.25xk − 0.5xk−1

Loc€J(OgrMF u cL m EGF*ETH F m c _DaKu L x = 1/3 Loc x = 1/12 S ^ HKL u FI|~d _gfG_ NzLo|šNKF a d h™_ {KcYOšH FM{KL u FIE nlXb'FIHKL™rIc _ E m E _ NzcL4cYO u O"crIc _DaKu L m_ WLob _ cYO aKh F m cgNKF u _ r }GF%È h™_gf O"HKL u FIb a D} O a }DO h O É 1

2

xk =

4 3

· 4−k

S


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Z"w

0

−5

−10

−15

−20

−25

−30

−35

−40

5

10

15

20

25

30

35

40

45

50

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 k 1 + β2k . 4

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1 0 (4 + 2−56 ) 3 1 −1 (4 + 2−55 ) 3

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S

x

x

10−5 10−6 10−7 10−8 10−9 10−10 10−11 10−12 10−13 10−14 10−15 10−16

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_ WHOMETcYOMEDO"b _ d _gh Loc _ b q(x) = p(x)/(x − x ) }TL>NKFlJ(O‰FIc _ cL Ÿ NKF aKu _ dcgNKFDS § _ d _ cYOME h N O"b _  m_ } h FIH cF LoJMHOgrIXcYO"b _ E a FI|€cL™rMF h S ž O aKu O"WL h c _DaKu NKFd _gu H FIWc _ cL™r h F*LoJ h™_ rpO u L4EídHOMETL h cFIbETH aKu cFIbãH F m X4S †[J#HKTX UnfMNO†O~!V b _gŸ c _DaKu NKF m O~LoJ%d _gh Loc _ b%O p(x) = a x + a x + · · · + a a F aKu OMETLob _ u S L@S ]MNOQW!MNH £ KNRV 1

0

0 0 1 0  0 1 A=  

··· ··· ···

n

1

0 0 0

n−1

n

−an /a0  −an−1 /a0   −an−2 /a0   

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0

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p(x) = (x − 1)(x − 2) · · · (x − 20) = x20 − 210x19 + · · · + 20!

1, 2, . . . , 20

g(x) = p(x) − 2−23 x19

x9 = 8.91752 x10,11 = 10.0953 ± 0.64310i

x16,17 = 16.7307 ± 2.81263i x18,19 = 19.5024 ± 1.94033i x20 = 20.8469

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-

f (z) = a0 z n + a1 z n−1 + · · · + an

α1 , . . . , α n

f (z) = a0 (z − α1 ) · · · (z − αn ).

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S1 (z) =

Loc

n X i=1

S2 (z) =

n X

f 0 (z) 1 = z − αi f (z)

f 02 (z) − f (z)f 00 (z) 1 = −S10 (z) = . 2 (z − αi ) f 2 (z)

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n

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b(z)

1 , z − αn

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b(z) =

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di (z) =

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1 − b(z) z − αi

d(z) =

n−1 X

d2i (z).

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S1 = a + (n − 1)b, n−1 X S2 = a 2 + (b + di )2 = a2 + (n − 1)b2 + d   q 1 2 = S1 ± (n − 1)(nS2 − S1 − nd) . n

αn = z −

n p . S1 ± (n − 1)(nS2 − S12 − nd)

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S2 |S1 |, |S2 |  0

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S1 = zr+1

_ JMLoH _ b%O

zr+1 = zr −

nf (zr ) p . (n − 1)((n − 1)f 0 (zr ) − nf (zr )f 00 (zr )) r) ±

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1

1

n

1

n

1

1

1

1

n

n

n

2

2

n

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n

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=

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(r)

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k=1 k6=i

p(zi ) (r)

(r)

(zi

− zk )

(0)

(0) 1

,

i = 1, . . . , n.

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zi

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− Qj−1

i = 1, . . . , n.

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n−1

0

n−1

1 h(x) = xn f ( ) = an xn + an−1 xn−1 + · · · + a0 , x h(x) = 0 f (1/x) = 0

cYO u O"c} _eu F m O(Nš} _ NKF

J(O­}DO u FIH F f OšEGF h N O W É _ WHO u cYOšH F m X}G„IL N OQP

S

d FJ(O"dL™{ FIb _ FIcYOgrIWsFJ(O  m_ WLob _ O hofG_ HKL u FIb~P

f (x) = (−α + x)(c0 + c1 x + · · · + cn−1 xn−1 ) + cn xn . ci

c0 = −an /c0 r = 1, . . . , n − 1 cr = (cr−1 − an−r )/α cn = a0 − cn−1

T¥ F m O(NBNKF c = SId FkNKF β cL™r h O g(x) = c + . . . + c x GNKF β cL™r h O'd _gh Loc _ b%O f (x) − m _ _ Lob'F h L*b%O(Nz|cFÇb _gu cgNKF f dHKLEGF h Lo}TL |α| [}DO"H•d _ b'FIcLF m O~NKF _ WHO u cYO x Sæ¥TF O(N•W b H F m X}G„IL N O aKu O"WL h cYOrMF*LoJ h™_ rpO"b _ cL™r h F*d _ cYO"HOg{ rpO(N _ rIL6O"W a _gh X u cL6ETH F m c _DaKu L@S „ É } _ b‰WLocLoHO"cYO'H F m X}G„IL N OQP n

f (α) n αn

f (α) αn

0

n−1

n−1

Loc~rMF NKF u _ %b O(Nz|c _ W _ d D_ aKu _ sd FI} Ka u "O WL h FIc4S

f (x) = (x − α)(b0 xn−1 + · · · + bn−r−1 xr + cr−1 xr−1 + · · · + c0 ) + Axr .

¥TF m O(N m _ W Lob _

A=

f (α) αr


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a11 x1 + · · · + a1n xn = b1 an1 x1 + · · · + ann xn = bn

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ij

n×n

n

i

1

1

n

T

xn

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A = [a1 · · · an ] = 

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ai , α i ∈ C n

n

eij = δij

k

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k

wy

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αT1 αTn

,

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n


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w>]

AT

¥>}DO h O"HKcL4dH _Tm X} u EGFI} u _ H©NKFIE x Loc y J(O"dL™{ FIb _ E _ W h Lo}TL P h • x, y H FpO cYOQP y x = xy P _ h a • x, y } bed FI} cYOQP y x = xy Æ c _gŸ FIcgNKFEGFI} u _ H©N OšJ*b%O u HKLo} _ y = Ax a L h O"|} _ dH F maKu OME h N O"b _ cYO m EDO­cYOgrILocYOQP u h u _Tm u u aKu u _ • y = α x P i L4F FIb'FIc y NKFdH X} i F*ETH L™„MF A Loc~EGFI} H©N O x  P h _ aKu _gh ds„MFIEíb%O u HKLo}GF A S • y= x a P y NKF LocFpO"HKcYOš} b‰WLocYOg„IL N O ^`_Tm_ Wc _'a L h O"|} _ bec _gŸ FIcgNKFqb%O u HKLo} C = AB dH F maKu OME h N O"b _ cYO u HKL4cYOgrILocFDP u h u _Tm u u aKu u f aKu _gh ds„pO B  • c = α b P (i, j) L<F FIb'FIc C NKFdH X} i F*ETH L™„MF A Loc j  F O u aKu _gh dsFM„ C NKFdH _Tm X} u A Loc i u F f O aKu _gh ds„pO B  • c = Ab P i L P a _gu O n dH _Tm X} u _ E i u F f O aKu _gh ds„pO A Loc i u F*ETH aKu L™„MF B S • C= a β P C NKFE ÆÊO u HKLo} _ J _ W h Lo} _ xy Q}(NKFIH<NKF x, y 6= 0 Lob'FIc>XgNKFIb "_ W!OQT W T Loc~Lob%O­HO"c f FIcYOQS u R?K8˜NOR›!H Z T!MNRT rMFLoJMd _gh cgNzXgNKFqFIcF f OšLoJMb'F m FI}TETLoEDO h FIc u cLo|~d _gfG_ NKFIEsP n × n b%O HKLo}DO A NKF _ aKu u _'m OlNKF AA = A A = I  • W O(N O­Loc>EGFIHKJ A  O"} n i=1

T

n i=1

H

i

i i

i i

T i

n i=1

ij

T i j

i

i

n i=1

i i

T i i T



−1

−1

−1

ÈÔb%O"} a Lob%O h c _ { u FIETL h™_eh LocFpO"HKc _ cF _Tm ETL a cLo|~ETH aKu L™„qO h L aKu _gh ds„MFIE É  _ aKu m • cF W O(N O x 6= 0  OlNKF Ax = 0 S ÆÊO u u HKLo}DO A ]NKF VtbS˜NOnOUnJLO™XRKSUnV|MNOQWPSRKê T R+OnrMUnFqRT EGF h N O A =a A Sžd FqEGF h N h O A = A DNKFqb%^`_TO mu HK_ Lo}DO _ ®KNMNJ=OnU˜8!T u S[a ¥>Lob'F u HKu L™rIcYO b%_ O HKu Lo}DO[_eNKF m u rMFqJ(OšE O"} 0 EGF N O S x Ax > 0 S Wc NKF|FIHKbeL }DOeb%O HKLo}DO d JML LoETc FâácL cYO YrMF*J(O­E a O"} x 6= 0 EGF h N O xx6= Ax >0 d FJ(O a }DO h O"H λ Loc€cFIcL™rMF h cL6EGFI} u _ H x EGF h N O • det(A) 6= 0

• rang(A) = n

T

H

T

H

d _gua FI_ bNKF h λ Z T˜SUnRTu ™XMAaKK^uW!RV˜SU f  x dY_gOlh dHK_ LodYO m O(N _ rIL Z T˜SUnR+O™!K8UFV!M S<À a O"}DOkb%O u HKLo}DOlLob%O n h O aKu cLo|šETH F m c _DaKu LF }TL cL™r F*}DO"HO"} FIHKL L™rIcF Oed Loc b%O p(λ) = det(A − λI) S aKu m _DaKu h a u u a _ h h aKu u _ hx6O"O |} _ cF•LoJMETWsH FIF H FIbc _‰u L O"} λ_ , .m .O . ,u λE _ HKL H N Fp_­O _ cH u F _ c Lo_ b'HKbeF LoHKHL™O"rIc c_ F€WYb%O"OJ _ HKTLo}G}DFO"Hd _ H b'FpO FIccL Fx Ox =cF•δEGFIS } ž O H©NKh F O aKxu c,Fk. ET. .H F, mxc _DaKdYu O L a Lob'F u HKL™rIcF*d _ JML u LoETc _'m FâácL u cF*b%O u HKLo}GFqEGF h N O λ > 0 S Ax = λx,

1

n

T i j

i

1

ij

n


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wTR

¡

0/

k.k : Cn → R

kxk ≥ 0 kxk = 0 ⇐⇒ x = 0 kαxk = |α| · kxk

¢

kx + yk ≤ kxk + kyk x, y ∈ Cn

α∈C

nO(NzW _gh N JMcYO"cF*EGFI} u _ H a }GF*c _ HKb'F a _ P P i _ • kxk = |x | È @c HKb%O É   P ÈVtZ @c _ HKb%O%O h L6FIET} h L ma }DO‰c _ HKb%O É  • kxk = |x | _ h _ • kxk = max |x | È ∞@c HKb%O'O L4b%O •c HKb%O É S Àö a F h™m u HKL a _ d _Da FIWcL_DaKdu HKLob'FIHKL ö h™m FIH©NKFIEGF p@c _ HKb'F kxk = (P FIH©NKFIEDO­cFMFIcYO"} dHOMETL 1 2

n i=1

n i=1

i

i

2 1/2

i=1,...,n

i

¥¤

p

¥¤

|xH y| ≤ kxkp kykq ,

1 1 + = 1, p q

n p 1/p i=1 |xi | )

k}(NKFIHqNKF

1 ≤ p, q ≤ ∞.

1 ≤ p ≤ ∞

^`_Da FIWcL4dHKLob'FIHBNKF  O"X„ |>¨z¥T„ |>ê O"H u JpFIEDO•cFMFIcYO"} _DaKu |x y| ≤ kxk kyk . ^`_gh NzXWcL m EGFeEGFI} u _ H a }TL c _ HKbeL k.k Loc k.k aKu O•FI}TETLoEDO h FIc u cLF6}DO"H*d _ b'FIcLF m O _ W aKu O(N O u Oí} _ c aKu O"c u L m a h C , C > 0  O­J(OšE O"} x ∈ C EGF N O H

1

n

2

a

2

2

b

C1 kxka ≤ kxkb ≤ C2 kxka .

ž O­cYO(Nzd _gfG_DaKu F›NK{ Fc _ HKb'F*EGF h N O(N _'_ „MFIcFDP

√ kxk2 ≤ kxk1 ≤ nkxk2 √ kxk∞ ≤ kxk2 ≤ nkxk∞ kxk∞ ≤ kxk1 ≤ nkxk∞

€´ ·`¶ >¶ ïM¸¿Ðƨˆ² .ë TUnMNOQPSRT#RV!MNJ#T7`XK[]MAK8˜ Z O†!T!™T ñ ¸ ¬ ¬ î¸ ¬ ¸ ² UnMNO†!VUnR+O ¤ !T#R?KAKNRT!V˜SU ³t¬ Ö¸ ²Q˜NH+—NJ=H Z UnO ] Z O†!TUnO™XRV˜SU ³t¬ bXT#™š˜šT ¬ O R ¸ 0/

k.k : Cm×n → R

¢

kAk ≥ 0 kAk = 0 ⇐⇒ A = 0 kαAk = |α| · kAk

kA + Bk ≤ kAk + kBk kACk ≤ kAk · kCk

A, B ∈ Cm×n C ∈ Cn×p

α∈C

¬«bXT#!TUKNM^Va™!K Z `šT

S


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N1 (A) :=

|aij |,

i,j=1 n X

N2 (A) := (

i,j=1

N∞ (A) :=

1

|aij |2 ) 2 ,

max |aij |,

a FLoJM}DO Ÿ F m O N cL6b%O u HKL™rIcYOec _ HKb%O a O(N cF*LoJMd _gh cgNzXgNKFqd _gfG_ N O a XWb‰X hou Lod h Lo}DO u LoETc _DaKu L@Sl… Ka u O h L m EGF aKu O QdHKL4rMFIb'FIH a F N a FcF*Xd _ HO"W h N O N (A) dYO[NKF ð?M^V—AKNR+OH˜šV!™TaRV!MNJ#T kAk S ^`_ b'FIb‰WcF›NK{ F a _V8]+KNM^TUFV!M8˜8K´Vtbt¸7ORW!H_SOM^T!R?KµRV!MNJLK }TL a _em FâácLoHO"cF*J ∞

1

i,j=1,...,n

2

F

kAxk = max kAxk. kxk kxk=1

kAk := max

À m âF _ácL™_ „IL NzL _g_ u dsFIHO _ u _ H h a }Gh F%c _ HKaKb'h F m h O"|} _Êa u L™„MFIH‰E{ _u FIEG„IXËLoc¡Lob'FIc m_ _ EDO h „IX_ LoJMWsFIH FIb _ HO"J h L™rIu _cL a c _ HKbeLF_  beL dYO¿W b d H FIW EDO L F~cYO F cgNKF~b%O HKL™rIcFÇc HKb'F }TLBNzLo| WLob LoJ JMcYO"cLo|÷a EGFIu} H _ }TLo|c a HKb~m P  _gkAk  kAk := max S<À F HKLsc HKb'F F O kAk := max := max u h u Ÿ _ K a h m h LoJMHO"JML L4cYO FId{KL4cYOgrILocc} }DO F›N cYO F cgNKF FIb'FDS ÑB´TÖ¸ÆÐ ¨™Ð kAk = max ( X |a |) ²‘ñՍRV!MNJ#TN³ ¸ x6=0 x∈Cn

1

kAxk1 kxk1

x6=0 x∈Cn

1

»eV!Ttbt¸

2

j=1,...,n



n X

x6=0 x∈Cn

kAxk∞ kxk∞

ij

i=1,...,n

A = [a1 · · · an ] Ax = kAxk1 = k

kAxk2 kxk2

x6=0 x∈Cn

xi ai k1 ≤

Pn

i=1 xi ai

n X

S

|xi |kai k1 ≤ max kak k1

n X

|xi | = max kak k1 kxk1 .

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1

j 1

i=1

k=1,...,n

k 1

k=1,...,n

j

n×n

H

H

2

ÑB´TÖ¸Ðƨo×

»eV!Ttbt¸

k=1,...,n

k 1

H

1

i=1

n

H

n

2 1

2 2

q kAk2 = σ1 (A) = max λi (AH A) i=1,...,n

2 n

²Q˜F]+K8UnM^T Z RTaRV!MNJ#TN³ ¸

…lW aKu O(N O _ H u u __ c _ HKbeLoHO"cYO¡WY_ O"J(_gO¿u J(O CP }TLBN _a F aKmu _ OME h N O(_ N _¿h O aKu cLlEGFI} u _ H©NzL S d F*EGFI} H x J(O"dL™{ FIb } x = α u WLob i = 1, . . . , n f kAxk22 = xH (AH Ax) = (

n

n i=1

n X i=1

αi ui )H (

n X i=1

AH Aui = σi2 ui

i i

αi σi2 ui ) =

n X i=1

|αi |2 σi2 ≤ σ12

n X i=1

|αi |2 = σ12 kxk22 .




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À/dHKLob'FIHKX

A = AH

ÑB´TÖ¸ÐƨˆØ

»eV!Ttbt¸

NKF

x = u1

i=1,...,n

žO

A=

§ _ d _ b'FIcL kAk

S

kAk2 = max |λ(A)|.

kAk∞ = max ( 

w>

² ՍRV!MNJ#TN³ ¸

X

j=1,...,n

|aij |) ∞

SS  m _ W Lob _

αT1

αTn

kAxk∞ = max |αTi x| ≤ max kxk∞ kαi k1 .

FIcYO"} _DaKu dYOlNKF m_Da F Ÿ FIcYOedHKL

i=1,...,n

≤ maxi=1,...,n kαi k1

xk =

i=1,...,n

 ajk  |ajk | 

0

za ajk 6= 0 sicer,

}(NKFIH`NKF kα k ≥ kα k J(O k = 1, . . . , n S ž O _ dsFIHO u _ H a }GF*c _ HKb'F*EGF h N O kIk = 1 b'F mQu FIb-} _ NKF kIk = √n. §`_ O"} _ } _ga u EGFIu } u _ H a }GFBh c _ HK_'b'_ F a _u X m_ L>b%O u HKL™rIch FŸ c _ HKb'Fb'F mh a FIW _ Nzc _ _ FI}TLoEDO h FIc u cFDS `^ _ 'b FIWcF a _ Yc O aKh F m cgNKF „MFIcF }DO FIHKLobeL O"|} „MFIcLob kAk J O NKF*LoJMHOgrIXc NzLoETLobeL4c HKb%O"beL@P j 1

k 1

F

2

√1 kAkF n √1 kAk1 n √1 kAk∞ n

≤ ≤ ≤ ≤

N∞ (A)

kai k2 , kαi k2

§ µÔ´>—àÐƨ©‘ Ê«T ¦^bžbA› V!MNRX`NO†®|R?KAKNRT!V˜SUnO·VX_8KNR+OJ#V

kAk2 kAk2 kAk2 kAk2 kAk2 ≤ kAk2

kAkF √ nkAk1 √ nkAk∞ nN p ∞ (A) kAk1 kAk∞

¬ OR ¸ ¬Æ]M^T!™Ta™XMAK^W!RV˜SUc]T7`XK ¸ jld _ HO"W h N u O _ h L6W _ b _%hh F u S L@SX a } h O(NKFIcF‰dYO"H FqEGFI} u _ H a }TLo|ÇLoc b%O u HKL™rIcLo|Êc _ HKb}(NKFIHkJ(O'E a O"}•dYO"H[b%O u HKLo}GF Loc€EGFI} H©N O EGF N O 

3  A= 4 1

A

−1 1 −5

 2 −8  0

W V—NOJ#V

≤ ≤ ≤ ≤ ≤

kAk1 = 10 kAk∞ = 13

9 ≤ kAk2 ≤ 11

kAkF = 11

kAk2 = 9.02316

x

kAxk ≤ kAkkxk.

[–YÓ(µÔ´>—<¶ G¸Ðƨ™ç Ê«T#™š˜šT!VaJ#TUnMNOQPSRV#RV!MNJ#V#OR#]V Z `NH+—NRV Z T˜SUnRV#™XMAK^W!RV˜SU J#TUnMNO†K !™ K Z `šT λ

|λ| ≤ kAk.

A


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y\

x 6= 0

|λ| · kxk = kλxk = kAxk ≤ kAk · kxk.

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−1

H

»eV!Ttbt¸

T

2

nO(N W _

A = [a1 · · · an ]

S<ÀBF h N O

kU Ak2F

ž O a dsFI} u HO h c _ c _ KH b '_ m_ W Lob _

=

n X i=1

kU ai k22

=

n X i=1

kxk2 =1

kxk2 =1

aÜ @ : o+>^Dkk  A> ­ sz>Aãä \>   <  D OR ¬¿]VUKNJ `XK ÑB´TÖ¸Ðƨ™èº© KI`XK OR i=1

kai k22 = kAk2F .

kU Ak2 = max kU Axk2 = max kAxk2 = kAk2 .

½µ6½

P∞

2

L

Xi

kXk < 1 kIk = 1 1 k(I − X)−1 k ≤ 1−kXk .

I+X

R?K8˜NOR›!H Z T!MNRTÎJ#TUnMNO†!T¬

(I − X)−1 =

»eV!Ttbt¸ d FlWLYWL h O Loc

a Loc f X h O"HKcYOTWL _ W aKu O(N O hYu O"}'EGFI} u _ H z  m O NKF (I + X)z = 0 S ^`_gu FIbÂWLYWL h™_  u _ H F›N kXk ≥ 1 S z = −Xz À H aKu O aKu PaKh m X_ NKF J(O"H O Pm L kXk < 1 } P_ c>EGFIH f FIc u cYO TFIcYO"} _DaKu dYO m_ WLob _ GrMF ETH aKu _ d _ bec _gŸ Lob _ J I −X S ‚ƒJETH F F L „MFIcYO X ≤ kXk = S ž O"cLob%OšcYO a }DO"} _ NKFH FM{KL u FIE h LocFpO"HKcF f O a L aKu FIb%O Ax = b _ WsrIX uKh NzLoEDO­cYO a dH FIb'FIb‰WsF A Loc b SBnO(N W _ m•u _Tm€m_ WLob _ Ax = b Loc (A + δA)(x + δx) = b + δb S … I +X kzk = kXzk ≤ kXk · kzk

∞ i=1

i

∞ i=1

i

∞ i=1

i

1 1−kXk

δx = (A + δA)−1 (−δAx + δb) = (I + A−1 δA)−1 A−1 (−δAx + δb).

d FdH F m d _DaKu OMETLob _  m OlNKF † _ WLob _

kA−1 k · kδAk < 1

k(I + A−1 δA)−1 k ≤

kδxk ≤ kδxk kxk

≤ ≤

Qd _gu FIbÄNKF

I + A−1 δA

cF a Loc f X h O"HKcYOeLoc€EGFIb _

1 1 ≤ . −1 −1 1 − kA δAk 1 − kA k · kδAk

kA−1 k (kδAk · kxk + kδbk) 1 − kA−1 k · kδAk   kA−1 k kδAk kδbk · kAk · kAk + 1 − kA−1 k · kδAk kAk kxk · kAk   −1 kA k · kAk kδAk kδbk + . kbk 1 − kA−1 k · kAk · kδAk kAk kAk


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kδxk κ(A) ≤ kxk 1 − κ(A) kδAk kAk



kδAk kδbk + kAk kbk



yQi A

S Å _ crIcYO

.

ž O _ WsrIX uKh NzLoE _DaKu EGF h N O 1 ≤ κ(A)  a O(N6NKF 1 ≤ kIk = kAA k ≤ kAkkA k SBÛ m LocF*b%O u HKLo}GF}TL3Lob%O(N _ _ WsrIX uKh NzLoE _DaKu 1  a _ J*cFIcL™rMF h cLob a }DO h O"H©NKFIb-d _ bec _gŸ FIcF*XcL u O"HKcFqb%O u HKLo}GFDS ž O a dsFI} u HO h c _'_ WsrIX uKh NzLoE _DaKu EGF h N O −1

−1

σ1 (A) . σn (A)

κ2 (A) =

ÀBF h N OlcYO aKh F m cgNzLLoJMH FI}?G}TL f O[W _ b _*m_ }DO"J(O h LE‰cYO m O h NKFIEDO"cgNzXcG} _ W _ b _*_ WHOMETcYOMEDO h L a Loc f X h O"HKcLHO"Jp„MFId4S SG¹"´>òÆÐ ¨™ºû © KC`XKµJ#TUnMNO†!T A R?K8˜NOR›!H Z T!MNRT¬?]VUKNJæ`XK min



kδAk2 ; A + δA singularna kAk2



=

1 kA−1 k

=

2 kAk2

1 . κ2 (A)

…lWsrIX uKh NzLoE _DaKu NKF u O"} _ H FM„ILodH _ rIcYO _Tmm O h NKFIc _DaKu L _Tm€a Loc f X h O"HKcF f OedH _ W h FIb%OQS § µÔ´>—àÆÐ ¨ˆ² ¦ ¤ P8KNJ#V´!VtKê[_SOFKNR\UK¿]V Z ORV!J#T p(x) = a + a x + · · · + a x ¬EtOCRT

bS™!KAbSRV7œNHR!_SO `šV UFT!V¬·W T7`XKµRT8]T!T Ë\V!MAK`J#V!M^TL™!K Z `šTUnO bXT ¬C`XKNM `XK f

E=

∂E ∂ai

=0

1

f (x)xi−1 dx =

0

Z

1 0

0

1

2

(f (x) − p(x))2 dx

J=OR+OJ#T Z RT¸ n

n−1

[0, 1]

T8]M^VX˜NOJ=OM^T

i = 1, . . . , n Z 1 ∂E =2 (p(x) − f (x))xi−1 dx. ∂ai 0

¶ WLUFVXW|W V—NOJ#V Z

R1

 

n X

aj xj−1 xi−1  dx =

n X

aj

Z

1

xi+j−2 dx =

0

n X

aj

1 . i+j −1

e»V—^VPS—NHOU J#`NOVa™XO†˜N®¹O†˜SJ#UKNTJ¡UnMNO†˜eX¬CèI˜šO T‘Z —A`KN™!MSUFK V!`š™TLVR!J#]MXT¸ UnMNO†!V ¬·`XKNM¿`XK ¬ èIO Z —AKNMSUFV!™!ORK#J#TUnMNO†Kz˜šVz]MNOJLKNMNOf¸ btK Z V Z Z ¦^© bNK#!T —N£ O·KµR˜XT!K8JL¬¿W K8˜STµUFV|]M^˜SVUF— T!Z RKNJéW T!M^R+W!O R?™=Ka˜š—^T!TtJ=btK8O ¬7RtTO?Z `XVAKe›!O†bt¬[K tVO\`XVK=—^PSW HVU —N`NM^OV™TW¬iKê O btR+—NOM^M^T T!ROžT—^¬·TtbXUKNVJ=V!™!MSK^UFP´VA› ™´V!RO btT —NR+M^T!O†®LR+OÆ]]V V OZ RORV!V!J#JzV!˜8™št¬7O¿R?—^KLTtbS—NOFO¸ Z Z Z Z Z OJLK Z OERV—AKNR+O†®UK £ T!™!¸ § Ë\VLµÔR+´>—àO MAÐÆK8˜e™¨ Ð OR"° !PAVTUf˜NO†!®T ™!KK `šZ `šVaT R]TMN˜O ]K^MNW!OQRXPA`NT!OÆRX`X]K8MN¬cOJLW T«KN`XMNK[O†¬fVR+—^O PSHRU VZ `N—AOKN™R?V˜SK7U]]V!V!™!™!KAKAbXbXT!T!™!RKXT7¸bfË\™!TK !Z O†VL!VOJ#˜SU `šTLVR!W]KSMXU¸KNMNJ#J=TORUnMNT!O†R\!UT KfJ#TUnTMNO†® !KXV ¸ b]tKT#Z V#OJ#™!KT#Z O†R!!]V¹MXT¸7Z O«J#btTK UnZ MNVLO†!J#£ T T‘`8®RVaZWKSUKNMNJ=ORT!R\UFV ¬ RX`XKNRVI]VA› V‘`XKNRV˜SUnRV ¤ UKN™XO Z V]Ti`XK ¸¿­¿VaW!MNH›!OC˜SZ UnM^T!R+O ¸¸ ¸¸ ¸ ¸ ¸ ¸¸ WKSUKNMNJ=ORT!R\UFV ¬[V—^PSHU Z `NO™V˜SUc]T ¸ j=1

j=1

Hn hij = xi−1 j . 4 κ(H4 ) = 1.6 · 10 κ(H7 ) = 4.8 · 108

j=1

κ(H10 ) = 1.6 · 1013



αI

αn

1 0 Bn =   0

1

n2n−1

1

−1 1

··· ···

0

···

 −1 −1    1




gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q– ½µ6FÚ  k‰> _ m a _ _ h h a aKu ņFIO"c} Lo_eb h O"| } _ O X fGb _gu _ c>ETXLob b'_ FIHKL™O rIh c LQNKF H FMm{K_ L W L h NKFILoccYFpOšO"H HKFMc{KL L u FIL E FImb _ WHO

yZ

L

Loc m _ WL h LBdHKLoW h L Ÿ FI} xe J(O u _ rIc _ H FM{KL u FIEsS _gfDh F m O"b _ cgNKF fG_ E _ c _ HKb _ krk S Å FIH a F _DaKu O"cFI} a dH FIb'FIcLF ‚ƒJMHOga rIaKXu cYO"b _h O"|} __DaKu _ O"cFI} _grŸ =_Ëb−Ae x Locšd a g _ h rMF L h FI_eb fDh Axm u = MAKb Td UnO™XbeR+O[c V˜SLoUFb T!R?K8 d NzXWcLob a }DO h O"H©NKFIb H FM{KL u FIE x dYO _DaKu O"cF cF a dH FIb'FIcgNKFIcYO<NKF dHOMETL c F O L Z Ax = b 

krk . kAk · ke xk

nO"dYO"}DO­dHKL x NKFJ*H F h O u LoETcLob D_ aKu "O c} _ bd _ EGFIJ(O"cYOšJ _ „MFIc _

Èt Si É § _ u d _ b'a FIcLF _em u OÊm LoJ€b%O(_ Nz|cF f OÊu H F h O u LoETcmF _ f O _ _DaKu _gO"fGc_ }DO h O"|} _a } h FIdYO"b _ cYOÊb%O(Nz|c _ H F h O u LoETc _ cYO"dYO"} _ H FM{KL EGF O"b F O(NšQ} NKFb%O HKLo}DO A WH d NKFIcYOQS ^ HKLO"cYO h LoJML _ WHO u cF aKu O"WL h c _DaKu LYcYO a J(O"cLob%O>J(O} _gh Lo} _ b _ HO"b _‰a dH FIb'FIcL u L A  m OW _ xe u _ rIcYOH FM{KL u FIE a L aKu FIb%O (A + δA)ex = b S ‚ƒJ*J(O m cgNKFqFIcYOgrIWsF aKh F m L krk ≤ kδAk · kexk _ JMLoH _ b%O kδxk krk ≤ κ(A) . ke xk kAk · ke xk

kδAk krk ≤ . kAk · ke xk kAk

Å FIH a _ O hofG_ HKL u beLFG}TL(NzLo|­EqdHO"} a LTXd _ HO"W h N O"b _ J(OkH FM{ FIEDO"cgNKF h LocFpO"HKcLo| a L aKu FIb _ E?gE a L _ WHO u c _aKu O"WL h cLF b_ _ HO¿u W_ L u L[H F h O u _ LoETcL _DhoaKfGu _O"cFIu }EGF m a c _ u b%O(Nz|FIca ½_¡ÀBmF h_ Lo}¬h H F h O u LoETu cL _DaKu O"_ cFI}u WLl_ÌcYaKO"u beH FMh r d _ b'ž FIcL h m EGF u h Lo}f _ WHO c cYO"dYO"} Loc½O HKL FIb }DO FIHKLob b WL L[H FM{KL FIE?cL WHO c O"WL FIc4S O"HO L F O H F h O u LoETcL _DaKu O"cFI} a O"bãcL4dHOME _ b'FIHKL h™_ J(O u _ rIc _DaKulm_ W h NKFIcF*H FM{KL u EGFDS dH F FÊh O u| Lo_ETrMcFIF b f O _ _DLoaKb'u O"F cu Lq}DOeJ(O { fGF_gb%u _ O ETu L HKh™Lo_ }D O m mOÇ_ WNKHFÇ_ LoJMd H_gOgfGrI_ XNKcYFIO"cYcYOQOS H FM{KL u FIE m_ WHOb _ HO u _ H F›N•WL u Lqd _gh F f b%O(Nz|F f O ^`_gfG_ _DaKu u h _ fG_gu _ _ _gu _ a _*_ W aKu O(N O(N _ O h  k S<nO H FMr fG_ HKL u beNKFIFL c  a }DO b%u FIO HKLoHKbeLo}GL F h O"O"||} }_*mX _ WLob _*ETLomb _ E _gLoh J N κ(A) m_ WH _*>}(_ NK„MFIFIH c dY_ O[J(dO kAH FIWXgkNKFIWb H FIJkA HOgrIXcYO"cgN O SCd F WLQcYO"beH FMr LoJMHOgu rIXcYOh h L A 3f WLBa J(aKO u u _ d _gu H FIW _ EDO h LBEGFMr _ dsFIHOg„IL šN 4} _gu dYO‰NzLo| a L™„MFIH*d _gu H FIWXgNKFIb A_  m OíLoJMHOgrIXcYO"b _ H FM{KL FIE LocFpO"HKcF O L FIb%O Ax = b S ‚ƒJ'È t S i É aKh F m L4JpF h™_ Xd _ HO"WcYO'O"d _DaKu FIHKL _ HKcYO _ „MFIcYO >

−1

−1

−1

kδxk kA−1 k · krk ≤ , ke xk ke xk

}TLTN _ Xd _ HO"W h N O"b _ Eí} _ b‰WLocYOg„IL NzL6J‰O hofG_ HKL u beLF}TL _ „MFIcL N _

kA−1 k

−1

WH FIJqHOgrIXcYO"cgN O

A−1

S

½µ6qÝ  < <: 7¢«>o k <  Y?>Kk > ¾  < l s­  A> < > ­7¢«>o þEÔõ 5Eõô Á ;Ï(ø ADC = Èg9": Á Ï÷CAD;>=@: Á SS b%O u HKLo}DO< J(O"dL a O"cYO~d _ ETH aKu L™„pO"|c nO(N*W _ dYO€dsFIHKb‰X u Og„IL N O a }DO u FIH _ Ÿ F h Lob _ XH F m L u L ETH aKu L™„MF S~§ _ cYO"H F m Lob _ u O"} _  m O J h FIEGF%d _ bec _gŸ Lob _Êa]+KNMNJ=HUFT _SO `8˜8!V ¦

5

A = 

αT1

αTn

 

σ =

A



A

1 σ1

2 σ2

··· ···

n σn




gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q– J#TUnMNO†!V

 Pσ = 

SS  €S † _ W h NKFIcYO€b%O u HKLo}DO

eTσ1

 Pσ A = 

yDt 

SS

αTσ1

Lob%O a dsFIHKb‰X u Og„IL N _ σ XH F›NKFIcF%ETH aKu L™„MF

 

α b%O u HKLo}GF A S e dd _ F beŸ cF h _gLobŸ Lob_'_aKu J*_gh b%ds„MO F[u HKb%Lo} O _ u HKLo}GF  Aa O(=N`NK[F a · · · a ] XH F m SL u L a dsFIHKb‰X u Og„IL N _ σ Qd _gu FIbãb%O u HKLo} _ A J m F a cF P (P A ) = AP T σn

1

T σ

T T

σ

n

þEõÔ5EõÔ1 Ç Á Ï Á Å6ADCs;Å ÁÂÁ ǃ=Ïà=ÅBC = È Á  m OlNKF †FIcLob _  m O­Lob%O"b _'u O"}íEGFI} u _ H §

ljk =

xj xk



SB‚›{ rMFIb _ cF a Loc f X h O"HKc _ b%O u HKLo} _

xk 6= 0  

SS

x1

SS

x1

j = k + 1, . . . , n

X _gaKuu H FIJMcYO~b%O u HKLo}DOQS~ÆÊO u HKLo} _ } L = I − l e }(NKFIH`NKF T k k

Lk

 m OšW _

    Lk =    

Lk

1

SS

SS

xn

0

Qd _gu FIbÄNKF SSS

1 −lk+1,k

SS

1

SSS

       

Lob'FIc>XgNKFIb _(K Z KNJLKNR\UFT!MNRTÍK Z OJ=ORT _SO `šT `J(O"dL™{ FIb _ dYOšN _~h O"|} _ u X m L −ln,k

SS

0

1

     0    lk =  l .  k+1,k     

SS

‚ƒc>EGFIHKJqb%O u HKLo}GF

Lk

         x     k   xk  Lk   =  .  xk+1   0         

k

T σ

5

x ∈ Rn

d F m FâácLoHO"b _

T σn

ln,k

NKF

L−1 k

    = I + lk eTk =    

1

SSS 1 lk+1,k

SS

ln,k

 1

SSS 1

    .   


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q– Å FIHJ(O EGF h N O p NKFdH _Tm X} u b%O u HKLo} (I + li eTi )(I + lj eTj ) = I + li eTi + lj eTj

i<j

½µ6ó

 l21   l31  

1 l32

SS

SS S S S

ln1

s7á«¢` E¾

¨K©

1

ln2

1

SSS

···

ln,n−1

1

yw −1 −1 L−1 1 L2 · · · Ln−1

FIcYO"}

  .  

ÀBFMrILocYO'O hofG_ HKL u b _ E€J(O­H FM{ FIEDO"cgNKF*b%O u HKL™rIcLo|~dH _ W h FIb _ E m F h XgNKF*cYOšcYO aKh F m cgNKFIb-dHKLoc„ILodX4P i É ^ H _ W h FIbcYO(NzdH F›N dH FIEGF m FIb _ cYOeFIc _DaKu OMETcF›NK{KL3dH _ W h FIb}TL3Lob%Ošd _Da FIWc _e_ W h Lo} _ S Z É Û } _ c _ beL™rIc _ H FM{KLob _ FIc _DaKu OMETcF›NK{KL4dH _ W h FIbdHKL6rMFIb'FIH a L3d _ b%O f O"b _%a d _Da FIWc _'aKu HKX} u XH _ S t É ‚ƒJ*H FM{KL u EGFqFIc _DaKu OMETcF›NK{ F f OšdH _ W h FIb%O m_ WLob _ H FM{KL u FIE _ HKL f LocYO h cF f OedH _ W h FIb%OQS Û c _DaKu OMETcF›NK{ F _ W h Lo}GFDP h f a aKu u _gu _ h • H FM{ FIEDO"cgNKF LocFpO"HKcF O L FIb%OQP HKLo} cYO W Lo}DO h aKu m _DaKu L@P ö F a a FIc>WsFIH fG_ EDO'O h L u HKL m LˆO fG_ cYO h cYO _ W h Lo}DO • HOgrIXcYO"cgNKF O cLo|~ETH F c a f h m _DaKu L@P WL m LˆO fG_ cYO h cYO _ W h Lo}DOQS • HOgrIXcYO"cgNKF Loc X O"HKcLo| ETH F c †_ _ u _gH fGF _ m X}Gh „IL NKF­u cYO%FIa c _D_ aKu OMETcF›NK{ _í_g_u W h Lo} _ _ dHKL m FIb _•h a d _ b _ r›N _ dsFIHKb‰X u Og„IL šN sF h FIb'FIc u O"HKcLo|¡F h LobeLocYOg„IL N*O h L H cYO cLo| HO"c U HKb%Og„IL N‰ÈÔH Og„IL NKF JGS JMH „pO NKFIcgN O É S ^`_gfDh F›Nzb _ 4}DO"} _ J'F h FIb'FIc u O"HKcLobeL F h LobeLocYOg„IL N O"beLBb%O u HKLo} _ A J(O"dL™{ FIb _ E _ W h Lo}TL A = LU s}(NKFIH NKF L a d _Tm cgN O u HKLo} _gu cYOeb%O u HKLo}DOšJqFIcL™„pO"beL4cYO m LˆO fG_ cYO h LF U dYO­J fG_ HKcgN O u HKLo} _gu cYOeb%O u HKLo}DOQSBnO(N W _ 

Loc

a11 6= 0

}(NKFIH`NKF

a11  a21 A= 

S ž OšF h LobeLocYOg„IL N a } _ b%O u HKLo} _

l21 =

SS

a21 a11

3SMSMSš 

ln1 =

an1 a11

SS

SS

an1

0

an2

···

1  −l21 L1 =  

SS

−ln1

EGF h N O

   a11 a11  a21   0  = , L1     

··· ···

SS

an1

a12 a22

torej

1

 a1n a2n   

SS

ann

SSS 1

 ,  a

11

SS

 0 A(1) := L1 A =   0

a12 (1) a22

··· ···

(1)

···

SS

an2

a1n  (1) a2n  . 

SS

(1)

ann


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q– ¥TF m O(N`NKF A(2) := L2 A(1)

l32 =

a11  0   = 0  

SS

a12 (1) a22 0

a13 (1) a23 (2) a33

··· ··· ···

0

an3

(2)

···

SS

0

}(NKFIH`NKF

Loc

(1) a32 (1) a22

3SMSMSš

ln2 =

(1) an2 (1) a22

SS

1 −l32

SS

1

SSS

−ln2

1

nO­} _ c„IX m _ W Lob _

a

11

 U := Ln−1 · · · L2 L1 A =   | {z }

a12 (1) a22

SS

(2)

ann

  ,   a1n  (1) a2n  , 

··· ···

SS

SSS

=L−1

Loc

 a1n (1) a2n   (2) a3n  ,  

1

  L2 =   

.

yDy

(n−1)

ann

1

 l21  l =  31 

1 l32

· · · L−1 n−1

SS

  .  

SS S S S S S S l l ··· l 1 §`M^TtO"bX} _8_ KF]òa b —NMA_~KAb¿m]_ OW™L Vh L UnOAM^T!RX=`šT LU íS † _ W h NKFIcLHO"Jp„MFIdËLob'FIc>XgNKFIb _ M^TtbX_8KF]̗NMAKAbµ]O™VUnOM^T!RX`šT O h L T!H˜8˜šV!™ S †[LˆO mqfG_ a cYO h cLF h FIb'_ FIc a u _ L a a , a_l_ , . . .h , am _  _a }DO u u FIHKLobe_gL u m F h Lob _  ha FBLob'a FIc>XgNKF›N i_ u _T]m O™V_TUnO m  l _ dYO t™VXaK_Sh OFKNm R\UnO S ÆÇF O"beLob/dH „MF b b dYO"JML FL  O b HO(N WL L>dLoE L>cFIcL™rMF cFL  L™„MFIH4b'F O d EGFDS`nO F cgNzL LoJMH FI}íd _ EGF} m O(N h O"|} _ HO"Jp„MFId~LoJMHOgrIXcYO"b _ WH FIJ u F Ÿ OMEsS SG¹"´>òÆÐ ¨©‘Qó Ê«TLJ#TUnMNO†!V A `XKzK8t™XO™T Z KNR\UnRV ¯ L=

−1 L−1 1 L2

n1

11

(1) 22

1

n2

n,n−1

ž-«ª

(n−2) n−1,n−1

­¬

ij

8ñ ³ O¶ R —8˜SUFT‘`šR?T#K8˜NKNORR›!V ZH OQPST!R+MNORM^T=TtbXbA_8› KFV!] MNRX`šTaUnMNO†!VUn¬«RT#`XKNJ#M`XTKUnMNO†!T˜F]¸ VXW!RX`šT´UnMNO†!VUnRTzJ#TUnMNO†!TµbµKNR+OQ_AT!J=ORT´W!OQTX› V!RT Z O Z  ° ˜NOC™VXW!O Z R?K·]VXW!J#TUnMNO†K ˜šV#R?K8˜NOR›!H Z T!MNR?KX¸ »eV!Ttbt¸ _ ^`_ }DO Ÿ Lob _  m OlLoJ i É aKh F m LZ É S £ O"Jp„MFId J(Okd _gh NzXWc _ E _Tm L h c _ d _Tm b%O u HKLo} _ W h™_ rIc _ A = LU

L

U

A(1 : k, 1 : k)

J(O"dL™{ FIb

A = LU



A11 A21

_Tm } T_ m IF H Ka h F m L A

A12 A22 11





L11 = L21

= L11 U11

0 L22

S † _ WLob _



U11 0

U12 U22



A11



L11 U11 = L21 U11

det(A11 ) = det(U11 ) 6= 0

 L11 U12 , L21 U12 + L22 U22

 a O(N`NKF

U

cF a Loc f X h O"HKcYOQS


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q–

ž O m_ }DO"JqE _ WHO u cL a b'FIHKL4Xd _ HO"WLob _ Loc m X}G„IL N _ S ^ KH L4b%O u HKLo}DO"| 1 × 1 cL u F Ÿ OME? a O(N`NKF nO(N W _ A = LU Loc Ae = A b S £ O"Jp„MFId€J(O Ae b _ H OšLob'F u L _ W h Lo} _ cT



A cT

b δ

δ 



L = T l

Å FIH Ka u O L Loc U cF a Loc f X h O"HKcLF m_ WLob _ a η 6= 0  (O N6NKF 0 6= det A = η · det U S

0 1



U 0



  u LU = T η l U

u = L−1 b l = U −T c

§ µÔ´>—àÐƨo× ¦^bSM^T PSHRT‘`NJ#V ×M^TtbX_8KF]ÎbXT

 2 2 3 <-«ª A = 4 5 6. 1 2 4    1 0 0 2 A(1) =  −2 1 0  ·A =  0 −1 0 1 0 | 2 {z } L1    1 0 0 2 A(2) =  0 1 0  ·A(1) =  0 0 −1 1 0 | {z }

Loc

 Lu . lT u + η

η = δ − lT u

−1 L = L−1 1 L2

1  = 2

Ý oh fG_ KH L u FIb h O"|} _ J(O"dL™{ FIb _ E _ W h Lo}TL

1 2

a11 = 1 · a11

S ^ HKL u FIb b _ HO•WL u L

2 1 1 2 1 0

L2

y]

0 1 0

 0 1   0 0 1 0

0 1 1

  0 1   0 = 2 1 1 2

 3 0 5 2

 3 0  = U. 5 2

 0 0. 1

0 1 1



j = 1, . . . , n − 1 i = j + 1, . . . , n aij lij = ajj k = j + 1, . . . , n aik = aik − lij ajk

^ KH L4b%O u HKLo}TL L cYOe} _ c„IX b%O"cgNz}DOe{ F I E•J fG_ HKcgNKFIb u HKLo} _gu cLo}TX A dYOšcYOe} _ c„IX _DaKu O"cF U S Û h FIb'FIc u F h _'a |HO"cgNzXgNKFIb _ E a d _Tm cgNzL u HKLo} _gu cLo} A Loc u O"} _ cF*d _gu H FIWXgNKFIb _'m_Tm O u cF f OedH _DaKu _ HOQS L "O |} ¥ u FIETL h™_e_ dsFIHOg„IL NMP n−1 X

n X

j=1 i=j+1

1 +

n X

k=j+1

2 =

=

n−1 X j=1 n−1 X j=1

(n − j)(1 + 2(n − j)) = X  n−1 2(n − j)2 + n − j = (2l2 + l) = l=1

(n − 1)n(2n − 1) (n − 1)n = 2 + = 6 2 2 3 1 2 1 2 = n − n − n = n3 + O(n2 ). 3 2 6 3

S


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q– yR §  µÔ´>—àÐƈ¨ Ø ëÎKSUFVXW TaVXW8]V!™!K8¬¿P8K `XKf]O™VU·KNRT ¬ R+HJLKNMNOQPSRVI]TaVXW8]V!™!KzUnHW!O†¬fP8K `XK[]O™VU·— Z O bSH ¸ © K RTÎUnMNOWK^_SOJ#T Z KUFVXPSRVÍM^T PSHRT!J#V M^TtbX_8KF] ¬=W V—NOJ#V OR ¬iRT8]T!T ° K Z `šT ]T7`XKzVA›!M^V!J=RT¸ ]£ OFM™{KVL Unu OFIM^E T!RX_`XWsK ÉFI| h O"u |F }Ÿ _OMEËdYONKFÊu X dm LoE L _gaKu u Lo_gHhO"dscg„MNKFIFE¿}(È !NKFIV!H€Ji]b'Z KSF Unm RVO ho]fGO_ ™HKVL uUnOb M^T!_ RXb `XK É mS _ dX{ rpO"b _ J(O"b'FIcgN OME _ ETH aKu L™„¬È WK Z RV ^ HKL m F h cFIb dLoE _gu LoHO"cgNzXdH F m F h LobeLocYOg„IL N _ E  u FIb aKu _gh ds„IXdHKLob'FIH©N O"b _ Loc J(O"b'FIcgN O"b _  u _ ETH aKu L™„ _eau L aKu _ Q}TL3E a FIWXgNKF[b%O"} a Lob%O h cL6F h FIb'FIc u S§`O"} _ NKF[dHKL4cF a Loc f X h O"HKcL4b%O u HKLo}TL 0

U =



0.0001 0

1 f l(1 − 10000)



  0.0001 1 1 A = L = 1  1  10000 1 0.0001 1 . LU = 6 A = −10000 1 0

LU  0.0001 = 0



j

0 0 1



|ajj |, |aj+1,j |, . . . , |anj |

E%E a O"}GFIb-} _ HO"}TX~dLoE _gu cFIcL™rMF h FIc4S Å _gu H FIJMX hou O u m_ WLob _ P A = LU D}(NKFIH3NKF P dsFIHKb‰X u Og„IL N a }DO*b%O u HKLo}DOQS ž O"HO m LYdLoE _gu LoHO"cgN O a _ E­b%O u HKLo}TL a h u _ a _gh X u cL6ETH F m c _DaKu L _ b'F›NKFIcL4J 1 S L E L3F FIb'FIc L6d O"W SG¹"´>òÆÐ ¨©‘Yª‘ © K `XK A R?K8˜NOR›!H Z T!MNRT¬c]VUKNJéV—8˜SUFT‘`šTaUFT!Tµ]+KNMNJ=HUFT _SO `8˜8!T|J#TUnMNO†!T P ¬·W T¹V—8˜SUFT‘`šT LU j

^MJ#TtTbX_8UnMNKF]O†!T¸ ¬Æ`XKNMÆ`XK ˜F]VXW!RX`šTµUnMNO†!VUnRTµJ#TUnMNO†!TžbžKNR+OQ_AT!J=O?RTeW!OQTX› V!RT Z O?OR bA› V!MNRX`šTµUnMNO†!VUnRT »eV!Ttbt¸ ^ H F m XcL™rMFIEDO"cgNKFIbF h FIb'FIc u _ E€E  u FIb aKu _gh ds„IXeNKF a L u XYOg„IL N O­cYO aKh F m cgN OQP P A = LU

L

U

j

A

(j−1)



U11 = 0

 U12 (j−1) . A22

Å FIHNKF‰b%O u HKLo}DO A cF a Loc f X h O"HKcYOQNKF‰cF a Loc f X h O"HKcYO u X m L A  u _ Yd O'd _ b'FIcLF m O%b _ HO%WL u L cF a Loc f X h O"HKcYOšLoc~cFb _ H F›N _ WL u L4E a L4F h FIb'FIc u L6EídHKEGFIb aKu _gh ds„IX A | }THO u L6FIcYO"}TL 0 S Ý hofG_ HKL u FIb-J(Ošx4jÞHO"Jp„MFId~J m F h cLobdLoE _gu LoHO"cgNKFIbÄNKFDP (j−1)

(j−1) 22

(j−1)

A22

d _ L™{ rIL aKu J(O"b'FIcgN O(NETH L™„IL Loc

j = 1, . . . , n − 1 |aqj | = maxj≤p≤n |apj | q j i = j + 1, . . . , n aij lij = ajj k = j + 1, . . . , n aik = aik − lij ajk

† _Tm O u c _em F h™_ NKF

O(n2 )

dHKLob'FIH©N O"cgNMS

§ µÔ´>—àÐƨ™ç ¦^bSM^T PSHRT‘`NJ#V òM^TtbX_8KF]Íb=WK Z R+OJ ]O™VUnOM^T!RX`XKNJbXT GµT#bXT P8KSUtH"™SbXT!Õ J®LM^KNT!J#R+OV J#V#™e˜F]¬?VXW!]RXV`XUKNKNJ J Un]MNTLO†!™šV˜šUnTR+tO†tOQPNH ¬ !V´¸ bXT!JLKNRX`šT!J#V´™XM8˜SUnOQ_SO†¬EbXT!JLKNRX`šT!™V#RT!MAK^W!OJ#VaUnHW!OC™ ¸¿ë.TUnMNO†!V 

0  A= 1 1

-ǻ

P =I

A

A(1)

1 =  0 1

2 1 −2

 3 2 , −2

0 P = 1 0

1 0 0

 0 0, 1

1 2 0

 2 3. 1

P

L


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q– 

1  1 =  0

A(2)

 3 −2  ,

2 −2

− 12

yv 

0 P = 0 1

1

1 0 0

 0 1. 0

EVt™XOM`XKNR+OiK Z KNJLKNR\UnO·˜F]T W T‘`šV™aJ#TUnMNO†!V ¬7RT¹RX`NO†®V!™XO†®"JLK8˜SUnO†®"™ ]T¹˜šV¹R+OQP Z KX¸L»eV—NO Z O[˜NJ#V ¬ OR ¸ ª

1 1 0

1 − 12

1

1

2 −2

 U =

 3 −2  1

L

A

P A = LU

£ FM{ FIEDO"cgNKF a L aKu FIb%O Ax = b P i É P A = LU  Z É Ly = P b =: b  t É Ux = yS ¥>L aKu FIb Ly = b H FM{KXgNKFIb _e[a ]MAKNJ#V#˜NH+—8˜SUnOnUnH_SO `šV SB‚ƒJ 0

0

SS S 1S S

 l21  

ln1

m_ WLob _ i = 1, . . . , n P yi = b0i − i−1 j=1 lij yj Pn

¥ u aKFIu ETL h™_e_ dsFIHOg„IL N<NKF Ux = y

i=1 (1

H FM{KXgNKFIb _ J

ln,n−1

1

  b0  y1 1   y2   b02    =      

SS

SS

yn

b0n

i = 1, . . . , n,

S

V—NM^TUnRVL˜NH+—8˜SUnOnUnH_SO `šV S ‚ƒJ u11

S S S u SS

···

1n

unn

 

SS

x1 xn

=

uii xi + ui,i+1 xi+1 + · · · + uin xn = yi ,

i = n, n − 1, 1  . . . ,P  1 xi = uii yi − nj=i+1 uij xj

¥ u FIETL h™_e_ dsFIHOg„IL N<NKF

SSS

+ 2(i − 1)) = n2 

m_ WLob _ _Tm•u _Tm dYOeO hofG_ HKL u FIb

···



li1 y1 + · · · + li,i−1 yi−1 + yi = b0i ,

_Tm•u _Tm dYOeO hofG_ HKL u FIb

¥>L FIb

1

Pn

i=1 (2

+ 2(i − 1)) = n2 + n

SS

y1 yn

 

i = 1, . . . , n,

È©{ F m F h NKFIcgN O­cYO m LˆO fG_ cYO h L É S

L= 


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ž O¿x4jHO"Jp„MFIdJ m F h cLob dLoE _gu LoHO"cgNKFIb u _ H F›Ned _ HO"WLob _ n + O(n ) _ dsFIHOg„IL Nš } _ dYO L Loc U Ÿ F d _ JMcYO"b _ J(O­H FM{ FIEDO"cgNKF Ax = b d _ HO"WLob _ { F m_Tm O u cLo| 2n + O(n) _ dsFIHOg„IL NMS ž O­H FM{ FIEDO"cgNKF a L aKu FIb%O Ax = b cLo} _gh L4cF*Xd _ HO"W h N O"b _ Loc>EGFIHKJMcFqb%O u HKLo}GF A  a O(NMP _gŸ _ _ 2n _ dsFIHOg„IL Nš}DO"HcL4„MFIcF›NKF _Tm H FM{ FIEDO"cgN O u HKLo} _gu cLo| L Loc U • J(O­bec FIcgNKF A b d HO"WLob _gu _ 2n _ dsFIHOg„IL šN }DO"H<NKF u HKLo}THO uku _gh Lo} _ } _gu x4j/HO"Jp„MFId • J(O­HOgrIXcYO"cgNKF A d H FIWXgNKFIb a _ }TEGFMr›NKFIb‰X~EGFMr›NKFDS • c>Xb'FIHKL™rIcFqcYO"dYO"}GF §4X m L4}DO m O"H NKF[d _gu H FIWc _ LoJMHOgrIXcYO u L A B  u _ cYO"H F m Lob _'u O"} _  m OšH FM{KLob _ea L aKu FIb AX = B S ^`_gh F f x4jÿHO"Jp„MFIdYO­J m F h cLobãdLoE _gu LoHO"cgNKFIbãd _ JMcYO"b _ { F ìM^TtbX_8KF]Ȟ!V!Ji] Z KSUnR+OJå]O™VUnOM^T!RX`XKNJ T}(NKFIH E j  u FIb aK_ u _gh dsaK„Iu X¿dLoE aK_gu u _gch L F h FIb'FIc u LoJM_ WLoHO"b m_ _ LoJ%„M_ F h Fed _Tm b%O u HKLo}GF A(j : n, jaKu : n) 3cYO u _ dYOíLoJMEGu F m FIb a _ J(O"b'u FIcgN OME ETH aKu L™„BLoc_ _ ds„MFIaKEsu S`_gh nO} cu „IX h™_[W_ Lob HO"Jp„MFId P AQ_ =_gu LU "_D}(a NKFI_ H O P Loc Q dsFIHKb‰X Ogu „IL N }Th™_ L b%O HKLo}TLTJ(OkETH L™„MF JMLoH b%O ds„MFDS ¥ FIETL dsFIHOg„IL NNKF FIcYO"} } dHKL c ETcFIbx4jHO"Jp„MFIdcX G{ FIETL dHKLob'FIH©N O"cgN dYO[NKF O(n ) S ^ HKL3} _ bed h F u cFIbdLoE _gu LoHO"cgNzX a L aKu FIb Ax = b H FM{KXgNKFIb _ P i É P AQ = LU  Z É Ly = P b =: b  t É Ux = y w É x = Qx S 2 3 3 2

2

−1

−1

2

®

−1

3

®

−1

-ǻ

3

0

0

0

§ µÔ´>—àÐƨԘ O†˜SUKNJ ¬C`XKNM˜SUFT ˜I!V!Ji] Z KSUnR+OJ ]O™VUnOM^T!RX`XKNJL¸ )

0  A= 1 1

Ax = b

 A(1) =  

 A(2) = 

3

2

1 3

1 3

1 3

1 3

 1 − 13  ,  2 3

3

1

2

1 3

2 3

1 3

1 3

− 12

1 2

 ,

 1 3 1

1 2 1

0  P = 1 0 

0  P = 0 1

OR

  2  b = 7 3  0 0, 1

1 0 0 1 0 0

 0 1, 0

¬[—^V!J#VaMAK ¤ O Z O]MAK8!V ×M^TtbX_8KF]T -«ª

0  Q= 0 1 

0  Q= 0 1

EVt™XOM`XKNR+O[K Z KNJLKNR\UnOE˜šV#O b ¬C™ ]T´˜šV#RT#RX`NO†®V!™XO†®aJLK8˜SUnO†®|R+OQP Z KX¸ ä TtbX_8KF]L`XK OR ¸ ª

1 L =  13

»eV—NOJ#V

1 3

1 − 12

1

 

L A 3 1 2 U = 3

¬

¬

2 1 3 1 2

 

      7 7 2 2 0 0     b = Pb = 3 y= 3 x = 1 2 0 0

OR

  1  x = 0 2

¸

 1 0, 0

0 1 0 1 0 0

 0 1. 0

P AQ = LU

¬C`XKNMI˜SUFT 


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¯

¨K©

f l(a b) =

: +, −, ∗, / f l(a b) = (a b)(1 + δ) |δ| ≤ u

|δ| ≤ u

a b 1+δ

^ KH LQH FM{ FIEDO"cgNzX Ax = b dH FI} _ 4x j¬HO"Jp„MFIdYO m_ WLob _ xbS ^ HKLYO"cYO h LoJML _ WHO u cF aKu O"WL h c _DaKu LL™{ rMFIb __ „MFIc _ J(O m ^ _ m _DaKu OMETLob _  m Oqb'F m HOgrIXcYO"cgNKFIbÂcFldHKL m F m_ dH FI} _ HOgrIL u EGF δA   + δA)b x = b S HKL M„ FIcLdH F d _ JGS<d _TO m NK} F _ (A u HOgrIL EGFDS   |a | · · · |a | S S …lJMcYO"}DO |A| d _ b'FIcL4b%O u HKLo} _ |A| =  S S   A ≤ B dYO m O[NKF a ≤ b J(OšE a O"} i, j S 11

1n

nOMEGF mQh L4W _ b _šh F*LoJMH FI}GF u _ rIcF m_ }DO"JpF h O"|} _ cYO(N m F u F*E v S ÑB´TÖ¸ÆÐ ¨©‘² GµT‘`#—^V L ˜F]VXW!RX`šT¹UnMNO†!VUnRT|J#TUnMNO†!T|™!K Z O†!V˜SUnO n × n ¸ |an1 | · · ·

|ann |

–

ij

ij



¬C`XO†˜SKNUM KNJ`XK MAK ¤ OJ#¸ Va˜¿]MAKNJ#V ˜NH+—8˜SUnOnUnH_SO `šV¸·¦^bSM^T PSHRT!RTLMAK ¤ OnUKN™ bXT W V ¤ PAT¹KNRT Pt—NO ÑB´TÖ¸ÐÆ©¨ ‘QÐ GµT‘`a—^V J#TUnMNO†!T™!K Z O†!V˜SUnO Ê«T#O bSM^T PSHRT!RT OR ™!K Z `šT ¬C`XKNM ¬C`XK]MNO·!TUKNMNO7˜XK´O bS™!K^W¸K M^TtbX_8KF]̗NMAKAb]O™VUnOM^T!RX`šT¸ ^`_DaKh F m L™„pOqNKF m O%J(O'LoJMHOgrIXcYO"cYO h N O G E F  }(NKFIHNKF sJ(O  O h L c _ HKb _ ÈÔ}(NKFIHEGF h N O É  dYO m_ WLob _ cdHpS x b

b L

A b U

F

)

Lx = b |δL| ≤ nu|L|

(L + δL)b x=b

n×n b b A = LU + E

L, U k|A|k = kAk

.-ǻ

b · |U b| |E| ≤ nu|L|

A = LU + E

kEk ≤ nuk|L|k · k|U |k

∞ 1

kEk∞ ≤ nukLk∞ kU k∞ .

¬«`XKNM S`X KG¹"´>òÐƨ©‘T× «Ê T´O bSM^T PSHRVtT!bSORM^V#V!J#MAK T ¤ OnUKN™ ˜NO†˜SUKNJ#T ]MAK8!¸ V ìM^TtbX_8KF]TL™!K Z `šT ž O"cLob%OšcYO a Q } m O(N`NKF  a O(N`NKFd _gu FIb-b'F u _Tm O _ WHO u c _%aKu O"WL h cYOQS Å _gh L™rILoc _ Lob'FIc>XgNKFIb _=]O™VUnRT#M^T˜SU S ¥ h F m L S ^ HKL m F h cFIbLoc } _ bed h F u cFIbÏdLoE _gu LoHO"cgNzXÇEGF h N O _DaKh F m L™„pO'dYO[NKF _ J(O m F h c _   d S ` § " O } Loc€} _ bed h F u c _ dLoE _gu LoHO"cgNKF*EGF h N O …[„MFIcY_ ONKF h O"|} _ JpF h™_ EGF h Lo}DO u X m LrMF`NKF S ¥>} _ HO(N<EGF m c _ NKF _ „MFIcYO aKh O"W{O _Tmšm F›N O"c a }TLo|šH FIJMX hou O u _ EsS ÀBFIb { FDP x b Ax = b kδAk∞ ≤ 3nukLk∞ kU k∞

|δA| ≤ 3nu|L| · |U |

<-ǻ

(A + δA)b x=b

3nukLk∞ kU k∞ = O(u)kAk∞

g :=

max |uij | max |aij |

kU k∞ ≤ ngkAk∞

|lij | ≤ 1

kLk∞ ≤ n

kδAk∞ ≤ 3gn3 ukAk∞ .

g=1

O É m F h c _ d LoE _gu LoHO"cgNKFDP ÑB´TÖ¸Ðƨ©‘Ø ­[MNO[WK Z R?KNJ ]O™VUnOM^T!RX`NHµ`XK[]O™VUnRTLM^T˜SU[V!JLK`XKNRT´b 2 ¸ • »eV!Ttbtu ¸ ž O"HO m a L a = _a − l _Tam _ Loc |l Å | ≤ 1a a F a h O"|} f _ ETH hF m c _DaKuu cYO(mNz_gEGu FMr›NKF f OíF_ h FIb'FIc u O Eb%O HKLo}TL*E¬u E O"}GFIbú} HO"}TX½d E NzL@S FIH FÇE O"}GF OÌF FIb'FIc O O"}TcFIb cYO(NzEGFMr S (n − 1)@}THO gNKF g ≤ 2 ' _ a _ g _ u F g WcYOg{Oš} n S • …lWL™rpO(Nzc n−1

jk

jk

ji ik

n−1

2/3

ij


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]i

x4j/HO"Jp„MFId~J m F h cLob-dLoE _gu LoHO"cgNKFIb NKFEídHO"} a L _ WHO u c _'aKu O"WL h FIc4S W É } _ bed h F u c _ dLoE _gu LoHO"cgNKFDP ÑB´TÖ¸ÆÐ ¨©‘Qç ­[MNOC!V!Ji] Z KSUnR?KNJç]O™VUnOM^T!RX`NH`XK·]O™VUnRT#M^T˜SU·V!JLK`XKNRT=b • •

1  1 ln n 2 g ≤ n · 2 · 31/2 · 41/3 · · · n1/(n−1) ≈ n2+ 4 .

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½µ6

L

‘

¢` ­ ;¾í ß  ­k  z>Aã B  ED>  LoJMHOgrIXcYO"cYO­_TH FMm•{KL u u _TFIm E? dYmO _ u _ rIcY_%O‰_ H FM{KL u _ FIE a L aKu FIb%O  dYO WLob „MFIc

n O(h NW _ xb ÀBF N O xb − x = A

x

Ax = b

S…lJMcYOgrILob _e_DaKu O"cFI}

−1 r

r := Ab x−b

S

kb x − xk∞ krk∞ ≤ kA−1 k∞ . kb x k∞ kb xk∞

^ _ h J(O¿Xd _ HO"W __ „MFIcFËÈ©rMF a F _ b'F›NzLob _ cYO i  ∞ Loc F c _ HKb _ É NKF u O m O¿cF€d _ JMcYO"b _ A S ^`_gH u H W FIWFIb¾ _Êm_ WH _Ç_ „MFIc _ J(O kA k WH FIJíHOgrIXcYO"cgN O A  a O(N*J(O~HOgrIXcYO"cgNKF A d _ HO"WLob _ 2n g X K N I F b _ dsFIHOg„IL šN QJ(OšH FM{ FIEDO"cgNKF Ax = b dYO h F n S F h Lob _'m_ WL u L4Xd _ HO"Wc _%_ „MFIc _ E O(n ) _ dsFIHOg„IL N O"|4S ž O"dL a O h LW _ b _ O hofG_ HKL u FIbÂJ(O _ „MFIc _ kBk dHKLd _gfG_ NzXc m OqJMcYO"b _ LoJMHOgrIXcYO u L Bx Loc B y J(Oqd _gh NzXWcYO EGFI} u _ _H©N O x, ym S u ^ HKL _ _ „M_FIcgNKFIEDO"_ cgNzX _ kA_ k u __ d _ b'FIcLF m O'JMcYO"b _ H FM{KL u L a L aKu FIb%OeJ A Loc A  u _ dYO JMcYO"b cYO"H F L L6FI} c beL™rIc Y} d JMcYO"b LU HO"Jp„MFId A S P ž a ÀBFIb _  m O NK_ F kBk = max = max |b | S O*E O"} kxk = 1 NKF kBxk ≤ kBk S †FâácLoHO"b f (x) = kBxk . m¥TF O(N L™{ rMFIb _ b%O"} a Lob‰Xb f cYO _ Wb _ r›NzX kxk ≤ 1 J f HO m L™FIc u c _ b'F u _Tm_ S _ a O  a O(N`NKF f NKF} c>EGFI} cY −1

2 3 Y± 3

−1

1

kBxk1 x6=0 kxk1

−1

−1

−1

3

2

T

1

−1

1

j=1,...,n

n i=1

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1

1

1

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j

n j=1 ij j

i

i

f (x) =

n X i=1

ξi

n X j=1

bij xj .

n j=1 ij j


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n

∂f (x) X = ξi bik ∂xk

Loc ∇f (x) = ξ B = (B ξ) S Å FIH<NKF f } _ c>EGFI} a cYOgNKF f (y) ≥ f (x) + ∇f (x)(y − x) S

i=1

T

T

SG¹"´>òÐƨ©‘˜ GµT‘`z—^V ñ8³ © K™!K Z `šT  © K™!K Z `šT »eV!Ttbt¸

T

¬ ¬ OR ² ³ ¸ ¬ ]VUKNJ OJ#T ™ Z V!T Z R+OCJ=OR+OJ=HJ ¸ ¬`XK ?¬ ]MNOfP8KNJLKNMf`XK ¸ z = B T ξ z T = ∇f

kxk = 1 w = Bx ξ = sign(w)

kzk∞ ≤ z T x kzk∞ > z T x

f

x

kwk1

f (ej ) > f (x)

|zj | = kzk∞

i É À m _ E _gh Nkb%O(Nz|cL _ } _gh L™„IL x  }(NKFIH ξ D_ aKu "O cF›N _ cF a d H FIb'FIcgNKFIcLFYEGF h N O x) = f (x) + z (y − x) S i

T

n X

z T (y − x) = z T y − z T x ≤

Z É Å FIH NKF

f (ej ) = f (−ej )

|zi ||yi | − z T x ≤ kzk∞

h O"|} _ dH F m d _DaKu OMETLob _  m O[NKF i=1

n X i=1

f (y) = f (x) + ∇f (x)(y −

|yi | − z T x ≤ kzk∞ − z T x ≤ 0.

eTj z = kzk∞ .

f (ej ) ≥ f (x) + ∇f (x)(ej − x) = f (x) + z T ej − z T x = f (x) + kzk∞ − z T x > f (x).

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1

T

j

T

1

j

2

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L

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−1

−T

1

−1 k

S

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]t

h _ _ _ ö f hofG_ u a hEGF N O "O |} Xd HO"WLob O FIH©NKFIE÷O HKL FIb O(NíJ(O r = [ r · · · r ] Loc k|A |rk = kA Rk , _d gh F feu F f OšdYO­Xd _ { u FIEDO"b _ { F m OšJ(O­E a O"} _ b%O u HKLo} _ EGF h N O kAk = kA k S § µÔ´>—àÆÐ ¨™è ° bXT!JLKNJ#V ¬ OR ¸ A OR b OJ#TUFTL_8K Z V ¤ UKN™XO Z ˜8K ž O _ „MFIc _

k|A−1 | · |r|k∞ R = diag(r1 , . . . , rn )

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)

ce0u T `

«³ u € · ³ e

kx−x0 k∞ kxk∞

A

err =

A0

err ≤ 3.5 · 2.6 ·

0

0

b0

= 9.3 · 10−9 kx−x0 k∞ kxk∞

+

krk∞ /(kAk∞ · kxk∞ )

10−3

Š e0u ³ e·¸ce

a k o

Š u € · Š e

x = A\b

]z¶ )

x0

j

r = b − Ax

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1.6 · 10−17

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err ≤ k|A−1 | · |r|k∞ /kxk∞

10−8

err ≤ 

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½

A

A ∈ Rn×n

A = AT

det Y 6= 0

A

¢

A

¿

4

⇐⇒ Y T AY

H = A(1 : k, 1 : k)

k ≤ n =⇒ H

H = A([i1 i2 · · · ik ], [i1 i2 · · · ik ]) =⇒ H

A

⇐⇒ A = AT

A

=⇒ aii > 0

A

A

xT Ax > 0

=⇒

-ǻ

A

∀i

maxi,j |aij | = maxi |aii |

uii > 0

∀i

x 6= 0


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V

A=VV

T

A =VVT

ZÉ tÉ wÉ

V

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T

n

T

T

T

T

ii

T

T

T

T i

n i=1

T

2 i i

1

i,j

T

n

ij

T

11 22

pp

pq

pq

qq

2 pq

pp qq

ii

kk

T

11

T

T

nn

T

T

d FLoJ

2

i

pq

T

−1

k

T

T

det Y 6= 0 =⇒ Y x 6= 0

1/2

1/2 11

1/2

T

1/2 nn

jk

ajk =

k X

vji vki =

_TmQu _Tm dYOeO hofG_ HKL u FIbJ(OšHO"Jp„MFId  | _gh F a }GF f OQP i=1

k = 1, . . . , n  Pk−1 2 1/2 vkk = akk − i=1 vki j = k + 1, . .. , n  Pk−1 1 vjk = vkk ajk − i=1 vji vki

k−1 X i−1

vji vki + vjk vkk ,

T

T


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(2k + 2(n − k)k) =

]y

1 3 n + O(n2 ). 3

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§ µÔ´>—àÐƨ™û ­[MNOCM^TtbX_8KF]H Ù ®V Z K8˜8KQ› TLbXT 

2  −1  V =  2  −1 2

3 1 −2 −1

2 1 −1

 4 2

    

2

4  −2  A=  4  −2 4

¸

−2 10 1 −5 −5

4 1 9 −2 1

 4 −5   1   7  14

−2 −5 −2 22 7

W V—NOJ#V

£ FM{ FIEDO"cgNKF a S d4S m S a L aKu FIb%O Ax = b P i É A=VV  Z É V y = b t É V x = yS ‚ƒJ d _Tm_ WcFO"cYO h LoJpF G} _guBa b _ N _ cYÅ O"H F m L h LJ(O[x4jz aKh F m LF m OlLoJMHOgrIXcYO"cYOH FM{KL u FIE xe J(O m_ { rpO b Q}(NKFIH<NKF |δA| ≤ 3nu|V | · |V |. FIHkdYOlNKF T

T

(A+δA)e x=

T

(|V | · |V T |)ij =

X k

|vik ||vjk | ≤ (

X k

|vik |2 )1/2 (

X k

|vjk |2 )1/2 =

√ √ aii ajj ≤ max |aij |, i,j

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G(x(r+1) ) < G(x(r) )

S

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x(r+1) = x(r) + λr vr ,


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r

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gradG(x) = JF (x)T F (x).

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f1 (x) = 3x1 − cos(x1 x2 ) − 0.6

f2 (x) = x21 − 81(x2 + 0.1)2 + sin(x3 ) + 1.1 f3 (x) = e−x1 x2 + 20x3 + 9.1.

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x(0) = (0, 0, 0) = (0.65079172132, −0.21305107190, −0.511385522215)


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s =

xi r xk . r

ÊÆ O u HKu Lo} _ RaKu Lob'FIž c>XgNKaKFIu b _ O™!KNR_g˜šu V!™TM^VUFT _SO `šT S ^ HKLbe_ c _gŸ h FIcgNzX_ b%O u HKLo}GF'h J R a aKF u a dH FIb'FIm cL u O h h F i_  u O Loc ku  O%ETH L™„pOQS _ a X H h FIJMcLobeu L<H _TOgm€„ILm N O"befG_ LF}Th™L_ NzLo|ÇXu d _ HO"W N O"b E€d_ HOMô[ET£ L cFIbÏETH cFIbH F Xc O"|} E b%O HKLo}TL3XcL™rILob E F*F FIb'FIc Fqd LˆO cYO Loc O"} LoJMHOgrIXcYO"b HO"Jp„MFId4S ^ HKL3b%O u HKLo}TL 4 × 3 u O"} _em_ WLob _ ¬

ik

T ik

      × × × × × × × × × T ·  T ·  T ·     R R R ×  12  0 × ×  13  0 × ×  14  0 × −→  −→  −→  × × × × 0 × × 0 × × × × × × × × 0 ×      × × × × × × × × × T    T   RT × × 23 ·  0  R24 ·  0 × ×  R34 ·  0 × ×  e −→  0 0 ×  −→  0 0 ×  −→  0 0 ×  = R. 0 × × 0 0 × 0 0 0

× × A= × ×

× × × × 

 × ×  × ×

ÆÊO u HKLo}DO Re NKF J fG_ HKcgN O u HO"_ dsFIu JM_gcYfG_OlE¬h J fG_ HKcgNKu FIbú}TEDO m HO u XàdYO¡E aaKFIu _gWh XgNKFÇb%O a u HKLo} _ R S u ^ H _T_ m X} u Qe =_ R R R R R R NKF H a b _em_ WL h L4HO"Jp„MFId A = QR S cYO cYO b%O HKLo}DO<}TLE¿dHKETLo| n ds„ILo|ËE FIWXgNKF'b%O HKLo} Q S §`O"} ^ HKL3bec _gŸ FIcgNzX~J R a F j  u OšLoc k u O­ETH aKu L™„pO a dH FIb'FIcL u OšE h LocFpO"HKcL4} _ b‰WLocYOg„IL NzL j  u F*Loc k u F*ETH aKu L™„MFDS d F NKFEícFI}GFIb aKu _gh ds„IX~E _ WsFI|~ETH aKu L™„pO"| 0  a F u _ cFb _ H F*d _ }TEDO"HKL u L3Jqbec _gŸ FIcgNKFIb-J R S ¥>}TL™„pOeO hofG_ HKL u b%OQP 12

13

14

23

24

34

T jk

T jk

Q = Im j = 1, . . . , n k = j + 1, . . . , m r = (a2jj + a2jk )1/2 c = ajj /r s = ajk /r


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vD]

 c s A([j k], j : n) = A([j k], j : n) −s c   c s b([j k]) = b([j k]) −s c  c s Q([j k], 1 : m) = Q([j k], 1 : m) −s c Q = QT

S ¥ u FIETL h™_e_ dsFIHOg„IL NJ(O­H FM{ FIEDO"cgNKF n X

 

m X

È©rMF*H FM{KXgNKFIb _ dH F m_gh™_ rMFIcL a L aKu FIb Ax = bÉ È©rMFd _gu H FIWXgNKFIb _ QÉ

Ax = b

NKF

(6 + 6(n − j + 1) + 6) ≈ 6

n X

(m − j)(n − j) ≈ 3mn2 − n3 .

_ u _ u _ _ a _gu _ _ m_Tm u ud HFO"c d a U _ H HKFIbeWLoXgHNKO"FIb b _ EQJ fGLob%_ KH O"cgb N _ u { HKF Lo} _gu c _ O JícLoX| d 6m _ HO"W n_ −2n3mn_ dsFIHdsOg„IFILHNšOg<„IJ(L NMO SBQÆÊO dYO~HKLo} d _gunH FI×WXgnNKFIb O"} _ { d F mõ[_Tm LoEGO FIu c cLo|X 3n S j=1

k=j+1

2

j=1 2

3

3

Ý6qÝ ã

ž O­EGFI} u _ H

 : Y Cã­kAà­ <" ED ¡á6?¢`ko  w ∈ Rn

Q}(NKFIH<NKF

w 6= 0

 m FâácLoHO"b _

P =I−

2 wwT . wT w

KN F a Lob'_ F u HK_gL™u rIcYO€Loc _ H u _gfG_ cYO h cYO€b%O u HKLo}DO a O(_ N NKF P_ = P Loc P = I SšÀ a _ O"}ÇEGFI} u m _ H x ∈ R hh O"|} _ J(O"dL™{ _ FIb } x = αw + u >_ }(u NK_gFIH4fG_ NKF uh ⊥ w S † WLob u P _x = −αw + u >_|}DO"èµHV!d H˜Xb'K8®FIV cLFW KNM`XO KNNK™F VLPbSM^_AJMT H „p`XO KNRXNK`XFIKcgNKF dH FI} |LodsFIHKHOMETcLocF}TLQNKF H cYO cYO'cYO w SBÆÊO HKLo} P Lob'FIc>XgNKFIb Z Z S Æ c _gŸ FIcgNKF a P LoJMEGF m FIb _'u O"} _  m O­LoJMHOgrIXcYO"b _ P x = x − (x w)w }(NKFIH<NKF m = w w S d FqLob%O"b _%S u O"}DO'cFIcL™rMF h cYO%EGFI} u _ H©N O x Loc y  m ONKF kxk = kyk Yd _gu FIbEGF h N O P x = y YrMF‰LoJMWsFIH FIb _ w =y−x nO(NW _ x 6= 0 S ‚›h{ rMFIu b _ _JMH „pO h NKFIcgNKF}TLsE x XcL™rIL3E a F[} _ beh d _ cFIu c u F*HO"JpFIm c€dHKEGF u _ H F›u N Pž x = ±ke T}(_ NKFIH   m _ NKF k = kxk S<ÀBF N O L4b HO w = x ∓ ke QETdHOg{O"cgNKF NKF F Q}DO FIHKL4dH F JMcYO"}•LoJMWHO L@S O m WLob T

P

2

1 m

2

n

1 2

T

T

2

1

2

1

m=

1 T 1 w w = (k 2 ∓ 2kx1 + k 2 ) = k(k ∓ x1 ). 2 2

ž O"HO m L m F h NKFIcgN O­J m Ÿ F h Lob _  m ­O W _

m

rILobãEGFMr›NzLFJ(O u _ LoJMWsFIH FIb _ 

 x1 + sign(x1 )kxk2   x2 . w=  

¥ETcYu FIO"ETdL H h™_šF›NM_S dsž FIOšHOgLo„IJMLHN OgrIJ(XO‰c LoJMHOgrILocXc~dHdY_TOšm dX_g} uu H O FIWXgNKFIb _ w

m

m = k(k + |x1 |)

SB‚ƒJMWLoHO[NKF u _ H F›N

SS

xn

_ _ dsNKF FIH4nOg„IL +NMS O(1) QdHKL4rMFIb'FIH m LoJMHOgrIXcYO"b

1 Pz = z − m (z T w)w 2n + O(1)


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vGR

ž bec _gŸ FIcgNKFIbüb%O u HKLo}GFkJkX aKu H FIJMcLobeLJMH „pO h NKFIcgNzLDN _*h O"|} _‰a dH FIb'FIcLob _ EšJ fG_ HKcgN _*u HO"dsFIJMc _‰_ W h Lo} _ S ^ HKL b%O u HKLo}TL 4 × 3 u O"} _em_ WLob _ 

× × A= × ×

^ HKL u IF bÄNKF

QT = Pe3 Pe2 Pe1

 u _ H F›N

¥>}TL™„pOeO hofG_ HKL u b%OQP

  × ×   e ×  P1 ·  0 −→  × 0 × 0

× × × ×

Q = Pe1 Pe2 Pe3

× × × ×

    × × × × ×     e e ×  P2 ·  0 × ×  P3 ·  0 −→  −→  × 0 0 × 0 × 0 0 × 0

i Pei = m−i



i

Q}TL3dH FIJMH „pO h L

S ¥ u FIETL h™_e_ dsFIHOg„IL NJ(O­H FM{ FIEDO"cgNKF n X i=1

m−i  0 . Pi

SBnO"b'F aKu _ HOgrIXcYO"cgN O Q HO(NKF a |HO"cLob _ EGFI} u _ H©NKF w S

Q = Im i = 1, . . . , n wi ∈ Rm−i+1 A(i : m, i) A(i : m, i : n) = Pi · A(i : m, i : n) b(i : m) = Pi · b(i : m) Q(i : m, 1 : n) = Pi · Q(i : m, 1 : n) Q = QT

m_gh™_ rIL

i Ii 0

 × × × × . 0 × 0 0

E ±ke S È©rMF*H FM{KXgNKFIb _ dH F m_g_guh™_ rMFIcL a L aK_ u FIb Ax = bÉ È©rMF*d H FIWXgNKFIb QÉ Ax = b

1

NKF

[2(m − i + 1) + 4(n − i + 1)(m − i + 1) + 4(m − i + 1)] ≈ 4 2 ≈ 2mn2 − n3 . 3

n X i=1

(m − i)(n − i) ≈

ž O Q d _gu H FIWXgNKFIb _ { F m_Tm O u cLo| 4m n − 2mn _ dsFIHOg„IL NMSÆÊO u HKLo} _ n × n a ö _ X a FI| _gh™m FIH©NKFIETLobeL JMmH _T„pm O h NKu FIcgNzL u HO"c a U _ HKbeLoHO"b _ EÊJ fG_ HKcgN _ u HKLo} _gu c _ J%Xd _ HO"W _ n _ dsFIHOg„IL Nš`J(O Q dYO€d _gu H FIWXgNKFIb _ { F O cLo| 2n S ^ HKLob'FIH©N OMEDO _ W aKu _ NKFMrILo|~b'F u _Tm P m_gh™_ rMFIcF f O a L aKu FIb%O Ax = b  m  n P £ • FM{ FIEDO"cgNKF[dH F c _ HKb%O h cL a L aKu FIb~P mn  Æ¿õq¥P 2mn  õ[LoEGFIc a P 3mn − n  ö _ X a FI| _gh™m FIHpP 2mn − n S m u f a aKu £ • FM{ FIEDO"cgNKF[}TEDO HO cF O L FIb%O Ax = b  m = n P x4j/HO"Jp„MFId4P n  ö _ X a FI| _gh™m FIHpP n  õ[LoEGFIc a P 2n S 2

4 3 3

3

2

Ý

2

Ý

2

Ý

3

2

Ý

2 3 3 4 3 3

Ý

Ý

Ý

2

3

2 3 3


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vDv

½

A ∈ Rm×n m ≥ n

A = U ΣV T ,

U ∈ Rm×m

`XKNMI˜šV

V ∈ Rn×n

Σ ∈ Rm×n 

σ1

  Σ=  

V— Z O†K

  σn  , 

˜NOR›!H Z T!MNR?Kµ™XMAK^W!RV˜SUnO ¸

σ1 ≥ σ 2 ≥ · · · ≥ σ n ≥ 0

A

¥ u _gh ds„IL U = [u · · · u ] a _ Z KN™XO  aKu _gh ds„IL V = [v · · · v ] dYO WK8˜NR+OE˜NOR›!H Z T!MNR+OC™!K8UFV!M`NO S »eV!Ttf bt¸ u Å FIHNKF A A a Lob'F u HKL™rIcYO•d _ JML u LoETc _~a FIbeL m FâácL u cYO•b%O u HKLo}DO a _ E a F­cgNKFIcF h O aKu cFšETH F m c _DaKu L cFIcF O LoETcFDP σ ≥ σ ≥ · · · ≥ σ ≥ 0. j aKu H FIJMcL _ H u _ c _ HKbeLoHO"cL h O aKu cL3EGFI} u _ H©NzL4cYO(N W _Tm_ A Av = σ v  i = 1, . . . , n S nO(N W _ σ > 0 Loc σ = · · · = σ = 0 S …lJMcYOgrILob _ V := [v · · · v ] Loc V := [v · · · v ] SB‚ƒJ 1

m

1

n

T

2 1

2 2

2 n

T

r

r+1

aKh F m L AV = 0 S ¥TF m O(N m FâácLoHO"b _

n

1

1

r

2

r+1

n

(AV2 )T (AV2 ) = V2T AT AV2 = V2T [0 · · · 0] = 0

2

ui =

1 σi Avi

uTi uj =



i = 1, . . . , r

1

1

r

T

U T AV =

†F a cYO m EDO­W h™_ }DO aKu O­J(O"HO m L AV Loc

S ÀBFI} u _ H©NzL u , . . . , u a _e_ H u _ c _ HKbeLoHO"cLF a O(N`NKF 1

r

σj 1 T T vi A Avj = viT vj = δij , σi σj σi

l… JMcYu OgrILob _ U u := [u · · · u ] Loce_ LoJMWsh FIH _ FIb _ b%O HKLo}DOQSBÆÊO HKLo}DO U AV Lob%O W Lo}

u _ H F›N

2 i i

i

2

=0

U2 := [ur+1 · · · un ]

r m−r



FIcYO"}DO 0 S ž O

r U1T AV1 U2T AV1

i, j = 1, . . . , r.

u O"} _  m O NKF

U = [U1 U2 ]

n−r  U1T AV2 . U2T AV2

i = 1, . . . , r

Loc

k = 1, . . . , m

_ H u _gfG_ cYO h cYO

EGF h N O

uk Avi = σi uk ui = σi δik ,

Loc

U2T AV1 = 0 U1T AV1 = diag(σ1 , . . . , σr ) S = diag(σ1 , . . . , σr ) Σ=

r m−r



S§`O"} _%a b _%m_ WL h LB¥>ÀHO"Jp„MFId

r S 0

A = U ΣV T

n−r  0 . 0

À/dHKLob'FIHKX n < m m_ WLob _ ¥>ÀÞHO"Jp„MFId u O"} _  m O u HO"c a d _ cLoHO"b _ ¥>ÀÞHO"Jp„MFId~J(O ^`_ b'FIc¿¥>ÀÞHO"Jp„MFIdYOQP

AT

S

}(NKFIH NKF


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q– a FXgNKFIb%OšJ  aKu _gh ds„IL u E _ HKL N _ WYO"J _ J(O  aKu _gh ds„IL u E _ HKL N _ WYO"J _ J(O  aKu _gh ds„IL u E _ HKL N _ WYO"J _ J(O  • r

vD

rang(A)

U1

imA

V2

ker A

U2

ker AT

1

aKu _gh ds„IL V u E _ HKL N _ WYO"J _ J(O imA S ¥ FcFI}DO(N h O aKu c _DaKu L a Loc f X h O"HKcF f OeHO"Jp„MFIdYOQP i Sžd F<NKF A = A Gd _gu FIb a F m O A m LˆO fG_ cYO h LoJMLoHO u Ls} _gu A = U DU  U U = I SB¥>Loc f X h O"HKcLHO"Jp„MFId J(O A NKFd _gu FIb A = U ΣV J(O σ = |λ | Loc v = sign(λ )u È u XšNKF sign(0) = 1É S ZQSkx6O aKu u _cF ETH F m c _DaKu L A A a _ σ , . . . , σ S …lH u _ c _ HKbeLoHO"cL h O aKu cLEGFI} u _ H©NzL A A a _m F a cL a Loc f X h O"HKcL EGFI} H©NzL v , . . . , v S t Skx6O aKu cF%ETH F m c _DaKu L AA a _ σ , . . . , σ , 0, . . . , 0 SʅlH u _ c _ HKbeLoHO"cL h O aKu cL EGFI} u _ H©NzL AA a _~h FIETL | {z } a Loc f X h O"HKcL4EGFI} u _ H©NzL u , . . . , u S ^`_gh f¬_ f _ _Ëu X m L‰¥>À HO"Jp„MFId _ W h Lo}GF A = Ue ΣV kfG_}(NKFIH‰h NKF Ue u b%O u HKLo}DO ma × n J  e _

u g _ _ H u _ F c _ HKbeb'LoFIHcgO"NKcFILocbeF L OËaKu _gd h dsJM„IcYFL  O"Σb e = diag(σ a dHKETLobeL n aKu _gh ds„IL3b%O u HKLo}GF U  Σe dYOlNKF,J .fG._ . HK,cgσNzL4)}T EDVO m dYHOO u NKFΣnE~×¥>ÀÞn HO"H Jp„MFIdcYX O AcYO%=b%UO ΣVHKLo}DOQS S Ue FqXgNKFIb%O maKu OME h N OšdH F aKh Lo}DOME _ LoJ R E R SõF _ b'F u HKL N a }TL3d _ b'FIcÇ¥>À/HO"Jp„MFIdYO[NKF  m O a F[J _ H u _gfG_ cYO h cLob%O A dH F u HO"c a U _ HKb%Og„IL N O"b%OšWYO"J U E R Loc V E R A a dH FIb'FIcL4E m LˆO fG_ cYO h c _ b%O u HKLo} _  a O(N d _gu FIb-EGF h N O T

T

T

T

T

1

i

2 1

i

i

i

T

i

2 n

T

n

T

2 1

2 n

T

m−n

1

m

T

1

n

r

T

m

m

n

Avi = σui ,

ÑB´TÖ¸¡Ø?¨o×

© KC`XK

¬

¬

A ∈ Rm×n m ≥ n rang(A) = n x=

»eV!Ttbt¸

i = 1, . . . , n.

n X uT b i

σi

i=1

nO(N W _

A = U ΣV T

Loc n U1

U=

¬?]VUKNJæ`XKJ=OR+OJ=HJ

m−n  U2 ,

kAx − bk2

W V˜XK £ KNRL]MNO

vi .

n Σ= m−n



 S . 0

  SV T x − U1T b . kAx − bk2 = kU ΣV x − bk2 = kΣV x − U bk2 = UT b T

Æ LocLob‰XbÄNKF m_Da F Ÿ FIc~dHKL SV

Tx

= U1T

T

T

_ _ b JMLoH b%O

x = V S −1 U1T b =

2

n X uT b i

i=1

σi

vi .

2


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A ∈ Rm×n m ≥ n rang(A) = n

A+ ∈ Rn×m

m<n

A+ = (AT A)−1 AT .

¬µWKê R+OM^T!J#V ¸

D\

È©Æ _T_ H FN ^ FIcH _Da F _ E É

A+ = AT (AAT )−1

rang(A) = m

£ FM{KL u IF E%dH F m_gh™_ rMFIcF f O a L aKu FIb%Ošd _gh cF f OšHO"c f O Ax = b h O"|} _ J(O"dL™{ FIb _ } _gu x = A b S d F A cL4d _gh cLoF c f OeHO"c f ODNKF*d a }(FINKE FImH`_ NKLoF c>EGFIHKJ m FâácLoHO"cÇdH FI} _ ¥>ÀHO"Jp„MFIdYOQS nO(NW _ A ∈ R rang(A) = r A = U ΣV +

T

r U1

U=

m−r  U2 ,

Loc S = diag(σ , . . . , σ ) S `^ _gu IF b NKF (} NKFIH`NKF 1

ÑB´TÖ¸¡Ø?¨™ç

8ñ ³  ³ Ö!³ ¢

d F NKF

r

r V1

V =

n−r  V2 ,

Σ=

r m−r

r S 0



m≥n

n−r  0 0

A+ = V Σ + U T ,

Σ+ =

r n−r



r

m−r  0 . 0

S −1 0

ë.TUnMNO†!T X` K·]\˜XKN™W V!OR+™!KNMAb RTUFT!R!VLUK^W T‘`8¬ !VLO b]V Z RX`NHN`XKIë.VXV!MAKNÕQ­7KNR+M^V˜XK^V!™!Kž]VA› V‘`XK ¯ ¬ ¬ ¬ ¸ ¥>ÀHO"Jp„MFId LoceNKF Q d _gu FIb m LoH FI} u c _ LoJ*HO"Jp„MFIdYO aKh F m L X

A

AXA = A

XAX = X

(AX)T = AX

(XA)T = XA A = U ΣV T

A

rang(A) = r A=

SG¹"´>ò¬Ø?¨Ô˜ GµT‘`z—^V VtbSOM^V!J#T



m×n

r X

σi ui viT .

i=1

A = U ΣV T

Ak = U Σ k V

T

¬C`XKNM `XK

)

° M^TtbX_8KF] OR A

Ak =

rang(A) > k k X

¸·GµT‘`z—^V

σi ui viT

i=1

      Σk =      

σ1

¸¸¸

σk 0

¸¸¸

      .   0  




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rang(B)=k

»eV!Ttbt¸ d FkNKF

i

kB − Ak2 = kAk − Ak2 = σk+1 .

Yd _gu FIb_ W aKNKuF O(N O

rang(B) = k dim imVk+1 + dim ker B = n + 1

SknO(NkW _

S Å FIHNKF _ _

S † WLob

Loc

dim ker B = n − k Vk+1 = [v1 · · · vk+1 ] 0 6= z ∈ imVk+1 ∩ ker B kzk2 = 1

^`_'m HKX f L aKu HO"cLNKF _ rIL u c _  m O[NKF kA − Ak = σ  a O(N6NKFb%O"} a Lob‰Xb m_Da F Ÿ FIc~dHKL v S § _ _ed _ m b'h FIcLF m O*NK_TF m A _DcYaKO(u Nz_ W _gh NK{O%u O"dH _ } a Lob%f Og„IL N O'b%O u HKLo}GF A J­b%O u HKLo} _ HO"c f O k  σ dYOecYO"bd _ EGF }DO"} O FMr NKF A dH HOšb%O HKLo}•HO"c O k S § µÔ´>—½?Ø ¨™Ð ° M^TtbX_8KF] Z T® !VLHX]V!M^T—NOJ#V#bXT#!V!Ji]MAK8˜NO `šV#˜ Z O†¸ Z O†!V Z T® !V]MAK^W˜SUFT!™XOJ#V#beJ#TUnMNO†!V A¬ 2 2 kA − Bk22 ≥ k(A − B)zk22 = kAzk22 = kU ΣV T zk22 = kΣV T zk22 ≥ σk+1 kV T zk = σk+1 . 2

k+1

k+1

k+1

k

k+1

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)

A

k [σ1 v1 · · · σk vk ]

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1

mn

?>o

<

(m + n)k

[u1 · · · uk ]

  

ž O­b%O u HKLo} _ A Q }TLQNKFHO"c f O r  m FâácLoHO"b _ κ2 (A) = kAk2 kA+ k2 =

SG¹"´>ò¬Ø?¨™è ¸·µG µG T‘`LT‘`z—^V —^V

r = Ax − b

­¿VUKNJÌ`XK

¬

σ1 (A) . σr (A)

¬ ¬ `XKNM `XK ¬

MAK ¤ OnUKN™µ]MAK^W V Z VXP8KNR?KQ› T.˜NO†˜SUKNJ#T¹OR

A ∈ Rm×n m ≥ n rang(A) = n x = A+ b x e = (A + δA)+ (b + δb)   kδAk2 kδbk2 1  = max , < . kAk2 kbk2 κ2 (A)

(A + δA)

M^T!R› T OR™!K Z `šT k

ke x − xk2 κ2 (A) ≤ kxk2 1 − κ2 (A)



krk2 2 + (κ2 (A) + 1) kAk2 kxk2



.

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2

2 2

2 2

2

2

2

2 2


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A

kAx − bk2

å

x

x

z ∈ ker A

kAx − bk2

A ∈ Rm×n m ≥ n rang(A) = r < n A = U ΣV T x kAx − bk2

OR

x=

r X uT b i

i=1

»eV!Ttbt¸

kAx −

ž Ošd _gh NzXWsFIc

x ∈ Rn

bk22

=

σi

m X

x = A+ b

vi

(uTi b)2 .

i=r+1

EGF h N O

kAx − bk22 = k(U T AV )(V T x) − U T bk22 = kΣa − U T bk22 =

r X

(σi ai − uTi b)2 +

m X

(uTi b)2 ,

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T

i=r+1

uT i b σi

i

r+1

r+1

m



å

n

m×n

ñ ¸ © KC`XK MAK ¤ OnUKN™i]MAK^W V Z VXP8KNR?KQ› T¹˜NO†˜SUKNJ#T x

î¸ © K ˜F]MAKNJLKNR+OJ#VL™ b

b + δb

Ax = b

]VLJLKSUFVXW!O RT‘`NJ#T!RX` ¤ O†®#t™T W!M^TUFV!™7`XK

kxk2 ≥

|uTn b| . σn

¬ ˜XK ˜F]MAKNJLKNR+OC™ x

n

x + δx

kδxk2 ≤

»eV!Ttbt¸

i Sk‚ƒJ

T

x = A+ b =

Pn

uT i b i=1 σi vi

¬ `XKNM `XK

kδbk2 . σn

aKh F m L kxk22

=

n  T 2 X u b i

i=1

σi

(uTn b)2 . σn2

m


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δx = A+ δb =

Pn

i=1

uT i δb δi v i

t

S¥ h F m L kδxk2 ≤

|uTn δb| . σn

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r

x=

r X uT b i

σi

vi ,

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2

kδbk2 σr

kbk2 σr

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• ai1 • ai2

• ai3 , . . . , ai9 • aij

j = 10, . . . , n

• bi

æ

bi

ai1 , . . . , ain

bj =

n X

ajk xk .

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Ax = b A ∈ Rm×n m < n dim ker A = n − m x

 _ u aKu _ ž m u f _ JMO"LoHH O _ b%L O F O~L™{ rMFIb L

x=

m X uT b i

i=1

σi

vi .

rang(A) = m kxk2

z ∈ ker A x = A+ b


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d _gu FIbNKxFL a }DO"cYO­H FM{KL u FIE•}DO"kAx H − bk

A

kxk2

2

x = A+ b =

r X uT b i

σi

rang(A) = r

vi .

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n

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yH

yH x = 0

H

¸

y H A = µy H

dYO­J m F a cF*J x S † _ WLob _

u _ dYO[NKF h O"|} _ LoJMd _gh cgNKFIc _eh FEídHKLob'FIHKXc} _ NKF y x = 0 S ÆÊO u u HKLo}DO A a F m O m LˆO fG_ cYO h LoJMLoHO u LFmTrMF _ W aKu O(N O u O[cF a Loc f X h O"u HKcYO[b%O u HKLo}DO X = [x · · · x ] Loc m LˆO fG_ cYO h cYO b%O HKLo}DO Λ = diag(λ , . . . , λ )  OlNKF A = XΛX S<À FIbãdHKLob'FIHKXeNKF Ax = λ x J(O i = 1, . . . , n S u u _ xh6O aKaKu u cF[ETH F mm c _D_DaKaKuu L a _ cL™r h F[}DO"HO"} u FIHKL aKu L™rIcF f O­d _gu h Loc _ b%O h p(λ)u = det(A − λI) S ÆÊO HKLo}DO A Lob%O O"} n _ u Å a _ h _gh _ h O"O |} c_ Lo|'JpF ETh™H _qF _ Wsc rIX uKLh NzλLoEGF, .cY.O. ,b λ_gu GcgdNKFHKL} rM_ FIFâb'áYFI„IHL™FIEGc FMu r _ }TE HO ÈÔccdFlHpS cL™r L Flh }T{ Loc F›a NK_FIbc _ EšEGdFMHKr Lo}Tb'HFIO H É S  u _ FIHcdYLOcYO(NzdcHKLoL™b'r FIFkHKd cF›NKLo{c Od b%_gO u J(O­HOgrIXcYO"cgNKF h O aKu cLo|~ETH F m c _DaKu L@S x6_gO haKu c_ FlETH a F m _ c _DaKaKu u L a _ cL™_r h Fl}DO"u HO"} _ u FIHKL aKu u L™rIcF f O‰d _gh u Loc _ aKb%u O det(A_gh −_ λI)  _ h O"|} _ dYO u X m L _ _gWh HO _u c _ J(Å OqE a O"a } d Loc h4a b h™_ } c f HKX_gLoHh O"b _ b%aKO u _ HKLo} 3}Dy O FIh H Fš}DO"HO"} m FIHKL L™rIcL d u Loc m bìf W }DO"_gHu LoJMWHO"cL d _ Loc u _ b~S h FIu H m F cL™rMhF aKu d {KcF m O­_Dd aKu Loc b%m O u dcgNKF u _TO m L3EGFMrh cF aKu O­LoJMHOgm rIX_DcYaKO u L u _ HKX OgrMF} c>Xb'FIHKL™rIc  EGF N O X L J(O O cF*ETH F c L3Loc LoH FI} cF*b'F F*J(O O cF*ETH F c L H F›N cL@S ÆÊO u u HKLo}DO A a F m O W!OQTX› V!RT Z O bSOM^TUnO mDrMF _ W aKu O(N O u O[cF a Loc f X h O"u HKcYO[b%O u HKLo}DO X = [x · · · x ] Loc m LˆO fG_ cYO h cYO b%O HKLo}DO Λ = diag(λ , . . . , λ )  OlNKF A = XΛX S<À FIbãdHKLob'FIHKXeNKF Ax = λ x J(O i = 1, . . . , n S aKu u ]VXW V—NR+O Loc•m Lob%a O u Ošh FIaKcYu O"}GF h O aKu u_ cF[ETH F m c _DaKu L@S ÀBF h N O  m OkNKF x md F Fa cNKF L det h O aKu ScL46=EGFI0}  u _ HO‰J(b%O AO HKcYLo}TO Lu O"Ac} Loc_'Su F m O(AS _ NšQ} NKF S x F cL O cL3EGFI} HkJ(O B S y y H Ax = λy H x = µy H x, H

1

1

n

1

1

−1

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i i

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B=

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1

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1

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 |

1 0

1

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     0 1 0 λ ×···× B = V∗ 0 V1 0 T1 {z 1 } | {z }

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n×n

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λi + δλi

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λi + δλi = λi +

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si :=

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λ()

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s−1 i

λi

si = 1

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A + E

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|z − λi | ≤ kXk · kX −1 k · kEk = κ(X)kEk,

i = 1, . . . , n.

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−1 X −1 EX

a Loc f X h O"HKcYOqb%O u HKLo}DOQS

1 ≤ k(Λ − λ()I)−1 X −1 EXk ≤ kΛ − λ()I)−1 kkX −1 kkEkkXk.


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1 mini=1,...,n |λi − λ()|

min |λi − λ()| ≤ κ(X)kEk.

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E = ET

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i

S

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yi

si = 0

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0

6= 0

k = 0, 1, . . . zk+1 = Axk xk+1 =

zk+1 kzk+1 k∞

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k → ∞ zk

λ1


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A = XΛX −1

· · · xn ]

1

^`_gu FIb-dHKL4d _gfG_ NzX

Λ = diag(λ1 , . . . , λn )

α1 6= 0 xk =

z0 =

EGF h N O

n X

z0

αi xi .

i=1

α1 w1 + α2 ( λλ21 )k w2 + · · · αn ( λλn1 )k wn Ak x0 = kAk x0 k∞ kα1 w1 + α2 ( λλ21 )k w2 + · · · αn ( λλn1 )k wn k∞

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k

1

1

1

1

2

3

1

1

2

3

1

2

2

1

k

1

1



k+1

k

2

ρ(A, x) =

xT Ax , xT x

m FâácLoHO"c~J(O x 6= 0 S ž £ ¨ h FIL f | _ E•}TE _ „IL™FIc u EGF h N O m O[NKF ρ(A, x) = ρ(A, αx) J(O α 6= 0 S…[rIL u c _ NKF u X m LF m OšLoJ Ax = λx aKh OF m L OMρ(A, x) = λ S ^ HOMETL h FIcÊJ(O"X aKu OMETL u EGFIcL }THKL u FIHKL N[J(O%d _gu FIcrIc _ b'F u _Tm_ NKF m O'cYO(NzdH F›N[J(O z LoJMHOgrIXcYO"b _ £ OM¨ h FIL f | _ E }TETL™„IL™FIc u ρ = ρ(A,m_z ) d _gu FIb dYO'd _ghfDh aKF u m O"b _ c _ HKb _í_DaKu O"c}DO kAz − ρ z k Sµd FkNKF‰c _ HKb%O m_ E _gh N h Ÿ b%O(Nz|cYO gNKF ρ , z WsFIHdHKLoW L FI}•J(O O cL4dYO"HpS k

k

k

k

k

k

k k


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i

T

B = A − λ1 x1 xT1 .

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k

k k

T

B=



bT C

λ1 0

Q

 m O­NKF

Qx1 = e1

,

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1

−1

k+1

−1

1

1

1 1

−1

k

−1

k+1

1 1

−1 n

k

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σ |λi − σ|  |λj − σ|

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zk+1 =

0

σ

u O"}

j 6= i

1 e0 ke z 0 k∞ z

= zk z e k+1 ke zk+1 k∞ 1

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0 6= 0 k = 0, 1, . . . σk = ρ(A, xk ) (A − σk I)zk+1 = xk zk+1 xk+1 = kzk+1 k∞

H FM{KL

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§ µÔ´>—àçƨˆ² GµT‘`"—^V ¬ OR ¬ ¸æ­[MNO=KNR?KNJ !V!M^TtH ä T Z KNO‚›®V!™!KeOnUKNM^T _SO `XKzW V—NOJ#V ¦^bI˜NO†˜SUKNJ#T W V—NOJ#V VXWLUFVXW]T#˜ Z K^W!O ¶ WLUFVXWL™i]MNOJLKNMNH ˜ Z K^W!OCtH+—NOQPSRT#!V!R+™!KNMQ›KNR_ATz]M^VUnO VtbSOM^V!J#T ¸ óµ6 Y  ß k­k ­¡z>  <s7¢«o>  nO"b'F aKu _‰u m_ _ beLocYO"c u cF f O‰EGFI} u _ H©N O> WLYHO m LYLoJMHOgrIXcYO h L m_ beLocYO"c u cLsLoc>EDO"HKLˆO"c u cLsd _Tm dH _DaKu _ H m Lob'FIcJML NKF J(Ošb%O HKLo} S A =



λ1 0

0 λ2



λ1 > λ 2

zr =



cr sr



c2r + s2r = 1

σr = zrT Azr = c2r λ1 + σr2 λ2 .



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p

1 = (λ1 − λ2 )c2r s2r 1

zr+1 = p c6r + s6r

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 c3r , −s3r

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k = 0, 1, . . . : Yk+1 = AZk

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Λ = diag(λ1 , . . . , λn ) |λp | > |λp+1 |

ß-


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Lin(Zk ) = Lin(Ak Z0 )

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1

k

0

k

k

T k

k

r

i

T k

k

k

k

k1

k2

k1

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T AZ Zk1 k1 = T AZ Zk2 k1

 T AZ Zk1 k2 . T AZ Zk2 k2

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k

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Z0 = I

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ZkT AZk = ZkT ( Zk+1 Sk+1 ) = ZkT Zk+1 Sk+1 = Qk Rk , | {z } | {z } | {z } | {z } |

ort. zg. trik.

{z

}

ort.

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zg. trik.

S¥ h F m L

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3

n×n

T

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× ×  Q1 A =  0 0 0

× × × × ×

×  (×)  A=  (×)  (×) (×)

× × × × ×

× × × × ×

 × ×  × , × ×

× × × × ×

× × × × ×

 × ×  × , × ×

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× ×  Q2 A1 =  0 0 0

× × × 0 0

Q2

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× ×  Q3 A2 =  0 0 0

Q3

× × × 0 0

× × × × 0

× × × × ×

 × ×  × , × ×

× × × × ×

× × × × ×

× × × × ×

× ×  × . × × 

× ×  A1 = Q1 AQT1 =  0 0 0 

× ×  A2 = Q2 A1 QT2 =  0 0 0 

× ×  H = Q3 A2 QT3 =  0 0 0

× × (×) (×) (×)

× × × × ×

× × × 0 0

× × × (×) (×)

× × × 0 0

× × × × 0

× × × × × × × × × × × × × × ×

 × ×  × . × ×  × ×  × , × ×  × ×  ×  × ×


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H = Q3 Q2 Q1 A(Q3 Q2 Q1 )T

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Q = I 6û i = 1, . . . , n − 2 wi ∈ Rn−i A(i + 1 : n − i) A(i + 1 : n, i : n) = Pi A(i + 1 : n, i : n) A(1 : n, i + 1 : n) = A(1 : n, i + 1 : n)Pi Q(i + 1 : n, i : n) = Pi Q(i + 1 : n, i : n) 6û

Q}TL3dH FIJMH „pO h L

E

±ke1

È É

È É P rMF*d _gu H FIWXgNKFIb _eu X m L Q ¥ u FIETL h™_e_ dsFIHOg„IL N<NKF n + O(n ) _ JMLoH _ b%O n + O(n ) rMF*d _gu H FIWXgNKFIb _eu X m L Q S ¹"—<¶ˆÕp´G”¬Æç ¨©‘º ² © KC`XK A bA› V!MNRX`šT=èeK8˜8˜XKNR?—AKNMQ› V!™T¬[˜XKzV— Z O†!TLJLK^W ä OnUKNM^T _SO `šV|V®M^T!RX`šT¸ »eV!Ttu bt_ ¸ ^ HKL m ô[£_ HO"m Jp„MFIaKdu X Am m_ WLob _ h J fG_ HKcgN _ ö _ F a a FIc>WsFIH fG_ aKE u _ _gh b%O u HKLo} _ Q LocíJ fG_ ö HKcguN _‰_[u a HKLo} m _gu c _ R S ^ u HKL Q NKF m OíNKF€dHH O"_TJMm ETXL } uc J fGLoJ _ HKcgF›NKN F uEDHKOLo} _guO cNKF F qLoc¬LoJ cfGFp_ O"HKHKcgcYNKOlF } ö Fb‰a aWFILoc>cYWsOg„IFILH N fGO _ EGF b%ds„MO FIu E HKLoa}GF , .a ds. .F ,u aJ SfG_ HKLcgN H O ö F F a aOkFIdc>H WsFIFIEGH FIfGHK_ L EDLFO  b%O u HKLo}DOQS d FqcYOeJ(cYOgO"rMb'F u F }TaKXu _ A H F m X„I_ LoHdsO"FIb HOg_ „IL cYNMSO ö F a a FIc>WsFIH fG_ E _•_ W h Lo} _ Yd _ HO"WLob _ J(O%FIc } _ HO"} ô[£ L u FIHOg„IL NKF h F‰{ F O(n ) O(n ) €´ ·`¶ >¶ ïM¸¿Æç ¨©‘QÐ èeK8˜8˜XKNR?—AKNMQ› V!™T=J#TUnMNO†!T H `XK¿OMAK^W!H_SOF—NO Z RT¬CP8K¿˜šVµ™š˜NOÆRX`XKNR+O˜NH+—^W!OQTX› V!RT Z R+O«K Z KNJLKNR\UnO 6û

10 3 3

14 3 3

2

2

å

i

2

0/

hi+1,i

d F NKF

1

i

3

R?KNR+OQP8K Z R+OF¸

H F m X„ILoWL h cYODNKFcdHpS

H

 × × × × × × × × × ×    H=  0 0 × × ×  0 0 × × × 0 0 0 × ×

LoaKc€u dH _ W _ h FIb m h O aKu cLo|~ETmH F m c _DaKh u L4HO"JMdYO m F*cYO m EDO h™_ rMFIcYOšdH _ W h FIb%OQS ž O"HO m L u F f O h O"|} _ EGF m c _ dH F m d _  OMETLob  OlNKF H LoH F X„ILoWL cYOQS 8Eõ EõÔ1 ; Á Ïà=@: = Å _ c>EGFIH f FIc„ _%h O"|} _ d D_ a sd FM{KLob e_ a d H FIbeLo}TL@P

¦

A0 = A k = 0, 1, . . . :

LoJMWsFIHKL4dH FIbeLo}

ÈÔLoJMHOgrIXcYO"b _ [ô £ H O"Jp„MFId É

σk Ak − σ k I = Q k Rk Ak+1 = Rk Qk + σk I


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i\y

üà

Ak+1

Ak+1 = Rk Qk + σk I = QTk Qk Rk Qk + σk QTk Qk = = QTk (Qk Rk + σk I)Qk = QTk Ak Qk .

Å "O } _ LoJMWsFIH FIb _ dH FIbeLo} dLob-W h L Ÿ NKF h O aKu cL3ETH F m c _DaKu LNKF u FIb-W _gh NKFDS ÑB´TÖ¸çƨ©‘Ø © K`XK σ Z T˜SUnRT͙XMAK^W!RV˜SUeOMAK^W!H_SOF—NO Z R?K|èeK8˜8˜XKNR?—AKNMQ› V!™!K.J#TUnMNO†K 

»eV!Ttbt¸

B = RQ + σI

¬?]VUKNJæ`XK

bn,n−1 = 0

OR

bnn = σ

¸

A

OR

A − σI = QR

¬

Å FIH NKF A LoH F m X„ILoWL h cYOQNKF­dHKETLo| n − 1 aKu _gh ds„MFIE A − σI h LocFpO"HKc _ cF _Tm ETL a cLo|4S*ÀüHO"Jp„MFIdX u _ EGF h N O r 6= 0 J(O i = 1, . . . , n − 1 S Å FIH‰dYOšNKF A − σI a Loc f X h O"HKcYO`b _ HO€WL u L A − σI = QR J(O _ _ m m aKu u u _ u r = 0 S<§ dYO*d b'FIcLF O NKFJ(O cgN O*ETH L™„pOEšb%O HKLo}TL RQ FIcYO"}DO 0  H F›N Ešb%O HKLo}TL B = RQ + σI hEGF N O b = 0 Loc b = σ S Å _ cYO(N m FIb _ FIc _šh O aKu c _ ETH F m c _DaKu cYO m O h NzXgNKFIb _ HOgrIXcYO"cgNKF*Jqb%O u HKLo} _ B(1 : n − 1, 1 : n − 1) S ^`_gu H FIWXgNKFIb _ rILob-W _gh NK{KLsdHKLoW h L Ÿ FI}€J(O h O aKu c _ ETH F m c _DaKu A J(O u _ Lob%O"b _ cYO­E _gh N _ HO"J h L™rIcF*dH FIbeLo}GFDP O É KNRh V‘Ÿ `NR+Oh]MAaKKNu J=O† P ž m O σ_DaKu LoJMWsFIH FIm b _ (A ) S À mu _ FIbdHKLob'FI_ HKX~Lob%h O"b a _ }Th EDaKO u m HO u L™rIc m _ }_D_aKu c>EGFIH f FIc„ _ E W L LocL O cF*ETH F c FL EGFIc O"HkdH FIbeLo}•cL WsFIHkJ(O­} bed FI} cF O cF*ETH F c L@S W É W!™V‘`NR+Os_ JGS ð?M^T!R_SO†˜šV!™i]MAKNJ=O† PBÀ J(O"b'FIb _ d _Tm b%O u HKLo} _ ii

nn

n,n−1

nn

k

k nn

k

Ak (n − 1 : n, n − 1 : n) =

T} LLob%O h O aK_ u cLsETH F m c _DaKu L σ E•FIcFIb-} HO"}TX4P

(k) (k) 1 , σ2

(k)

"

(k)

an−1,n−1 (k)

an,n−1

(k)

an−1,n (k)

ann

#

,

È h O"|} _šaKu O u X m Ls} _ bed h FI} a cL É S¥TF m O(N cYO"H F m Lob _šm EDOqdH FIbeLo}DO

ÈÔLoJMHOgrIXcYO"b _ [ô £ H O"Jp„MFId É ÈÔLoJMHOgrIXcYO"b _ [ô £ H O"Jp„MFId É

Ak − σ 1 I = Q k Rk (k) A0k = Rk Qk + σ1 I (k) A0k − σ2 I = Q0k Rk0 (k) Ak+1 = Rk0 Q0k + σ2 I

Å _ bed h FI} a cYO'O"HKL u b'F u Lo}DOecL3d _gu H FIWcYO a O(N EGF h N OQP Loc

Qk Q0k Rk0 Rk = = = = (Qk Q0k )(Rk0 Rk )

(k)

Qk (A0k − σ2 I)Rk = (k) (k) Qk Q∗k (Ak − σ2 I)Qk Rk = (Ak − σ2 I)Qk Rk = (k) (k) (Ak − σ2 I)(Ak − σ1 I) = (k) (k) (k) (k) A2k − (σ1 + σ2 )Ak + σ1 σ2 I =: N

NKF [ô £ H O"Jp„MFId~H FpO h cF*b%O u HKLo}GF N S Å FIHkEGF h N O u X m L@P

Ak+1 = = = =

(k)

(k)

(k)

0 0 Rk0 Q0k + σ2 I = Q0∗ k (Ak − σ2 I)Qk + σ2 I = (k) (k) (k) 0 Q0∗ k (Rk Qk + (σ1 − σ2 I)Qk + σ2 I = (k) (k) (k) (k) ∗ 0 Q0∗ k (Qk (Ak − σ1 I)Qk + (σ1 − σ2 I)Qk + σ2 I = 0∗ 0 Q k Qk Ak Qk Qk ,

d _gu H FIWXgNKFIb e_ h F H FpO h cL [ô £ H O"Jp„MFId H FpO h cF*b%O u HKLo}GF N S


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÷

ÞßZ

üýÓÎ

[q1 · · · qn ]

QT AQ = H

Q =

q2 , . . . , q n

q1

Ak = Qk Rk Ak+1 = Rk Qk = QTk Ak Qk

Qk

þà

Ak+1

Ak

Ak − σ k I

Qk Qk

Ak − σ k I QTk Ak Qk T Qk Ak Qk

ÊÆ O Ÿ u HKLo}m_ _gh™Q_ d _ L™{ a rMFIb _ } _guaKu d_gHh _Tm X} u õ[LoEGFIc a _ ETLo|í_DH aK_gu u Ogh „IL N Q m=_gh™_ R R¿_ u · · _ · Rm _ S ^ HKEDO*H _gu Og„IL N fGO _ R NKö F a aF fGrMFI_ cYO du HKETLob ds„MFIb A − σ I  O F€dYO rILob O"}  OÇW Q A Q J HKcgN O F FIc>WsFIH EDOšb%O HKLo}DOQS[d F NKFcYO"beH FMr k

12

k

k

R12

d _gu FIbNKF

23

n−1,n

k

s1 c1

c1  s1  =  

1

1

Qk = R12 R23 · · · Rn−1,n

k

×

SSS

··· ···

 × ×   .  

SS SS

Ý hofG_ HKL u FIbæ cYO(Nz}TmH_ O(NK{ F _ _ JMcYOgrILob _ } _gu ]MAKNJ=O†!T!RX`XK¿›!MX—AK S `^ _gfDh ›F Nzb š_ a L f O*cYOqdHKLob'FIHKXíb%O u HKLo}GF _ dHKEGFIb-} HO"}TX WLob   T R12 Ak R12

k

  ,  

× ··· × ···

c1  s1   =  

12

SSS

T k

×

×

× × × × × × × × × ×    = + × × × ×.  × × × × ×

5×5

S ^`_

n _ ET_ LYcFI_ cL™rMF aKh u cL3F _ h FIb'FIc u _ + NKF f KH WYOT}TL>N _ J[cYm O aKh F m cgNzLobeLH _gu Og„IL N O"beLsd _ beLo}DO"b _ cYOMETJ m_ghs_ W m LˆO fG_ cYO h L@S §`O"} d ETH L3d L™{ rMFIb R  R oL c R  O[NKF 23

34

45

T T R23 R12 Ak R12 R23

× ×  =  

× × × +

× × × ×

× × × × ×

 × ×  × , × ×


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Loc

i \GR

 × × × × × × × × × ×   = × × × ×    × × × + × ×

T T T T R45 R34 R23 R12 Ak R12 R23 R34 R45

 × × × × × × × × × ×   = × × × ×  .  × × × × ×

`^ _Tm_ Wc _€h O"|} _ cYO"H F m Lob _€u X m L J(O m E _ NzcL dH FIbeLo}sS Å FIH*EGFIb _ A = U A U 3}(NKFIH NKF U LoJ ô[£ HO"Jp„MFIdYOšu b%O u HKLo}GF _ Nh =_ A − (σ + σ )A + σ σ Ih "NKF mh _ E _gh Nu d _ JMcYO hu L h F*_ dHKETL aKh u _gh dsFM_¿„*m b%O u HKu Lo}GF LoH FI} cF N S¡x4F O Lob%O W Lo} [× × × 0 · · · 0] BJ(O cFIcL™rMF cF~F FIb'FIc F€dYO O"|} LoJMdsF NKFIb _U HKb‰X h FDS ¥TF m O(N cYO(NzdH F›N d _ L™{ rMFIb _ ö _ X a FI| _gh™m FIH©NKFIE _ JMH „pO h NKFIcgNKF _ W h Lo}GF 2 k

(k) 1

(k) 2

k+1

(k) (k) 1 2

k T

× × × × × ×  × × × P1 =    

1

k

k

k

k

SSS 1

   ,   

}TLLob%O dHKET_ L Ka u _gh dsmFM_g„•h6a FIcYö O"_ }ca _ HK_gbeh™m LoHO"cFIb‰X¬dHKEGFIb‰h X aKu _gh ds„IXËb%O u HKLo}GF N Sʆ _ WLo_ b Ç_ f KH W _ dH FIbeLo}DO"b cYOMETJ X FI| FIH©NKFIETLobeL6JMH „pO NKFIcgNzL@S À/dHKLob'FIHKX 6 × 6 Lob%O"b 

   P2 =    

Loc u O"} _ cYO"dH F›NMS

1

× ×  + P1 Ak P1 =  +   × × × × × × × × ×

1 1

   ,   

× × × +

× × × ×

× × × × ×

× × × × × × 

 × ×  × , ×  × ×

× ×   P 2 P1 Ak P1 P2 =    

× × × + +

× × × × +

× × × × ×

× × × × × ×

2×2

 × ×  × , ×  × ×

<}TL6N _


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ñ

3

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λ1 ≥ λ 2 ≥ · · · ≥ λ n .

A = AT

x1 , . . . , x n

kAk2 = max(|λ1 |, |λn |).

£ OM¨ h IF L f | _ EGF f Oe}TE _ „IL™FIc u Lob%OšdHKL a Lob'F u HKL™rIcL6b%O u HKLo}TL6{ F u _šh O aKu c _DaKu  m O­J(OšE a O"} § _ ETL m Lob _  rMFHO"JMETL NKFIb _

x=

x 6= 0

λ1 ≥ ρ(x, A) ≥ λn .

Pn

i=1 αi xi

S `^ _gu IF b m _ W Lob _

Pn α2i λi ρ(x, A) = Pi=1 n 2 . i=1 αi

¥ h F m L4cFI}DO(N d _ b'FIb‰WcLo| LoJMH FI} _ E•J(O a Lob'F u HKL™rIcL4dH _ W h FIb h O aKu cLo|~ETH F m c a _gu L@S SG¹"´>ò˜¨©‘ ‰–6¹"¸·Õ ¶ÔÓX gñ6´>¹KïM´G”Ö¡¶©·`¶©Ö¸ò3Óe¶G¹"´>ò "Ì

Hô3õ

λi =

min

S⊂Rn dim(S)=n−i+1

max ρ(x, A) = maxn min ρ(x, A). x∈S x6=0

x∈R R⊂R dim(R)=i x6=0

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A α b1 ≥ · · · α bn



EGF h N O

A, E A+E

i = 1, . . . , n

|αi − α bi | ≤ kEk2 .

i \Dv

α1 ≥ · · · ≥ α n

È@RTS i É

Z T˜SUnR?Kž™XMAK^W!RV˜SUnO


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i \D

Loc€d _Tm_ Wc _ ρ(x, A + E) ≥ ρ(x, A) − kEk . ¥TF m O(N d _ beLocLob%O"} a LoJMH FI}TX aKh F m L |bα − α | ≤ kEk S SG¹"´>ò˜ ¨™Ð ‰+¸  gñTïM´G”¶G¹"´>ò–î`¹"´>î<µÔ´GÕp¸·QïX © K`XK A ˜NOJLKSUnMNOQPSRTIJ#TUnMNO†!TIOR¿`XK ρ(x, A + E) = ρ(x, A) + ρ(x, E) ≤ ρ(x, A) + kEk 2 2

i

i

2

]VXW!J#TUnMNO†!TLJ#TUnMNO†K ¬?]VUKNJº™!K Z `šT Ì

A

Ar

› Z T!™XRT

r×r

λr+1 (Ar+1 ) ≤ λr (Ar ) ≤ λr (Ar+1 ) ≤ · · · ≤ λ2 (Ar+1 ) ≤ λ1 (Ar ) ≤ λ1 (Ar+1 ).

SJ#GT¹"Un´>MNòO†!T˜¨o× ™š© ˜šK¿T‘`=`XKKNR T˜N˜SOUnJLR+OKSUn]MNT!OQPSM RT¹J#TUnMN¬CO†!tTOE¬ bXT W V PAT OR ]MNOF— Z O £ K8abXT Z T˜S¸ UnRV™XMAK^W!RV˜SU¬ ]VUKNJ OJ#T Z ¤ »heV!Tt_ bt¸ m na O(N W _ h _Tm a SB†h _ aK}Du O"J NKFd _Tm m__DWsaKu FIc m_ }DO"_JMX€J(O ¼ m O"XFIH‘ `Lo}GF _ E•LoJMH a FI}sf S ^ h H F m d _DaKu OMETu Lob _ O"|a } m  mO F fG_ HO"h J Lo}TXgu NKF _gu E FI| O cLo|šETH u F _ c LFg}DO"H`d b'FIcLF OBNKF cF Loc X O"HKcYOQS ÆÊO HKLo}DO m SB¥TF O(NNKF  u _ H F›F N O LˆO cYO LoJMLoHO L}  _TmQu _Tm dYO  m_ H WF›LoN b _ A

A

(λi , xi )

6*

r = Ax − βx

β

A

kxk2 = 1 β |β − λi | ≤ kAx − βxk2

A−βI A−βI = X(D−βI)X T

A = XDX T −1 1 ≤ k(D − βI) k2 krk2

x = (A−βI)−1 r

min |λi − β| ≤ krk2 .

I6qŸ

i

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i = 1, . . . , n

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k=1

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x − xi 1

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Ii,i+1,...,i+k−1 (x)

. Ii+1,i+2,...,i+k (x)

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I0123 (x) I123 (x)

I23 (x)

x3 x − x 3 y3

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k

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f

f [x0 , x1 , . . . , xk ]

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»eV!Ttbt¸

f [x0 , . . . , xk ] =

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f [x1 , . . . , xk ] − f [x0 , . . . , xk−1 ] . xk − x 0

f [x0 ]

XgNKFIb%O­J

f (x0 )

S nO(N W _

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Pi+1 = Pi (x) + c(x − x0 ) · · · (x − xi ). f [x0 , . . . , xi+1 ]

S

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x1 , . . . , x k

x0 , . . . , xk−1 p x − xk x − x0 p(x) = p0 (x) + p1 (x). x0 − x k xk − x 0

p1

d FdHKLob'FIH©N O"b _ E _Tm L h cF*} _ FâáY„IL™FIc u F m_ WLob _ JMEGFIJ _ ÈVQS w É S …lW h Lo} _ ÈVQS t É Lob'FIc>XgNKFIb _|GeK  UFV!RV!™=OR\UKNMF]V Z T _SO `8˜8tO?]V Z ORV!J S 32

n[FI}DO(N dH FIdH _DaKu Lo|~U _ HKb‰X h P • f [x0 ] = f (x0 )



f (x1 )−f (x0 ) x1 −x0

S

Å O(N a FJ fG_Tm LFQ} _ E h LobeL u L4d _ { h NKFIb _ • f [x0 , x1 ] =

x1 → x 0

‚ƒJ 

p(x) = f [x0 ] + f [x0 , x1 ](x − x0 ) = f (x0 ) +

f (x1 ) − f (x0 ) (x − x0 ) x1 − x 0

m_ WLob _ p(x) = f (x ) + f (x )(x − x ) S § _ _ d _ b'FIca LF m O_gh h O"f |} _'m m Fâá_DcaKu L™„IL N _ Loc u FIHK_šd u _gh Ogm „IL N a }G_TF m f Oš_Tdm _gh Loc _ b%Oša Loc u _ m F h NKFIcLo| m L‡UVFI_ H FIc_g„u HO"Jp{Ku LoHK_ Lob _ _ cYO­d _gm h L c b'F _Q}TL Fd F u ETH m F c _Tm L4_TXgm NKFIb%O(N _ X L4E E u _ Lo|4S[d F F r }GF*XgNKFIb%O(N Qd FIb _ _gd h b'_ FIcLF O cYO(u NW f XgNKFIb%h O"cgNKF X L6E E Lo|4S §`O"} cdHpS dHKL r }DO"| x , x , x , x , x , x L™{ rMFIb d Loc b p J(O }DO FIH F OíEGF N O p(x ) = f (x )  p(x ) = f (x )  p (x ) = f (x )  p (x ) = f (x )  p(x ) = f (x )  p (x ) = f (x ) S d F m_ dX{ rpO"b _'u X m L3XgNKFIb%O"cgNKF u _ r ?} d _gu FIbãJ(O m F h NKFIcF m L‡UVFIH FIc„MF*EGF h N O­H FI}TXHKJMLoETcYOšJMEGFIJ(O 0

0

0

3

0

1

0

0

1

2

0

2

2

1 0

2

2

2 2 00 2

3

3 00

2

3

3

3

   

f k (x0 ) k! f [x0 , x1 , . . . , xk ] =  f [x , . . . , x ] − f [x0 , . . . , xk−1 ]  1 k  xk − x 0

x0 = x 1 = · · · = x k

.

sicer

†_ F h NKFIcF m L‡UVFIH FIc_ „MF HOgfGrI_ XcY_ O"b _ m E u _DHKaKLou} _gu ca L a |FIbeu LQ_ LoJ }DO u FIH F cYOl} _ c„IXe|L u H _ dH FIWsFIH FIb _ FIcYOgrIW _ d _gh Loc _ b%O JGS<LoJMHOgrIXcYO"b cgNKF E ETH F c EíL }DO"cL r }TL@S § µÔ´>—àÆû ¨ˆ² Ê«T8]O ¤ O·KNRT Pt—^Vz]V Z ORV!J#T#˜SUFV8]RX`XK ¬ bXTL!TUKNMAKQ› T¹™!K Z `šT ¯ p(0) = 1¬ p (0) = 2¬ p (0) = 3¬

¬

¬

p(1) = −1 p0 (1) = 3 p(2) = 4

¸

¿

0

xi f [.] f [., .] f [., ., .] f [., ., ., .] f [., ., ., ., .] f [., ., ., ., ., .] 0 1 2 3 0 1 2 2 − 11 2 29 0 1 −4 2 −2 9 − 79 8 21 1 −1 5 −4 3 − 32 1 −1 2 5 2 4

00


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q–

i ZGR

3 11 29 79 p(x) = 1 + 2x + x2 − x3 + x3 (x − 1) − x3 (x − 1)2 . 2 2 2 8

SG¹"´>òûƨˆØ

f [x0 , . . . , xk ] =

»eV!Ttbt¸

Ê«T ÕFtM^TUCbS™!KAbSRV|VXW!™!K^W Z `NO™VžœNHR!_SO `šV ™!K Z `šT

€´>¹gÖ¡¶ˆÕp´ • ´T·`·6–Æ gñ`¶ ïM´G”¸Îƒ–Y¹gÖ.`µÔ¸

ŒÌ

Z

1

dt1

0

Z

Z

t1

dt2 0

···

jld _ HO"WLob _ Loc m X}G„IL N _ S·d F NKF

tk−1

f 0

x0 6= x1

(k)



f



tk (xk − xk−1 ) + · · · + t1 (x1 − x0 ) + x0 dtk .

 m_ WLob _

t1 =1

1 0 f (t1 (x1 − x0 ) + x0 )dt1 = f (t1 (x1 − x0 ) + x0 )

= x1 − x 0 0 t1 =0 f (x1 ) − f (x0 ) , x1 − x 0

Z

f [x0 , x1 ] = =

a L™„MFIHdYO

Z

k

1

0

f [x0 , x0 ] =

Z

1

f 0 (x0 )dt1 = f 0 (x0 ).

¥TF m O(N*dH F m d _DaKu OMETLoh b _  _ m O•LoJMH FI}Ê_EGF h N O•u fJ(O 1,h ._g.u . , k − 1 S¹d F m FâácLoHO"b _ Loc F HO } t (x − x ) + x  O"|} (J O"dL™{ FIb 1

1

0

0

0

f [x0 , x1 , . . . , xk ] =

Z

1

dt1

¥TF m O(N ka a X W aKu L u X„IL N _'a dH FIb'FIc h NzLoET} m _ W Lob _ P 0

Z

t1

dt2

0

Z

···

Z

tk−1

ξ = tk (xk − xk−1 ) + · · · +

f (k) (ξ)dtk .

0

dξ = (xk − x0 )dtk ,

Loc

tk = 0 =⇒ ξ0 = tk−1 (xk−1 − xk−2 ) + tk−2 (xk−2 − xk−3 ) + · · · + t1 (x1 − x0 ) + x0 tk = tk−1 =⇒ ξ1 = tk−1 (xk − xk−2 ) + tk−2 (xk−2 − xk−3 ) + · · · + t1 (x1 − x0 ) + x0 Z

tk−1

f

(k)

0

1 (ξ)dtk = xk − xk−1

^`_ Loc m X }G„IL N a }TL3dH F m d _DaKu OMET}TL4EGF h N O Loc }GFIHdYOšEGFIb _

Z

1

dt1

0

Z

1

dt1 0

Z

Z

t1

dt2 0

t1

dt2 0

Z

Z

···

f [x0 , x1 , . . . , xk ] =

NKF m _ D} O"Jq} _ crpO"c4S

Z

··· Z

tk−2

Z

ξ1

ξ0

f (k) (ξ)dξ =

f (k−1) (ξ1 ) − f (k−1) (ξ0 ) . xk − xk−1

f (k−1) (ξ1 )dtk−1 = f [x0 , x1 , . . . , xk−2 , xk ]

0

tk−2

f (k−1) (ξ0 )dtk−1 = f [x0 , x1 , . . . , xk−2 , xk−1 ],

0

f [x0 , x1 , . . . , xk−2 , xk ] − f [x0 , x1 , . . . , xk−2 , xk−1 ] , xk − xk−1


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i ZDv

f

`XKNMC`XK

1 (k) f (ξ), k!

f [x0 , . . . , xk ] =

min (xi ) < ξ < max (xi ).

i=0,...,k

SG¹"´>òûƨԘ ™!Ê«K TZ `šT

(n + 1)

x0 , . . . , x n

»eV!Ttbt¸

i=0,...,k

ÕFtM^TU7bS™!KAbSRVVXW!™!K^W Z `NO™VeœNHR!_SO `šV ORÂOR\UKNMF]V Z T _SO `8˜8tO]V Z ORV!J RTUFVXPN!T® f

In

f (x) = In (x) + f [x0 , . . . , xn , x](x − x0 ) · · · (x − xn ).

nO(N W _ q d _gh Loc _ b}TL3Loc u FIHKd _gh LoHO f E u _ r }DO"|

x0 , . . . , x n , t

Å FIH u _ GE F h N O­J(OšE a O"} t TU _ HKb‰X h O ÈVQS y É m H Ÿ L@S [–YÓ(µÔ´>—<¶ G¸ûƨ™è Ê«T (n + 1)ÕFtM^TUzbS™!KAbSRV(VXW!™!K^W Z `NO™V¹œNHR!_SO `šV

ÈVQS y É

S<ÀBF h N O u L4b _ HO

q(x) = In (x) + f [x0 , . . . , xn , t](x − x0 ) · · · (x − xn ).

FU VXPN!T® `XKNMC`XK

x0 , . . . , x n

™!K Z `šT

f (x) − In (x) =

f

OROR\UKNMF]V Z T _SO `8˜8tO7]V Z ORV!J RT In

f (n+1) (ξ) ω(x), (n + 1)!

min(x, x0 , . . . , xn ) < ξ < max(x, x0 , . . . , xn )

¸

^ H F m c _DaKu c _ GE F _ „MFIcF NKF m OšdHKL m F*E%d _ { u FIE u X m LFrMF*E a F u _ r }GF*cL a _ b'F ma FIW _ Nzc _ HO"J h L™rIcFDS

µ6FÚ ²

Ï

kz@`­ ªàz> M <s z¢` zä

a _ a u _ K8t™XOQW!O†˜SUFT!R\UnR?K _ mea _Da m

u _ hd O"F|} _ E U _ HKF b‰X r h Fq}GF { Fd _ FIc _DaKu OMETLob D_}DO"SBH<nd O(N b'W FI_TcmL _ xy ==xf (x+ih) DET}(H NKF FIm HYc NK_DF aKhu L@S HO"JMbeLo}­b'F F cgNzLob%O r }DO"b%O €´ ·`¶ >¶ ïM¸¿ûƨ™û ØzV!RPSR?KzW!O œXKNMAKNR_8Kµ˜šVaWKê R+OM^T!R?KµMAK8tHMAbSO™XRVL!VU i

0

i

i

0/

m

∆ yk =



∆m−1 y

‚ƒJETH F m c _DaKu L y a F aKu OMETLob _W!O œXKNMAKNRPSRVaUFT—AK Z V P

yk , m−1 y , k+1 − ∆ k

i

xi yi x0 y0

∆ ∆y0

x1 y1 ∆y1 x2 y2 ∆y2 x3 y3

∆2

∆2 y0 ∆2 y1

∆3

∆3 y0

m=0 . m>0


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q–

i ZD

ž O­dHKEGF u HKL m L‡UVFIH FIc„MF m_ WLob _ P ∆y0 = y1 − y0 , ∆2 y0 = ∆y1 − ∆y0 = (y2 − y1 ) − (y1 − y0 ) = y2 − 2y1 + y0 , ∆3 y0 = ∆2 y1 − ∆2 y0 = (y3 − 2y2 + y1 ) − (y2 − 2y1 + y0 ) = y3 − 3y2 + 3y1 − y0 .

…lWHO u c _ EGF h N OQP

y1 = y0 + ∆y0 = (I + ∆)y0 y2 = y1 + ∆y1 = (I + ∆)y1 = (I + ∆)2 y0

° K Z `šTUFTiœšV!MNJ=H Z O ¯

ÑB´TÖ¸ûƨ©‘Qó

TN³

m

∆ y0 =

—³

m X k=0

  m (−1) ym−k , k k

m

ym = (I + ∆) y0 =

m   X m k=0

ÑB´TÖ¸ûƨ©‘Y‘ª©

O R »eV!Ttbt¸

∆m p(x)

k

∆k y0 .

KbX`XT K ] V Z OR¸ V!J ˜SUFV8]RX`XK žb ™VXW!O Z R+OJ !VtKê[_SOFKNR\UFV!J ¬]VUKNJª™!K Z `šT

=0

p m>n

n

a0

∆n p(x) = n!hn a0

^ KH L`E a O"}TL m L‡UVFIH FIc„IL a F aKu _ dcgN O'JMb%O"cgNK{O'J(O'FIcYOE _Tm L h cL<} _ FâáY„IL™FIc u dYO'd _ bec _gŸ L [a aKu _ d cgN _ n Loc a h S ‚ m F›N OlNKF m O­J(O x = x + sh J(O"dL™{ FIb _

∆p(x) = (a0 (x + h)n + a1 (x + h)n−1 + · · · + an ) − (a0 xn + a1 xn−1 + · · · + an ) = nha0 xn−1 + · · · .

0

s

In (x0 + sh) = (I + ∆) y0 =

∞   X s

∆k y0 .

Å IF HLoc u FIHKd _gh LoHO"b í_ a d _gh Loc _ b _ b _Tm dYO m F›N _ E a 4L r h FIcL3E a _gu F*}(NKFIH`NKF k > n S ÑB´TÖ¸ûƨ©‘²   k=0

In (x0 + sh) = (I + ∆)s y0 =

k

∞ X s ∆k y0 k

`XK´OR\UKNMF]V ¸Z T _SO `8˜8tOE]V Z ORV!J±˜SUFV8]RX`XK´t™!K^Pn`XKNJ=H bXT"UFVXPNK ¬ »eV!Ttbt¸ …[rIL u c _ NKF aKu _ dcgNKF}TEGFMr›NKFIb‰X Loc _ rIL u c _ EGF h N O J(O _ HKb‰X h™_ ÈVQS ] É Lob'FIc>XgNKFIb _=]MAKNJ#T´GeK  UFV!RV!™TLOR\UKNMF]V Z T _SO `8˜8!TiœšV!MNJ=H Z T S

ÈVQS ] É

k=0

n

xi = x0 + ih i = 0, . . . , n

¬žORì™XMAK^W!RV˜SUnO

y0 , . . . , y n

*

In

n

32

In (xi ) = yi

i = 0, . . . , n

S


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q– §  µÔ´>—àûƙ¨ Ð K8˜SUFT!™XOfW!O œXKNMAKNRPSRVaUFT—AK Z V#bXT]VXW TUK=W VLUnMAKSU `XK=W!O œXKNMAKNR_8KeOR"O bSM^T PSHRT‘` )

I3 (0.02)

ipt \

¸

x 0.0 0.1 0.2 0.3 0.4 y 0.00 0.11 0.28 0.57 1.04 xi yi 0.0 0.00

∆2

∆3

0.11 0.1 0.11

0.06 0.17

0.06

0.2 0.28

0.12 0.29

0.06

0.3 0.57

0.18 047

0.4 1.04 h = 0.1,

I3 (0.02) = 0.00 + 0.2 · 0.11 +

s=

x − x0 = 0.2 h

0.2 · (−0.8) · (−1.8) 0.2 · (−0.8) 0.06 + 0.06 = 0.02008. 2 6

§—NOJ#µÔ´>VL—àK8tûƙX¨oOQ× W!O†˜Sä UFT!HR\RUn›R?KKµ`XKNUF™žVXPN]M^K8V¬?UnO]]VMNUOKNJLJªKNMXbI¸7™!»zK^PAKNT!R+RXO`XJ#KNJV¬[˜SW UFTLV8]OR\RX`XUKIKNMFO]R\V UZ KNOM^MFT!]J#V T V _SO `8˜8tO\]V ORV!JªR™!T K^W!RV´˜ T—¸ ©KeKµT8]HXM^]VV!XM^˜NT!OÕÕ Z Z Z ¤ J=OM^T ¬E!VU !T £ KµRT˜ Z K^W!RX`šT#˜ Z O†!T² Z KN™TN³ ¸ f (x) =

1 1+x2

[−5, 5]

f

2

1.5

f p6 p12

1

f p6 p12 p18

1

0 −1

0.5

−2 0 −3 −4 −5

0

−0.5 −5

5

0

5

tOËR\O¿K U£ ˜šKNT!V"MF]™Tz™V ˜XT UK KN_SJO ]`8˜8V‘t`š]OT!MN™X]OJLOcV bXKNOT!RMNM^H×V!T J W!WO K™!ê K8K^tR+W!™XOROQM^W!VT!O†R?˜S—^K#UFV T!`XR\!KzVUnR+U T8O†]®aM^UFVVXXPN˜NOJ=¸ O© M^TiK7œNbXHTzRO!R\¬ _SUO KN`šMFV ]V Z T ²q_SWO `8K8˜8˜NRKµT#¬ UF˜ VX]PNO†V!UTNKIKN³ Jº™S¸ bXT!]JLTKNbaJ#™!V#K^PAUFVXT!PNRX`XKKNJí© KX˜S—NUFO V8¤ ]KNRX™`XTK ¬ Z Z Z Z ­¿bXTLV!R]T!V ™`NT H+W!—NOER˜XV.KebXUFT8VX]PNV!MAK K^W‘© `XKXK—NOO¤R\KNU™KNTLMF]˜NV OQ_8T KN_SMµO `8MA˜8K8t˜zO†®×V—NUFVXRPNT Í¤ T‘V`šV|—8˜SUF—^T‘V `šZ `XT´KµœN!HVRU¿!_SK8tO `š™XTOQ¬¿W!O†]˜SMNUFT!OžR\!UnTR?UKeKNMNUFOžVXPN˜XKLK8¬¿bA› T´VXW!˜XO†Kz¬´W W TIT]b¹V!™!TtK^bXPATT!UnRXO†`X¬·KNW J T ˜]SUFV V8]ORRXV!`XZ J#K´V!OR\JLUKN¸ MF]V Z T _SO `8˜8KQ› TL]V Z ORZ V!J#T´]MNO†®T‘`šT"W V™š˜XK#™!K^Pn`NO†®ÍM^Ttb Z O†JLK^WeœNHR!_SO `šVORìOR\UKNMF]V Z T _SO `8˜8tOJ Z xi = 5 cos

iπ n



i = 0, . . . , n f


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q– µ6qÝ l  <í¾ k 7¢«o>  ½kk Y <  í¾ kA > ­ < > < > : ‰7¢«>o < > ²

iptQi

ä

÷

naKu O"_ b'F aKu _ FI_cF f O¡_ d _g_Dh Loa c _ _ b%O¡ET_gL h a _ }G_ F a Ka u __ dcgNKF Loc u _eFIHK_ d _gh Og_ „IL N a } b _ KF]+U©XK8c }G„IL N _ J h FIdLob _ LoJÇd _gh Loc _ b _ EcLoJM}TLo| dsFIcgNMS † WLob } b%Ošd Loc b } ©U Xc}G„IL N JMLoH b%O Z S

p1

p2

p

p3

0

p

4

x2

x

x0

1

x

x4

3

x

5

n OlE a O"}GFIb Loc_ u FImHKEDO h X _¿[xa ,aKxu h ] Lob%O"b _ d _gh h Loc _ b p (x) DfDJ(h Olm }DO u FIH F f O[m EGF h N _ O p (x _gh) =_ y a Loc _ p h (x _ h ) =_  h _ g h F f OME NKFIcYOÊh }THKLoETX N u OÊ_ rILob W N O }DOÊLoc OÊW rILob a aKW u NKFh dL EDO O h W Loa } y S F Lob  OÊW u T _ m u _ Å J(OgrMF cLo|•a d O h } EsS FIH H F[}THKLoETX N O‰rMFIJ r }GF (x , y ) (J O i = 1, . . . , n pNKF F OME NKFIcYO‰}THKLoETX N OqE O(N JMEGFIJMcYO  O(N EGF N O i+1

i

±

i+1

i

i

i

i+1

i

i

i+1

i

… _Ta mc _ _ ETcYOÇEDO"HKLˆO"c u _ OíNKF_%_!V˜ša V!J#h T tH+Z O—NR?OQK^PST!R?MNKeRTÍb KFO]\R\UKKNMF]V Z T _SO `šT S Å EDO m HO u L™rIc_gLlh J h _ FId}TLcL a _ cYO(NzW _gh N­ž Xd _ HO"WcLF d H WcF›NKFqdYOeW b dL O L Z }(NKFIH NKF }TXWL™rMFIcÇd Loc bcYO S Oe}TXWL™rIcL Loc u FIHKd _gh Og„IL N a }TL4J h FIdsFI}íd _ cYOMEDO m L4J(O"| u FIEDO"b _ { F m O[NKFpE a O(N JMEGFIJMc _'_Tm EGF mQh NzLoEsS [x , x ] ^`_Da O"b'FIJMcL6}TXWL™rIcL4d _gh Loc _ b p NKFcYO u O"c} _'m_gh™_ rMFIc~J*ETH F m c _DaKu beL6Loc _Tm E _Tm L3Eí}THO(NzL™{ rILo|4P pi (xi+1 ) = pi+1 (xi+1 ).

i

i

i+1

i

d F _ JMcYOgrILob _



pi (xi ) = yi ,

pi (xi+1 ) = yi+1

p0i (xi ) = di ,

p0i (xi+1 ) = di+1 .

yi+1 −yi hi

hi = xi+1 − xi δi = xi

yi di

xi

Loc€Xd _ HO"WLob %_ m F h NKFIcF m L‡UVFIH FIc„MF m_ WLob _

yi

δi − d i hi

δi xi+1 yi+1 di+1

Loc

di+1 − δi hi

di + di+1 − 2δi h2i

xi+1 yi+1

pi (x) = yi + di (x − xi ) +

δi − d i di + di+1 − 2δi (x − xi )2 + (x − xi )2 (x − xi+1 ) hi h2i


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q–

ipt Z

Å XWL™rIcL J h FIdsFI}•NKF u O"} _Çm_gh™_ rMFIc a dYO"HO"b'F u HKL d , . . . , d `}TL4NzLo|¡b _ HO"b _Êm_gh™_ rILob _ u O"} _  m O€W _ Lob'F h J h FIdsFI} Ÿ F h NKFIcF h O aKu c _DaKu L@S ^ èeKNMNJ=OnUFV!™!KNJ tH+—NOQPSR?KNJb Z KF]\tH LoJMWsFIH FIb _ }DO"H d = y  i = 0, . . . , n + 1 S • HKL ^ W!™TtM^TUfbS™!KAbSRV|VXW!™!K^W Z `NO™!KNJ tH+—NOQPSR?KNJ b Z KF]\tH dYO"HO"b'F u H F d , . . . , d m_gh™_ rILob _'u O"} _  m O • HKL NKFJ h FIdsFI} m EDO"}THO u JMEGFIJMc _%_Tm EGF mQh NzLoEsS<ÀBF h N O 0

n+1

0 i

i

0

p00i (xi+1 ) =

Loc ‚ƒJqFIcYOgrIW m_ WLob _'a L aKu FIb

1 (2di + 4di+1 − 6δi ) hi

1

p00i+1 (xi+1 ) =

hi+1

(−4di+1 − 2di+2 + 6δi+1 ).

p00i (xi+1 ) = p00i+1 (xi+1 ),

h n LocFpO"HKcLo|ÇFIcYOgrIW~J(O

n+1

u_ n + 2 dYO"HO"b'F H EsP

i = 0, . . . , n − 1

hi+1 di + 2(hi + hi+1 )di+1 + hi di+2 = 3hi+1 δi + 3hi δi .

ÆÊO(Nz}DO(N _ rIL m EGFqFIcYOgrIWL m _ W Lob _ P !V!Ji] Z KSUnR+O«b Z KF]+K8 PrMF*d _ JMcYO"b _ ETJ(O"b'FIb _ d = y Loc d = y  RT!M^T!™XR+O«b Z KF]+K8 P a d _gfG_ NKFIb%O p (x ) = 0 Loc p (x ) = 0  b Z KF]+K8—NMA_'KAb#a ™Vtb_ Z V!h™™_ PlJ(O"| u FIEDO"b _  m OqNKF­J h FIdsFI}~cYO [x , x ] Loc [x , x ] }TXWL™rIcL<d _gh Loc _ b ÈÔJMcFIWLob FE J E x Loc x É S ^ tH+—NOQPSR?KNJ b Z KF]\tH¬´tO=V®M^T!RX`šT×VXW Z O†!V cYO"} h™_ cF d m_gh™_ rILob _Ëu O"} _  m O¿J(O 1/d ETJ(O"b'FIb _ • HKL d _ ETdH FMr›NKF*H FM„ILodH _ rIcLo|~ETH F m c _DaKu L a b'FIHKcLo|~} _ FâáY„IL™FIc u _ E€dH FIbeL™„ a } _ JML (x , x ) Loc (x , x ) S Ý

00 0

Ý

0 0

0 00 n

0

n+1

Ý

0

1

0 n+1

n+1

2

n−1

n

n

i

i

i−1

d F aKaKuu O a b'FIHKcYO­u } _ FâáY„Ih™_ L™FIc u O



i

i



1 1 1 1 = + . di 2 δi δi+1

i+1

cYO a dH _gu c _ dH F m JMcYOgrMFIcYOETJ(O"b'FIb _ d = 0 S d F_ h Oe{KLoHKLocL3Loc FIHKEDh O E [x , x ] u Loc [x , x_%_Tm ] HO"mQJ h h L™rIcLF a F[a U __ HKb‰X h m OšJ(fO d a _Tdm H FI_Tb'm FIcL@S h _ † m W NKFIcL4aK}Th XWL™rIcL6J h FIdsFI}­aKh NKFqFIc}Tm HO JMEGFIu JMc _%EGF _Tm NzLoE?mQh dH F } h }€E HKX m FIb a aKE h X'NKF h FId HO"JS ET_ L FIc~LoJ aKh Lo}G_ FDS nO FIETL Lo}TLNKF EDO"}THO JMEGFIJMc EGF NzLoE•J FIdsFI?} cYO F cL Lo}TL4dYOeJ FIdsFI}?}TL |HO"cgN O Lo} S

p

δi

Loc

i−1

δi+1

i

i

i

i+1

i

p

p

1

p

1

2

p

p

p3

0

2

p3

0

p4 x

0

x

1

x

2

x

3

x4

p4 x

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x

0

x

1

x

2

x

3

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x

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iptDt

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^ HKL`Loc u FIHKd _gh Og„IL NzL }THKLoETX h NlE~HOMETcLocL a _ d _ b'FIb‰Wc _€_ H _Tm NKF KAbSO$#NM^V!™!K=tMNO™XH Z `XK S*À a O"}DO ¼ FIJML IH _ EDOí}THKL ETX h N OlNKF m_gh™_ rMFIcYO a } _ c u H _gh cLobeL u _ r }DO"beL p = (x , y )  i = 0, . . . , n S _ HKb‰X h OeJ(O u _ r }GF*cYOš}THKLoETX h NzLQNKF P

i

i

i

*

P (t) =

n   X n

tk (1 − t)n−k pk ,

}(NKFIH`NKF t ∈ [0, 1] SnO(Nzd _gfG_DaKu F›NKF a F Xd _ HO"W h N O(N _ }TXWL™rIcF ¼ FIJML IH _ EGFq}THKLoETX h NKFDS k=0

k

n[FI}DO(N h O aKu c _DaKu L ¼ IF JML IH _ ETLo|~}THKLoETX h N`NKFDP • P (0) = p Loc P (1) = p  h h Ÿ _gu _ a _gf HKLocgN OgrMF p , . . . , p  • }THKLoETX N O F L4JMc HO(N } c>EGFI} cF h™_ h™_ a _ _Tm_ Wc _*a F cYO"} h™_ c'dHKL t = 0 XgNKFIb%O • nO"} cedHKL t = 0 NKFFIcYO"}­cYO"} c>X'dH FIbeL™„MF } JML p Loc p Dd ™ h _ _ a _ J*cYO"} c b-dH FIbeL™„MF } JML p Loc p S n^`_gO"fGb'_ F aKu _ ¼f FI_ JM L IH _u ETLo|¬a }T_ HKLoETX h N­cY_DaKOÇušEG_TF m h Lo} _T_¿m u _ r }DO"|m a F aKu m OME h N O"mb _ }Tu _XWL™rIcF ¼ FIJML IH _ EGh F€}THKLoETX h NKF•m E¡J h FIu ds_ FI}sS m HKX f F*Nq}TJ(HKO LoETX F h NKFb'h F F Ÿ HKL LN N _ } cYOšJML EGaKuFIL3JMc dH FIbeL™„IL@ES O­NKF  O€J(O cgNKF EGF r }TLdHKEGF'}THKLoETX NKF'Loc¡dHKETL EGF r }TL n[FI}DO(N d _ Nzb _ ?E }TL a Fd _ N OMETL N _ dHKL3Loc u FIHKd _gh Og„IL NzL6HOMETcLoc a }TLo|~}THKLoETX h NMP ›K^V!JLKSUnMNO `8˜8!TÂbS™!KAbSRV˜SU PÊ}THKLoETX h NzL a F m_gu Lo}DO u OàÈÔJ(O m cgN O u _ r }DOdHKEGF }THKLoETX h NKFeNKF J(OgrMF u cYO u _ r }DO • m HKX f F É  ›K^V!JLKSUnMNO `8˜8!TebS™!KAbSRV˜SUCVXW!™VXW V!™ ñ P u O"c f FIc u LdHKEGF m HKX f Fk}THKLoETX h NKFLob%O u O*E a }TXdcL u _ r }TLYFIcYO"} _ • a b'FIHX ]T!M^T!JLu KSUnMNOQPSRT¹bS™!KAbSRV˜SUžVXW!™VXW V!™ Ù ñ P[dHKEDO _Tm E _Tm O•dHKEGFšLoc m HKX f Fš}THKLoETX h NKFšE a }TXdcL u _ r }TL a F • XgNKFIb%O OQS 0

n

0

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0

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beLocLob%O h cYOQS·d FLoJMWsFIH FIb _ m ≥ n Qd _gu FIb m_ WLob _ Loc u FIHKd _gh Og„IL N a }TL4d _gh Loc _ b~S m c _DaKu } _Da _ b%O d _gh Loc _ b a }GFíLoc u FIHKd _gh Og„IL N a }GFíU©Xc}G„IL NKFíE u _ r }TL    e P T E K H  c % F T E

H F • TS Å _gu rMF u H u L`O"H f Xb'FIc u[h O"|} _ d _Tm O"b _ ETH aKu _ Loc u FIHKd _gh Og„IL N a }GF*U©Xc}G„IL NKFcYO­E _gh N _'a _  IP<_D} a _D_ a _ b%Ošh } _ c aKu O"c u cYu OeLoc _gu FIh HKd _gh Og„IL N O IP<} b%O LocFpO"HKcYOšLoc FIHKd Og„IL N O  IP m EDO"}THO u h JMEGFIJMc _%_Tm_ EGF mQh NzLoEí_}TXh WL™rI_ cL4J h FIdsFI} IP<}TXWL™rIcL4J FIdsFI?} Q}TL |HO"cgN O W Lo} i=0

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ω(x) (n+1) (n+1)!

n

f 0 (x) = In0 (x) +

n

0

n

ω 0 (x) (n+1) ω(x) df (n+1) (ξ) f (ξ) + · . (n + 1)! (n + 1)! dx | {z }

ƒ‚ JMHO"J%m€J(u O€_ cYO"dYO"} _ cL cYO(N h FIdm {KLF a O(h NqcF'_Tm d _ JMm cYO"b _ m_T_ m ETL a c_ _DaKu L ξ _Tm x Sd F'dYO€HOgrIXcYO"b Ç_ _Tm E _Tm E FIcL LoJMb'F r } x , . . . , x J(O cgNzL4r FIc dYO F*Loc WLob ω (x ) È i \QS i É f (x ) = I (x ) + (ξ). f (n + 1)! §4XeNKF I (x ) _Tm E _Tm Loc u FIHKd _gh Og„IL N a }GF f Oed _gh Loc _ b%OšE u _ r }TL x S _ HKb‰X h™_ J(O I (x ) LoJMdsF h NKFIb _ dH FI} _ x6O f HO"c f FIETLo|€} _ FâáY„IL™FIc u _ EsS‚ƒJ I (x) = P f (x )L (x) aKh F m L P ž _Tm EDO(N O"cgNKFIb I (x ) = f (x )L (x ) S napaka

0

n

0

0 n

P

L0n,i (xk ) =

P

L0n,k (xk ) =

i 6= k i=k

0 n,i

k

(n+1)

n

n i=0

k

Ln,i (x) =

m_ WLob _ WÉ

0

k

k

0 n k n i i=0

k

0 n

k

*

0 n

k

(x − x0 ) · · · (x − xi−1 )(x − xi+1 ) · · · (x − xn ) (xi − x0 ) · · · (xi − xi−1 )(xi − xi+1 ) · · · (xi − xn )

ω 0 (xk ) , (xk − xi )ω 0 (xi ) n X j=0 j6=k

1 . xk − x j

iptDy

i

n,i


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q–

ipt ]

¥%d _ b _ r›N _šu FI|€U _ HKb‰X h4h O"|} _ J(O"dL™{ FIb _ In0 (xk ) = ω 0 (xk )

n X j=0 j6=k

n

X f (xj ) 1 + f (xk ) . 0 (xk − xj )ω (xj ) x − xj k j=0

ÀÄ_ dHKLo_Db'aKu FIHKXcs} _•a _'h u _ r _ }GFšFIaK}Tu ETL m _ L aKu O"c mu c_ F­LocÊ_ EGF h N Oh™_gxfG_ =m_ x +_ ih  i = 0, 1, . . . , n  a FqU _ HKb‰X h OÊÈ i \QS i É { F d FIc OMETL@S ‚ƒJMdsF N OME dX Lob J(O b%Ogr cYO  WLob dYO j6=k

i

f 0 (xk ) =

1  h

(−1)k n k

n X (−1)j

n j f (xj )

k−j

j=0 j6=k

+ f (xk )

n[FI}DO(N dHKETLo|€U _ HKb‰X h }TLQNzLo| m _ W Lob _ J*E aKu OME h N O"cgNKFIb • n = 1P f 0 (x0 ) =

• n=2

f 0 (x1 ) =

P f 0 (x0 ) = f 0 (x1 ) = f 0 (x2 ) =

0

n X j=0 j6=k

n

Loc k P

1  (−1)n−k hn (n+1) f (ξ). + k−j (n + 1) nk

1 (f (x1 ) − f (x0 )) − h 1 (f (x1 ) − f (x0 )) + h

1 00 hf (ξ0 ) 2 1 00 hf (ξ1 ) 2

1 1 (−3f (x0 ) + 4f (x1 ) − f (x2 )) + h2 f 000 (ξ0 ) 2h 3 1 1 2 000 (−f (x0 ) + f (x2 )) − h f (ξ1 ) (sim. diferenca) 2h 6 1 1 (f (x0 ) − 4f (x1 ) + 3f (x2 )) + h2 f 000 (ξ2 ) 2h 3

^ HKL a Lob'F u HKL™rIcL m L‡UVFIH FIc„ILYJ(O"HO m L a Lob'F u HKL NKFkdHKL m_ WLob _ H F m cYO u H FI| u _ r }DO"|~cYO"b'F aKu _ cYO m EGFI|4S

h2

cYO"b'F aKu _ h  u KH Lo}edYO NKF m O*Loc u FIHKd _gh LoHO"b _

5 µ6qŸ Û  z: ß >€­7@«> z>>Aá¾í o Dk b¼

_ h _'_Tm h _ h ­_ u m _ h™_ _ Ka u _ _Tm_ u _ r HK}Gb‰FqX FI}TFETJ(L O‰m L c>aKuXO"b'c uFIcHKL™F*rIc Loc y =EDO(N fO"(xcgNKF ) SBO"‚ƒ|J*} HO"JMLoJME ds_ FNKFINzE XgNKFIb X LsLoJHO"JME N O­E%§`OM¨ H©NKFIE ETH S nO(NW *

i

i

1 1 1 y0 = y1 − hy10 + h2 y100 − h3 y1000 + h4 f (4) (ξ0 ) 2 6 24 y1 = y 1 1 1 1 y2 = y1 + hy10 + h2 y100 + h3 y1000 + h4 f (4) (ξ2 ) 2 6 24 1 1 f 00 (x1 ) = 2 (y0 − 2y1 + y2 ) − h2 (f (4) (ξ0 ) + f (4) (ξ2 )). h 24 2f (4) (ξ)

aa FM{ u FIEDO"cgNKFIb m _ W Lob _ U _ HKb‰X h™_ h _ _ f (ξ ) + f (ξ ) O"|} dL™{ FIb (4)

0

(4)

2

f 00 (x1 ) =

Loc m_ WLob _ } _ crIc _ U _ HKb‰X h™_

1 1 (y0 − 2y1 + y2 ) − h2 f (4) (ξ). h2 12

nO"b'F aKu _

Ý hou FIHKcYO u LoETPcLcYOgrILoc'LoJMdsF h N OMEGF<NKF b'F u m_T_gm h™O[_ cF m_ _gh™_ _ rMFIcLo|'} _u FâáY„IL™FIu c u _ _ EsS m ^ HKL u F›N`_ b'F u _Th m Lu _ cYO aKu OMETLob _q_gh a L aK_ u FIb f (x ) = m_ W)Lob Loc _ LoJ*d rI_gLohb Loc _ b%} OšFâáYET„IL™{©L™NKFIFc aKF u _ αdcgNKFDO"S}  OkNKFkU HKb‰X O rIcYOqJ(O‰d Loc b'F rILobãETL™{©NKF aKu _ dcgNKαFDS fn(xO"dY)O"+} _'R(f 0

k

n j=0

j

j

j


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ipt R

f 00 (x1 ) = af (x0 ) + bf (x1 ) + cf (x2 ) + R(f )

1, x − x1 , (x − x1 )2 , . . .

1 x − x2 (x − x2 )2

: : :

0 0 2

= = =

 a+b+c  −2 1 1 −ha + hc ⇒ a = 2 , b = 2 , c = 2 .  h h h h 2 a + h2 c

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C

p

r

r

f (x) = xr

¢

r=3

r=4

(x − x2 )3 (x − x2 )4

: :

0 0

= 6 =

−h3 a + h3 c h 4 a + h4 c



⇒ r = 4, p = 2.

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1 1 2 (4) (ξ) h2 (y0 −2y1 +y2 )− 12 h f

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a

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k=0

n−1 X

Bk yk + Rn (f )

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k=1

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f (x)dx =

x0

h h3 (y0 + y1 ) − f 00 (ξ). 2 12

x − x1 h dx = , x0 − x 1 2 x − x0 h dx = , x1 − x 0 2 Z 00 f (ξx ) f 00 (ξ) x1 h3 (x − x0 )(x − x1 )dx = (x − x0 )(x − x1 )dx = − f 00 (ξ). 2 2 12 x0

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1

1

5

0

1

2

(4)

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3

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x2

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5

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P

x0

Z

8

x4

f (x)dx =

x0

0

1

2

3

(4)

80

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Ai f (xi ) + R(f ),

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x0

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k=0

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xk

n−1 X k=0

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h h4 (xn − x0 ) (4) (y0 + 4y1 + 2y2 + 4y3 + · · · + 2yn−2 + 4yn−1 + yn ) − f (ξ). 180 |3 {z }

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j1 X

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Z

Z

b a

x a

f (n+1) (t)(x − t)n dt,

f (n+1) (t)(x − t)n+ dt.

È iDi S i É


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a

n

Z

b

a

f (n+1) (t)L((x − t)n+ )dt,

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b

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f

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áª

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x1



h K0 (t) = (x − − ((x0 − t)0+ + (x1 − t)0+ ) 2   Zx0x1 h 0 = (x − t) dx − (0 + 1) = 2 t h x0 + x 1 = x1 − t − = − t. 2 2 t)0+ dx



=

­7K^T!RV!™Vi`XK^W!M^VaR+O«!V!R˜SUFT!R\UnR?KQ› Tµ]MAK^WtbSRT!T¬ bXTUFV Z T® !V Z KzVX_8KNR+OJ#V |R(f )| ≤

57576ó Z

Z

x1 x0

x0 + x 1

h2 0

|f (t)|

− t

dt ≤ kf k. 2 4 0

>A¢?ã­?à­ Yk­ D ì <*  Ys¿¾ík 7¢«>o

ž HO"JMd _gh OME h N O"cgNKFIb h dHKL m FIb _ dHKL a F aKu OME h NKFIcLo|€U _ HKb‰X h O"| m_šu _ rIcF›N a Lo|€H FIJMX hou O u _ EsS‚ƒJ*dHKLoW h L Ÿ } _ E€dHKL HO"J h L™rIcLo| h h O"|} _'_ „MFIcLob _ cYO"dYO"} _ Loc~X fG_gu _ ETLob _ YrMF NKFd _gu H FIWc _ { FcYO m O h NzcF*HO"JMd _gh OME h N O"cgNKFDS


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q–

i wDw

ž _ „MFIcYO"beL a F*NKF'dHKETLX}TEDO"H©N O h £ L™„ |YO"H ma _ c4S nO(NqW _ S (f ) ¥>Lobed a _ c _ EDO€U _ HKb‰X h O a } _ HO"} _ b a _ _ _ h _ _ m h R (f ) cYO"dYO"}DO%¥>Lobed c EGF*U HKb‰X FqdHKL4} HO"}TX h S ÀBFIb  OšEGF N O h

h

Loc

h

I(f ) =

ž O­cYO"dYO"}TL4EGF h N O

a

Rh (f ) =

d FdH F m d _DaKu OMETLob _  m OlNKF ‚ƒJ

Z

b

f (x)dx = Sh (f ) + Rh (f ) = Sh/2 (f ) + Rh/2 (f ).

−(b − a)h4 (4) f (ξ1 ), 180

f (4) (ξ1 ) ≈ f (4) (ξ2 )

Rh/2 (f ) =

 m_ WLob _

−(b − a)h4 (4) f (ξ2 ). 16 · 180

Rh (f ) ≈ 16Rh/2 (f ). Rh/2 (f ) = I(f ) − Sh/2 (f ) = Sh (f ) + Rh (f ) − Sh/2 (f ) ≈ Sh (f ) + 16Rh/2 (f ) − Sh/2 (f )

m_ WLob _

Sh/2 (f ) − Sh (f ) . 15

§a _ U a _ HKb‰_ X h™_lh O"|} h _ _TXm d _ _THm O"WLob _ J(O _ „MFIc _ cYO"dY_ O"}GFa ¥>Lobe_kd a m_ _ c _ EGF_[U _a HKb‰_TmX _ h FDSB…[„MFIcYOkhou dYOku cL>^`d_gH FIh EGfFMr _ J(O"cF aKh h NzLoEDO_  O(N b LoJpFIcYOgrIL L E K8FDXS6˜SUnÀËM^T8]EGFMV rIT Loc_SL>O `šdVíHKLomb'_ FIH E*_ E FMFIc _gh WLob h Ÿ d a WcF H FIJMX O FDS F „MFIcF O"|} LoJ S (f ) Loc S (f ) J Z WLob { FW NK{KL3dHKLoW L FI}? O(N6NKF Rh/2 (f ) ≈

h

h/2

I(f ) = Sh/2 (f ) + Rh/2 (f ) ≈

16Sh/2 (f ) − Sh (f ) . 15

^`_Tm_ Wc _eh O"|} _ cYO"H F m Lob _ dHKL u HO"dsFIJMcL4Loc a H F m Loc a }TL3U _ HKb‰X h L@S

57576



¯

à­¿¾‰ >^D*­ < ` à­

^ KH LQO m O"d u LoETcLo|­b'F u _Tm O"| a dH _gu L _ „MFIcgNzXgNKFIb _ cYO"dYO"} _ Loc­rMF3NKFBd _gu H FIWc _[m_Tm O u c _[m F h Lob _ d _Tm Loc u FIHKEDO h FDS

u _Tm a h f _ m _ h _ _ h mÆÇLoE(F NKFIb O _ W FícYOgd{HKO"L cgO NzX~O(N dYOÇO­U©b%XcO"cg}GNK„I{KL NzL LFhS}DO"Hed b'FIcLF OÇdHKLd | FIETcFIb WcYOg{O"cgNzXXd HO"W N OÊEGFMr›NzL h dHKL 100 90 80 70 60 50 40 30 20 10 0 0

0.2

0.4

0.6

0.8

1


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qfhgiT `

u q

ydéIa AfUT v {

] 

ÑúŠ

o54 m

‰

] Uow

Q q

Qu

ydéIa AfUT v {

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u

qfhgiT `

u

u u

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ydéIa AfUT v {

u q

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k¸m

é

u

Ñ

ydéIa AfUT v k {

`



‰

] ‰azIa Š

Q q

axQ Š

a fUT vIaQIaQ q

ow

aQ Š

o

w o54 m ow

]]Hq¸Ñ0Š Q Š

ow

Ñßm ·AQ Œ q Ñ

…U·AQ îÑ

Q Š

ow

sé k o54 k Iw

 !!

v,j

a AfUT v mi{ ußé

QIa

fUT v

ußé mîю] é m

ydé Š



…U·AQ

] é m sAé k ow

AfUT v&6iu

é

o54 m

ÑúŠ

…U·AQ qŒÑ Ñ

] Q

fUT vIa

 !

4

] Q

a Š

è _U 0f c:j f

4,%'· ·

a fUT vUo

ow

]Š

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u

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Q Š u‰

}

AfUT viu Š j g Q

‰

Q

‰

é m

 ! u

]]  : Ñ q o54 m ow

u

é k

è _U 0f c

è _U 0f c:j f

sIw Š

w

u

iwy

Ñ

u u

 é Š

è _U 0f c:j f

] ‰a–Ia

è _U 0f c:j f

] ‰a

w

AfUT viu AfUT v k Ñ AfUT v m

q

a

fUT v,4 m a

fUT vUs AfUT v k

q

a Š

a

QIaxQ



axQ q

a

Q q ow a

Q

a

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ow

w

õ h OMETcYa _OÊL _ m F›_ NKFeNKFDu P _Tm^ _ HKETm LkdHKf LoW h L Ÿ FI} h Ÿ Q J(O¿Loc u F mf _ HO hm_q_ Wu Lob _ _m u O"} _  u m OÊcYh O¿„MFm h FIh b¾_Loc u FIHKEDm O h Xd _ HO"Wm Lob h _ ¥>Lobed a c E b'F _  HK_X LYdHKLoW a _ L FI_ } _ Q dYu O _Tm_ WLob h O"} _‰m OLoc fG_ FIHKEDO u _THmO"J_ F ^`Lob _gu cYO EDO*FIcYO"}DO _ F O Loc%cYOqE O"}GFIbüXd h HO"WLob ¥>Lobed c E b'_TF m ÈVO L}DO"}Qu {Kc h HKX m b'h F _ É S m FIm bÂh dHKLob'FIH©N O"b a Q Loc Q ÎS d _ FqNKF'HO"J Lo}DO _ |Q u −f Qh | EGaKFMu r›_ N O u _Tb'm_ F›NKF  <u Loc FIHKED_ O HO"mJ F Lob cYO a EDO m F O LocËa cYO~E _gh™O"_ }GFIb H FI_T}Tm XHKJMLoETc LoJMHOgrIXcYO"b Loc F HO J L b'u F f h<gmJ(_ O"| _gh FIEDu O"_ b _ dYO OBNKFcYO"dYm O"}DO F _ O(N6cYOlE hou O"u }TL>d _gETh L™„IL hd O"|} _ /2LoJ ·S Qd FqLoc dYO[QNKF J‰|QFIcLo−b-Q} _ |HO"≤} _ b gNKF£ LoL™c „ |YF O"H HmO a _ c _ E EGF­N FI} aKu rIHc O"d _gLoJMh HOgOg„IrIL NKXF cYmO"_ ccW Lob Oš_ dY{ OšFW W _gh NKH {KFILsJMX dHKLoOW h L Ÿ { FIF}sW S NK{KLF 1

2

1

2

2

2

2

1

1

1

57576  < íÜ < ß  D  < ` à­ äU _ V!HKb‰JLX —AKNh™_MQ›  V!O­™TòE%H JLF a KScUFVXL™„IW L T u FIJ b'fDh F F h mNzL3O¡cY}O _gmu HKXJ(O"f d OgrI_ cH FL m _DcYa O¡c _ XETdL@S _ HÆÇO"WYF mQO u FI£ bL™„ Q|Y} O"_ H m£ a _L™„ c |Y_ O"EGH Fma FI_ } c aK_ u E H_O"d FI_g} h aKOgu „IHL O"NKF d _gJ(h OOg„IuL N H_šO"dsh O"FI|JMc} __ cYO"H F m Lob _eh FqFIc}THO u  h O"|} _ £ _ b‰WsFIH fG_ E _ FI} aKu HO"d _gh Og„IL N _ LoJMEDO(N O"b _ cYO­EGFMr*cLoE _ NzLo|4S € ´ ·`¶> ¶ ïM¸Ë‘Y‘\™¨ Ð KNMNRV!H ZZ O `XKN™T ¤ UKN™XO Z T ˜šV|W V Z VXP8KNRT#bµM^TtbS™V‘`XKNJ 6°

0/

Z

P

Bk

X Bk x = xk , ex − 1 k! k=0

|x| < 2π.


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q– À¼ a O ¼ _ FIHKhoh c _ X hoh L NKFIEDu OÇ{ hu FIETL h O a _ HOg„IL _ cYO h cYOB E a O h Lo|YOÇ{ u FIETL h O~HO"JpFIc FIHKc X L NKFIETLo| { FIETL NKF 1 B1 = − , 2

B0 = 1,

B2 =

1 , 6

B4 = −

1 , 30

i w>] B1

dYO a _ FIcYO"}DO 0 SÊnO"}DO(N‰dHKETLo|

B6 =

1 , 42

....

S_ G¹"´>ò÷h ‘Y‘\¨o× Ê«T´R?K8˜8!V!RPSRVtM^TUCbS™!KAbSRVLVXW!™!K^W Z `NO™V7œNHR!_SO `šV !™ K Z `šT Û X h FIH‘ƒÆÊOg„ h O"XHKLoc _ EDO a Xb%Og„IL N a }DO U HKb‰X O f

I(f ) =

»eV!Ttbt¸ *

Z

b

a

È iDi S Z É

  ∞ X B2k 2k (2k−1) (2k−1) f (x)dx = Th (f ) − (b) − f (a) . h f (2k)! k=1

_ HKb‰X h™_ W _ b e_ m_ D} O"J(O h L ka a oL b‰W _gh cLob-HOgrIXcYO"cgNKFIb~S†FâácLoHO"b %_ _ sd FIHO u _ H©NKF Ef (x) = f (x + h), ∆f (x) = f (x + h) − f (x), Df (x) = f 0 (x).

ÀBF h N O(N _ cYO Ka h F m cgNKFU _ HKb‰X h FDP I +∆ E ∆ 1 D 1 ∆

= E = ehD (razvoj f (x + h) v Taylorjevo vrsto) = eZhD −I x = f (x)dt + C (nedoloˇceni integral) a

=

¥TF m O(N`NKF

∞ X 1 1 1 1 (hD)2m−1 B2m = hD = − I+ E −I e −I hD 2 (2m)!

(razvoj

m=1

(E n − I)

u _ dYOšd _ b'FIcL

ez

z ) −1

1 En − I = = E n−1 + E n−2 + · · · + E + I, ∆ E−I n−1

X 1 (E − I) f (x0 ) = f (xk ). ∆ n

^`_'m HKX f L Ka u H O"cLNKF 1 1 (E −I) f (x0 ) = ∆ h n

k=0

Z

xn x0

  ∞ X 1 h2m−1 (2m−1) (2m−1) f (t)dt− (f (xn )−f (x0 ))+ B2m f (xn )−f (x0 ) , 2 (2m)!

T_ m•u _Tm dYO aKh F m L3U _ HKb‰X h O~È iDi S Z É S À H aKu O­E¿È iDi S Z É dHOMETL h™_ b%OecL3} _ c>EGFIH f FIc u cYOGNKFdYO'O a Lobed u _gu a }DOQ} _ef H F }DO"Hd _ b'FIcL

I(f ) = Th (f ) −

m=1

h→0

SB† _ WLob _ ÈÔ} _

h2 0 h4 000 (f (b) − f 0 (a)) + (f (b) − f 000 (a)) + · · · , 12 720

I(f ) = Th (f ) +

∞ X k=1

ak,0 h2k ,

h→0

É


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q–

i wTR

dHKL4rMFIb'FIH a _ } _ FâáY„IL™FIc u L a cF _Tm ETL a cL _Tm h SB§`O"} _'m_ WLob _ k,0

I(f ) = Th (f ) + a1,0 h2 + a2,0 h4 + a3,0 h6 + · · ·  2  4  6 h h h I(f ) = Th/2 (f ) + a1,0 + a2,0 + a3,0 + ··· 2 2 2  2  2  6 h h h I(f ) = Th/4 (f ) + a1,0 + a2,0 + a3,0 + ··· 4 4 4

du _ F‰rIcFIF›cYNKOg{KrIL3W d_ HKLoWJ(Oh L Ÿ FI}sP d _ bec _gŸ Lob _•a h/2

4

Loc T_ m { u F›NKFIb _%_Tm FIcYOgrIWsF‰J(O

h

 a F*JMcFIWLob _ r h FIcYO

h2

Loc m _ W Lob _

(1)

I(f ) = Th/2 (f ) + a2,1 h4 + a3,1 h6 + · · ·  4  6 h h (1) + a3,1 + ···, I(f ) = Th/4 (f ) + a2,1 2 2

}(NKFIH Ka u O (1)

Th/2 (f ) =

4Th/2 (f ) − Th (f ) , 3

1 (f ) = Th/4

^`_DaKu _ dsFI} a F m O(NcYO m O h NzXgNKFIb _ E

4Th/4 (f ) − Th/2 (f ) . 3

(2)

I(f ) = Th/4 (f ) + a3,2 h6 + a4,2 h8 + · · · ,

}(NKFIH`NKF

(1)

(2) Th/4 (f )

À a d h™_ {KcFIb-d _DaKu _ d}TX u E _ HKLob %_ a | FIb _

=

(1)

16Th/4 (f ) − Th/2 (f ) 15

napaka O(h2 ) O(h4 ) O(h6 ) (0) Th (f ) (0) (1) Th/2 (f ) Th/2 (f ) (0)

(1)

(2)

(0)

(1)

(3)

.

O(h8 )

···

Th/4 (f ) Th/4 (f ) Th/4 (f )

}(NKFIH`NKF a d h™_ {KcYO­U _ HKb‰X h O

(j−1)

(j) Th/2k (f )

=

(0)

(0)

Th/2 (0)

Th/4

(j−1)

4j Th/2k (f ) − Th/2k−1 (f )

§W!™!KµµÔ´>M^—/Ttb]‘YV ‘\¨ˆT!² ™ `šÊ T!RXä`NOFV!¸ JL—AKNMQ› V!™V.JLKSUFVXW VO bSM^T PSHRT‘` Z Z »eV—NOJ#V ¯ Th

(4)

Th/8 (f ) Th/8 (f ) Th/8 (f ) Th/8 (f )

4j − 1

R 2.2 1

.

ln x = 0.5346062

¸"Ê«T PSR+O¿˜

h = 0.6

1 1 = 0.6( ln 2.2 + ln 1.6 + ln 2.2) = 0.5185394 2 2 1 (0) = Th + 0.3(ln 1.3 + ln 1.9) = 0.5305351 2 1 (0) = T + 0.15(ln 1.15 + ln 1.45 + ln 1.75 + ln 2.05) = 0.5335847 2 h/2

ORÂRT!MAK^W!O


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i w>v

Th/2k

1 h Th/2k = Th/2k−1 + k (y1 + y3 + · · · + y2k −1 ). 2 2

\UnË R?TKQ!› VT|™šW˜šK TTL!V=™iœN]HMNRO!JL_SOKN`8M˜8`š!T!V"™XO«™XbµMAK^M^W!T RPSVH˜SRUIT!O RXbS`XM^KNT J PSHRT!J#Vͬ KNMARKAbStH M^TUFUITUÆOR×]TiOJ#`XK T!J#T® V|!b VLä J=V!RJLVA› —AVLKNMQ› RV!TJ UFT!RbXPST!R?R?KKN` J#OFT!¸ M Z `NO™VÎW VXW T!Õ Z Z ¤ Z K^W T‘`žb ä V!JL—AKNMQ› V!™V"K8X˜SUnM^T8]V Z T _SO `šV¹W V—NOJ#V (0)

Th/2k

)

(1) Th/2

(0)

(0)

(0) 4Th/4

3 (0) − Th/2

4Th/2 − Th

=

(1)

Th/4 =

3

(1)

(2) Th/4

57576

â

‚ƒc u F f HO h R

= 0.5345337 = 0.5346013

(1)

16Th/4 − Th/2

=

15

= 0.5346058.

 : Y Y Dk æID 7à‰s : s­ M <:  Üä

b a f (x)ρ(x)dx

}(NKFIH<NKF Z

ρ

cFIcF f O u LoETcYOeX u F Ÿ YO"dH _ } a LobeLoHO"b %_ a T} EDO m HO u XHKc _ U _ KH b‰X h™_

b

f (x)ρ(x)dx =

n X

(n)

(n)

Ai f (xi ) + R(f ).

R u a _m_gh™_ _ h a h _ h u _ _gh _ Å_ aKu _ dFâáYcg„INKF L™FIEc a O(L N n S`¥[drMHKFILob'cLFIJHKc E _ JLoJMLFW LoH _O(N`E EG_ F J h™N _ O E Ah O"|} _k=m_Da F LŸ FIb (x)ρ(x)dx _  m O W _ U _ DHKU b‰HKX b‰h O X u O[_ rIdYcYOBONKFJ(O d rI_gcYh LoO[c J(_ Olb'd F aKu Loc_ dcgb'NKFF E a O(N 2n + 1 E _ J(O m NzX€dYO a _e_ H u _gfG_ cYO h cL6d _gh Loc _ beL@S ž O‰U©Xc}G„IL NKF h O"|} _ cYO [a, b] m FâácLoHO"b _%a }DO h O"HKcL4dH _Tm X} u } _gu a

i=0

(n) i

hf, gi =

b a

Z

n,i

b

f (x)g(x)ρ(x)dx.

_ Xu c_g}GfG„I_ L NzL h f Loc g_€aKmu _ O _ H u _•_gfG_ _ cYu _ O h c_ LF rMF‰NKF _ hf, gi_ = 0 S¡‚ƒJ aKu O"c m O"H m cF•WYO"JpF•d _gh Loc _ b _ Eg_ h 1,_ x, xaKu ,_ . . . J H cYh O LoJ(Og„IL N WLob H c HKbeLoHO"c WYO"J P (x), P (x), P (x), . . . }(NKFIHNKF P d Loc b dcgNKF i Loc€EGF N O *

a

2

0

1

2

i

n O(N4W _a F m O(N P u O"}‰c _ HKbeLoHO"cšd _gh Loc _ b m OBNKF hP , qi = 0 J(OlE a O"}*d _gh Loc _ b q aKu _ dcgNKF }TEGFMr›NKFIb‰X _ ^ HKL T!H˜8˜šV!™XOžt™T W!M^TUnHMNR+O œšV!MNJ=H Z O J(OÇE _ J h F n LoccYO(NšW P = k (x − x ) · · · (x − x ) S LoJMWsFIH FIb _ cL™r h F x , . . . , x  u _ H F›N`NKF ω(x) = (x − x ) · · · (x − x ). ^`_gh NzXWsFIcÌd _gh Loc _ b f aKu _ dcgNKF 2n + 1 h O"|} _ J(O"dL™{ FIb _ } _gu f (x) = q(x)ω(x) + r(x)  }(NKFIH aKu O q, r d _gh Loc _ b%O aKu _ dcgNKF}TEGFMr›NKFIb‰X n S § _ d _ b'FIcL@P hPi , Pk i = δik .

n+1

n+1 (n) 0

Z

n+1 (n) n

b

f (x)ρ(x)dx = a

n+1 (n) n (n) 0

(n) 0

Z

á¬

(n) n

b

q(x)ω(x)ρ(x)dx + a

= 0+

n X

§ _ H F›N<NKFdHOMETL h™_eu _ rIc _ J(OšE a Fd _gh Loc _ b'F Ka u _ d cgNKF

i=0

(n)

(n)

Ai r(xi ) =

Z

b

a n X

h 2n + 1 O L4b%O"cgNMS

i=0

r(x)ρ(x)dx (n)

(n)

Ai f (xi ).


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q– ÑB´Tָ̑Y‘\ˆ¨ Ø «UK £ O T!H˜8˜šV!™XO†®|t™T W TUnHMNR+O†®=]M^T!™XO Z ˜šVI]VtbSOnUnO™XR?KX¸ »eV!Ttbt¸ À JpFIbeLob _ ª

J(O

i w>

¬

S NKFd _gh Loc _ b aKu _ dcgNKF

Pi (x) =

i = 0, . . . , n Pi

Z

2n

ω 2 (x)

 u _ H F›NEGF h N O

b

Pi (x)ρ(x)dx =

n X

(n)

(x − xi )2

(n)

(n)

Ak Pi (xk ) = Ai Pi (xi ).

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i

i

i

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(n)

xi

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(a, b)

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(n)

(n)

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a

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i=0

Z

b a

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b

a

n X

(n)

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n X

(n)

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i=0

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a

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n X

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a

b

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ρ(x) = 1 1 ρ(x) = (1 − x2 )− 2 1 ρ(x) = (1 − x2 ) 2 ρ(x) = (1 − x)α (1 + x)β 1 ρ(x) = (1 − x2 )σ− 2 ρ(x) = xσ e−x 2 ρ(x) = e−x

ipy \

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r ! 1 1 (4) + f (ξ), 3 135

r ! 3 8 5 − + f (0) + f 5 9 9

r ! 3 1 + f (6) (ξ). 5 15750

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1

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k

k−1 k−1 0

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k k

k k+1

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b0 a1

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an−1 bn−1

bn−1 an

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n


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1

5 1 f (x)dx = f (−1) + f 6 6 −1

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[a, b]

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g(x) = Ps (x) + ostanek

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a

b

Ps (x) dx, (x − a)p

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Z

b a

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g(x) π √ dx = 3 1 − x2

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f (x)dx

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I = I1 +I2

a

I1 =

Z

b

f (x)dx,

I2 =

Z

f (x)dx.

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b

1

2

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1/b

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dx

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yj = c+jk

S




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Z

1

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0

0

ôY1EõÔ1Eõô 7ÿ ÷ø A Ç Èg=QÉ Ã 9(A~; Á\ù =VA!É ÁÂý CÆ÷ Á A Å Á 4C?; à ÿ[Ç Á Ï÷C †FIcLob _  m O X aKu _ H FIJ(O•x3Lod a „ |L u J _ EGFIb‰XÌd u_gfG_ NzX4Sš§ u _ rIcYf O•H FM{K_ L u FIh EÇJ(OgrMF u cF f_guO€dH _ W h FIb%O u _gfG_  Y cYO(NlW Sµd _guFkNKF _gH h FM{KL FIE~J(OgrMF cF O%h dH W FIb%OíJ­JMb FIcLobÏJ(OgrMF cLobìd NKFIb  d FIbãJ(Ošd NzXWsFIc EGF N O E

“

y 0 = f (x, y)

f

y(x0 ) = y0 y(x) y 0 = f (x, y) y(x0 ) = ye0

}(NKFIH`NKF

x ≥ x0

ke y (x) − y(x)k ≤ eL(x−x0 ) ke y0 − y0 k.

d FcYO"b'F aKu _ QEGJ(F Ogh rMN OF u cF f OedH _ W h FIb%O

b 0 ) = yb0 f(x

ye(x)



y 0 = f (x, y) y(x0 ) = y0

kb y (x) − y(x)k ≤ eL(x−x0 ) kb y 0 − y0 k +

kfb − f k = max(x,y)∈D kfb(x, y) − f (x, y)k,

H FM{KXgNKFIb _ W h L Ÿ cgNzL4dH _ W h FIb

eL(x−x0 ) − 1 b kf − f k, L

b yb) yb0 = f(x,

ôY1EõÔ1EõÔ1 ú ADC?ÿ =ÇÅ à 9(A~; Á\ù =VA!É ÁÂý CÆ÷ Á A Å Á 4C?; à ÿ[Ç Á Ï÷C S ˜ F U  T N — O  R T a O"} _ W aKu O(N O u O"} £ FM{KL u FIE~J(OgrMF u cF f O€dH _ W h FIb%O Z  Q  K N F 3  M r š F ( J í O E  m OšJ(O }TLQNKFH FM{KL u FIE  EGF h N O J(OšE a F S dd _gFqfGLo_ b%NzX Oe_DJ(aKOgu rMO"F cu F*cL`W dh LoH JM_ X W hu FI_ b rIcaKFu H O"FMW{KL L h u c EG_ FDS H FM{KL u FIE? h O"|} _ dHKL™rpO"}TXgNKFIb _  m O'H FM{KL u FIE€dHKL6JMb _gu FIcFIbJ(OgrMF u cFIb ^ HOMETLob _  m ONKF aKu O"WL h cYOqH FM{KL u FIE T˜NOJi]?UFVUnOQPSRVe˜SUFT—NO Z RT TrMF f H F _gu L >} _qf H F dH _gu L  d H cF_ga fG} _ _ crIc h _DaKu L@S _gu ^ HKLkO a _šLobef d u _gu L™rIc_gLku H FM{KL u a ETL _ h O"|_D} aKu_ dHKL™rpO"}TXgNKFIb _  m O W _ cYO"dYO"}DO JMb _gu FIcF f OÇJ(OgrMF u cF f O d N Oš{ O­dH L  } H F dH L4cF } rIc L@S §  µÔ´>—/‘²?©¨ ‘ ä K ¤ OnUKN™#W!O œXKNMAKNR_SOQT Z R?K=KNRT Pt—AK ¬ ¬C`XKNM `XK ¬+`XK ¬c˜SUFT—NO Z RT7]MNO ¬«ORaR?K8˜SUFT—NO Z RT¿]MNO ä ¸ K ¤ OnUKN™C`XKžT˜NOJi]?UFVUnOQPSRV˜SUFT—NO Z RT¿]MNO



“

δ>0

y 0 = f (x, y) y(x0 ) = y0 ye0 = f (x, ye) ye(x0 ) = ye0

ye(x)

>0 ke y (x) − y(x)k ≤ 

ke y (x) − y(x)k

0

0

x ≥ x0

x

x

y 0 (x) = λy y(0) = y0

λ = a + ib ∈ C

y(x) = y0 eλx = y0 eax (cos(bx) + i sin(bx)). Re(λ) < 0

0

Re(λ) = 0

Re(λ) > 


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q– i]i §OR ˜šµÔ´>V —/T‘˜SUn²?R?ˆ¨ Kµ² ™X¦8MAJ#K^W!T!RJ#V˜SVUnO ˜NO†˜SUKNJ b T¬ ˜SUnR+OJ=OE™!¬K8UF`XV!KNMM`N`XO K J#TUnMNO†!¬T ]VUKNJå¸ `XK© MAK¿K ˜XKIOnUKNW ™ T W!OQTX› V!RT Z O bSOM^TUnO Z Z ¤ y 0 = Ay y0 = y0 λ1 , . . . , λ k

A v1 , . . . , v k

y(x) =

`XKNMI˜šV W V Z VXP8KNR+OCb

k X

k×k

A

α i e λi x v i ,

i=1

αi

k X

˜SUFT—NO Z RT¬P8KL™!K Z `šT ä K ¤ OnUKN™`XKaTbX˜NT#OJi™š]?˜XKUFVZUnTOQPS˜SRUnR?V¹Kµ˜S™XUFMATK^—NW!O RZ RVT˜S¬UnO P8K´R?bXK8˜STUFT™š—N˜XO K Z RZ TT¬[˜SUnP8R?K=K#V™X—8MA˜SK^UFW!T‘`šRT V˜SUnOi™!K Z `šT ¸ 5fŸ6½ ­ks7`@ ­ < ` ­à Yô 1EõþEõô øfÇ Á ;8È Á sÉ CÓÏ Á A ÃCÄ C nO(NzdH FIdH _DaKu F›NK{OšFIc _ } _ HOgrIcYOeb'F u _Tm O[NKF K8X˜F] Z OQ_SOnUnRT Y H Z KNM`XKN™TaJLKSUFVXW T J*cYO aKu OMET} _ b y0 =

αi vi .

i=1

Re(λi ) ≤ 0

Re(λi ) < 0 F Re(λi ) > 0

F



G

§

yn+1 = yn + hf (xn , yn ) xn+1 = xn + h y2 y1

y0 x0

x2

x1

ÆÇF u _Tm O[Xd _ HO"W h N O[á} a FIc'dH FIbeLo} h GL m F›N O[dYO NKF m O a F E­E a O"}TL u _ r }TLdH FIb%O"}TcFIb _ E a b'FIHKL u O"c f FIc u FDS ¦8Jia ] Z OQ_SOnUnRT_ Y H Z KNM`XKN™uTLJLKSh UFVXW T NKF ya aKu = y + hf (x , y ) S ^ HKL3Lobed h L™„IL u cL4b'F u _Tm LTNKF[d _gu H FIWc _ E E O"}GFIbã} HO"}TX~H FM{KL L3cF LocFpO"HKcL L FIbãJ(O y S n+1

n

n+1

n+1

n+1

§ ¬ µÔ´>—/‘²?¨™¬cÐ ]MA»zK^WK ˜SZ V!UFT!™™T!RX`š`XTaKRK8TX˜ ˜F] K^ZW!OQRX_S`NOnUnOR?›!K M^T‘Y œXH¸ Z KNM`XKN™!KžJLKSUFVXWKiRT¿]MNOJLKNMNH Z Z

1 h = 0.2

¬

y 0 = cos(3x2 ) + sin(4x)y y(0) =


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q–

i ]DZ

1.4

1.2

1

0.8

0.6

0.4

0.2

0

−0.2

−0.4

0

0.5

1

1.5

2

2.5

3

3.5

4



§]MNOJLµÔ´>—/KNM`š‘T!²?™V|¨o× K8XÊ"˜F] J#OQT!_SOnRXUn` R?¤ KT!RX`XHKNJ KNM`XKN˜X™!KEKe]JLV!™!KSUFK^VXPSWHN`XKµKžRRTIT]UFT!MNROJLPSRKNVMNH˜SUW V— Z `XKNR?KiMAK ¤ OnUn™!KX¸[GµT˜ Z K^W!RX¬ `šTµ˜ Z O†!T·]MA¬ K^OWR\˜SUUFKNT!MN™ ™Z `šT T Z `XK ¸ Z Y Z h

y 0 = cos(3x2 ) + sin(4x)y y(0) = 1

[0, 4]

20 korakov

1 0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

1

2

3

4

80 korakov

1

0

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

1

2

0

1

3

4

0

2

3

4

3

4

160 korakov

1

0.8

0

40 korakov

1

0

1

2



ôY1EõþEõÔ1 û Cü Ç Ã ;8È Á ÉsCòÉ ;>9(ADC

d F*FIcYOgrIW _

y 0 = f (x, y)

_Tm EDO(N O"b _  m_ WLob _ y0 = f y 00 = fx + fy y 0 = fx + fy f y 000 = fxx + 2fxy f + fyy f 2 + fy (fx + fy f )

SS


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q–

i]t

`^ _ §`OM¨ h™_ H©NKFIETLTETH aKu L>LoJMHOgrIXcYO"b _ y(x ) = y(x +h) = y +hy + FIcYOgrIW _ H FM{KXgNKFIb _eu X m L a d _ b _ r›N _ HO"JME _ N OšE•§`OM¨ h™_ H©NKFIE _ ETH aKu _ S § µÔ´>—/‘²?¨ˆØ y = xy + 1¬ y(0) = 0¬ y(0.2) =? 1

0

0 0

0

h2 00 2 y0 +· · · .

§ _ H F›N h O"|} _[m L‡UVFIH FIc„ILˆO h c _

0

y 0 = xy + 1 =⇒ y 0 (0) = 1 y 00 = xy 0 + y =⇒ y 00 (0) = 0 000 y = xy 00 + 2y 0 =⇒ y 000 (0) = 2 1 y(h) = h + h3 + · · · . 3 y1 = 0.2 + 0.00267 = 0.20267

­[MNO

¸ © KO ¤ P8KNJ#V ¬#RT W T Z `NHN`XKNJ#V͘UFVXPN!V ¸ W V—NOJ#V €´ ·`¶ >¶ ïM¸ËO ‘bS²?M^T ©¨ ‘ PSH­[RT!M^T!R+™XO OJ#V¬´W HNT´`XKN`XJ#K¹T=OJ#bITÎM^TtbSJL™KSV‘`XUFVXKNW J T Z V!T Z RVÍRT8™¹]TË\!T VÂZ V!MAK^MW `XKNT ™V#¬=™XM8P8˜SK¹UFVL˜XKzVt]M^MNVAO› UFVXPSR+W OV´™X™šMAK^ZW!`NRHVPSR˜SUnV O P Z KNRT ¸ x _ }Da O h cYOecYO"dYO"}DO'dH F maKu OME h N O'cYO"dYh™O"_ } _ E€_ FIcFIaKb u _•a O"h b'FIb } _ _ HO"}TX4_ SkÆÇu F _Tu _Tmm _ OeLo_gJ‰h J(O m cgNKfF f O'J mfDh F m O'Lob%h OeH F m   _  m  L™„MFIHqdYO•J'HO"JMETL N O"cgNKFIbÚEʧ`OM¨ H©NKFIE ETH O"|} WLob b'F d NzXWcF O•H F OQSíÛ X FIH©NKFIEDO b'F u _Tm O€È©FI} a d h L™„IL u cYOeLoc€Lobed h L™„IL u cYO É Lob%OšH F m i S Yô 1EõþEõþ ö øfÅ Á ø AgADCÓÏ Á A ÃCÄ Á

h = 0.2 (0.2, 0.20267)

y(0.4)



0/

yn = y(xn ) hk

yn+1

y(xn + h)

k

xn

3

O

nO(NzdH F›N LoJMHOgrIXcYO"b _

ki = hf (xn + αi h, yn +

cYO u _ dYO

i X

βij kj ),

i = 1, . . . , m,

j=1

yn+1 = yn +

m X

γi ki .

^ KH L u FIbæNKF m aKu _ dcgN O £ Î Å b'F u _Tm F}DO"H[cF a b'FIb _ J(O"b'FIcgNKFIEDO u L<J­H F m_ bb'F u _Tm FDS Å _ c aKu O"c u F α  m_gh™_ _šu _ m a _gh _ h™_ _ aKu _ ^ β Loc γ u FIbãd _ cYOMEDO m L4rILoEGb F h N O O"α} = PO F yβ S rILobÂW N XgNKFIb%O­JHO"JME NKFIb y(x + h) Eí§`OM¨ H©NKFIE ETH S KH L À/dHKLob'FIHKXc} _ NKF β = 0 J(O i = 1, . . . , m gNKFb'F u _Tm OšFI} a d h L™„IL u cYO a L™„MFIHdYOšLobed h L™„IL u cYOQS § µÔ´>—/‘?² ¨™ç »™V˜SUFV8]+KNRX`8˜8!T¹K8X˜F] Z OQ_SOnUnRT äIH تJLKSUFVXW TLOJ#TaV— Z O†!V i=1

i

ij

i

i

i j=1

n+1 ij

ii

k1 = hf (xn , yn ) k2 = hf (xn + αh, yn + βk1 )

»eV—NOJ#V

yn+1 = yn + γ1 k1 + γ2 k2 .

k1 = hf

n


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q– O R !T!M[]MNOJLKNM`šT!J#V=b Z K^W!O

i ]"w

k2 = hf + αh2 fx + βhk1 fy + O(h3 ) yn+1 = yn + (γ1 + γ2 )hf + γ2 αh2 fx + γ2 βh2 f fy + O(h3 ), 1 y(xn + h) = y(xn ) + hf + h2 (fx + f fy ) + O(h3 ). 2

)

γ1 + γ 2 = 1

O†˜SUKNJOJ#T#™!K^PzMAK ¤ OnUKN™š¬f˜šT‘`ibXT]V Z `NH+—NR+O )

αγ2 =

W V—NOJ#V

βγ2 = γ2 6= 0

γ1 = 1 − γ 2 ,

α=

1 2 1 2.

1 , 2γ2

β=

1 . 2γ2 

†[EDO­dHKLob'FIHO m E _DaKu _ dsFIcgN a }TLo| £ Î Å 'b F u _Tm~m HKX f F f OeH F m O Ka u QO P •

èeKNHRV!™TLJLKSUFVXW T

k1 = hf (xn , yn ) k2 = hf (xn + h, yn + k1 ) yn+1 = yn + 12 (k1 + k2 ),

J#VXW!O ê[_SOM^T!RT Y H Z KNM`XKN™TaJLKSUFVXW T

napaka : O(h3 ).

k1 = hf (xn , yn ) k2 = hf (xn + 12 h, yn + 12 k1 ) yn+1 = yn + k2 ,

£ Xc f F Î Å X Ku u oL cYOšw Ka u _ sd FIcgN a }DO‰b'F u _Tm OšH F m OšwNKF

napaka : O(h3 ).

k1 = hf (xn , yn ) k2 = hf (xn + 12 h, yn + 12 k1 ) k3 = hf (xn + 12 h, yn + 12 k2 ) k4 = hf (xn + h, yn + k3 ) yn+1 = yn + 16 (k1 + 2k2 + 2k3 + k4 ),

§ µÔ´>—/‘¸¿²?»e¨Ô˜ V—NÊ×OJ#ÖV ÕF˜SUFV8]+KNRX`8˜8!V ä Ø JLKSUFVXW VMAK ¤ HN`XKNJ#V IH

h = 0.1

napaka : O(h5 ).

¬

y 0 = −y − 5ex sin(x) y(0) = 1

k1 = hf (x0 , y0 ) = 0.1 ∗ f (0, 1) = −0.1

ORì™SbtKNJ=OJ#V


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q–

i]y

k2 = hf (x0 + 12 h, y0 + 12 k1 ) = 0.1 ∗ f (0.05, 0.95) = −0.12127

k3 = hf (x0 + 12 h, y0 + 12 k2 ) = 0.1 ∗ f (0.05, 0.93936) = −0.12021 k4 = hf (x0 + h, y0 + k3 ) = 0.1 ∗ f (0.1, 0.87979) = −0.14315 y1 = y0 + 16 (k1 + 2k2 + 2k3 + k4 ) = 0.87898. 

Ä Cs?BAG=QÉ ÅBC à Á ÅBC: à ;GCs:C _ _gh __ _¿h™_ h h _ m u _ hf _ _ m _ aKd u F€_ dsLoFIb%cgO"N ab }TL £ cY OÊÅ E b'N F u _Tm „MLBFIH c F m Oíw•}DO LoJ cF~cYO"dYLoO"JM}GHOgFrI XcYO"|O"b} _ O O"d LoETc FIcd}THKHLO uqO a O(N } O"_ b HO"} S÷_ b †FIcLob m HK X f L™OÊred dYO%Ew m EGFI|€} _ HO"}TLo| a } _ HO"} _ b S ^`_Tm_ Wc _ } _gu dHKL £ L™„ |YO"H ma _ c _ ETL6FI} aKu HO"d _gh Og„IL NzL m_ WLob _ ôY1EõþEõ



J

5

h y(x + 2h)

y(x)

2h

h

y(x + 2h) = y (1) + (2h)5 · C1 + O(h6 ) y(x + 2h) = y (2) + 2(h)5 · C2 + O(h6 )

Loc

y (2) − y (1) 15

NKF _T_m „MFI_ cYO'J(_ O h™ž _ }DO h c _ cYO"dYO"} _ y S Å _g_ u NKF ∆ EGF h_ Lo}?iDi HO"JMd _gh™_ ET_ Lob _ h 3rMFkNKF m_ E _gh Nb%a O(Nz|_ FIcÊ£ dYÅ O h O"|} _ h da E _ Nz_ Lob m S O­LoJMu HOgrIXc ∆ Loc y d H FIWXgNKFIb LoJMHOgrIXc E f ÔÈ weLoJMHOgrIXcF*J(O‰E O"}  TdHKETL3dYO F[d c ETL EDO"}THO É S ¼ _gh NK{OkNKF ðK8® Z —AKNMQ› V!™T¹JLKSUFVXW T }(NKFIHETJ(O"b'FIb _ ] aKu _ dsFIcgN a } _ £  Å b'F u _Tm_ ∆=

(1)

(1)

d _gu FIbãdYOšLoJ*L aKu Lo|

ki = hf (xn + αi h, yn +

i−1 X

βij kj ),

i = 1, . . . , 6,

a F aKu OMETLob _ b'F u _Tm_ H F m O y j=1

k1 , . . . , k 6

yn+1 = yn +

Loc€b'F u _Tm_ H F m šO w

6 X

γi ki

i=1

∗ yn+1 = yn +

6 X

γi∗ ki .

[… „MFIcYOšJ(O­cYO"dYO"} _ y KN Fd _gu FIb-}DO"H y − y = P (γ − γ )k S ^`_Tm H _ WcF*U _ HKb‰X h F*J(O £ Xc f F Î Å X uKu O Î FI| h WsFIH fG_ E _ 'b F u _Tm_'a _ i=1

∗ n+1

n+1

∗ n+1

6 i=1

i

∗ i

i

*

k1 = hf (xn , yn ) k2 = hf (xn + 14 h, yn + 14 k1 ) k3 = hf (xn + 38 h, yn + k4 = hf (xn + k5 = hf (xn + k6 = hf (xn + yn+1 = yn + ∗ yn+1 = yn +

3 9 32 k1 + 32 k2 ) 12 1932 7200 7296 13 h, yn + 2197 k1 − 2197 k2 + 2197 k3 ) 3680 845 h, yn + 439 216 k1 − 8k2 + 513 k3 − 4104 k4 ) 1 8 3544 1859 11 2 h, yn − 27 k1 + 2k2 − 2565 k3 + 4104 k4 − 40 k5 )

16 135 k1 25 216 k1

+ +

6656 28561 9 2 12825 k3 + 56430 k4 − 50 + 55 k6 , 1408 2197 1 napaka 2565 k3 + 4104 k4 − 5 k5 ,

napaka : O(h6 )

: O(h5 ).


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q– À/cYO aKh F m cgNKFIb-} _ HO"}TX~ETJ(O"b'FIb _ HO"JMbeLo}  }(NKFIH<NKF Ÿ F h NKFIcYO­cYO u O"crIc _DaKu Loc qh

q=

5fŸ6FÚ ½



i ]D]



h ∗ 2|yn+1 − yn+1 |

1/4

.

¿Ü=>  ­k  æ> ¾kkžDk < ß z¢` ­ks7@`z>Aã

< ` à

À a O"} _ FIc _ } _ HOgrIc _ b'F u _Tm_eh O"|} _ J(O"dL™{ FIb _ E _ W h Lo}TL }(NKFIH`NKF

œNHR!_SO `šTI]MNOM^T˜SU!T S ^ HOMETLob _  m O[NKFb'F u _Tm O !V!R˜NO†˜SUKNR\UnRT rMF*EGF h N O yn+1 = yn + hφ(xn , yn , h),

φ

lim φ(x, y, h) = f (x, y).

§ µÔ´>—/‘²?¨™è ð?HR!_SO `šTI]MNOM^T˜SU!T´bXTLJ#VXW!O ê[_SOM^T!RV Y H Z KNM`XKN™VaJLKSUFVXW V h→0

`XK

φ(xn , yn , h) = f (xn + h2 , yn + 12 hf (xn , yn ))

h 1 , yn + hf (xn , yn )) 2 2

¸7ëÎKSUFVXW Ti`XKzVXPSOnUnRV´!V!R˜NO†˜SUKNR\UnRT¸

yn+1 = yn + hf (xn +



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n+1

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-

x, y, h

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õ ™h h™_ _ YW O hh cY_O¡cYO"dYO"}D_O¡cL[m dH FIdfDH _Dh™_ aKu _ h }D_O"HíE a _gu O _ ™h _ D} m O h cLo|àcYO"dYO"}?kaKE h f m H _ WsFIaKb h dYO¡EGF h N O m u OLom b%O¡b'F u _Tm O J _ }DO c fDcYh™O"_ dYO"}h H F h™O _ k h WYO c cYO"dYO"} H F O k − 1 S¿nO F cgNzL Lo}TL dHKLo}DO"JMXgNKF O EDO dHKLob'FIHO d EGFIJ(OMEGF WYO cF*Loc }DO cF*cYO"dYO"}GFDS


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1 0.9

7

0.8

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5 0.6

4

0.5 0.4

3

0.3

2 0.2

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0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0

0.2

0.4

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0.6

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0.8

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1.4

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2

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P<Lobed h L™„IL u cF

W T!Jz˜NÕFë.V!H Z UFV!RV!™!KfœšV!MNJ=H Z K

=

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αi y((k − i)h) + hβi y ((k − i)h)



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y

d0 = d1 = · · · = dp = 0 6= dp+1 . y

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r!

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P

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yn − yn−2 = h3 (fn + 4fn−1 + fn−2 )

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1 3

P yn+1

4



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K yn+1

y 0 = f (x, y)

Z

[xn−k , xn+1 ]

xn+1

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yn+1 = yn−3 +

HX]V!M^T—NOJ#VÍR¬T (P )

y4

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0.4 3 (2

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¬

£ K¹O bSM^T PSHRT Z O

· (−3.7472) + 3.8870 + 2 · (−2.9631)) = −0.27115

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y4 = −0.27913

y5




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y 0 = y + e2x y(0) = 1 C 6= 0

m_ WLob _

x

2x

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y 0 = −y − e2x y(0) = 1 C 6= 0

y(x) = Ce−x + e−2x

C =0

dYO


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k X

αi yn−i = 0.

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ρ(ξ)

1

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k X

k

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j=1

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j

j

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£ FM{KXgNKFIb _ y 00 = f (x, y, y 0 )


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i RgZ

y(x0 ) = y0 y 0 (x0 ) = y00

§ _eh O"|} _ dH FIEGF m FIb _ cYO a L aKu FIb-FIcYOgrIW dHKEGF f OšH F m O

y 0 = p, y(x0 ) = y0 , p0 = f (x, y, p), p(x0 ) = y00 .

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y 00 = x + y 2

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yi+1

β

α a

b

ƒ‚ c u FIHKEDO h [a, b] FI}TETL m L aKu O"c u c _ HO"J m F h Lob m _ _TcYm O n+1_ m a F h™_ E a<u _ _%r }DakO"a beL xu = a, x m , . . . , x , x }(NKFIH<NKF x = x + ih Loc h = S … E FqO"dH } LobeLoHO"b Lob'F HKL™rIcLobeL L‡UVFIH FIc„pO"beL i

0

b−a n+1

0

yi0 = yi00 =

† _ WLob _'a L aKu FIb-FIcYOgrIW

1

n

n+1

=b



yi+1 − yi−1 + O(h2 ), 2h yi+1 − 2yi + yi−1 + O(h2 ). h2

−yi+1 + 2yi − yi−1 yi+1 − yi−1 − pi + q i yi = r i , h2 2h

d HKL6rMFIb'FI_DH aKNKu F y = α Loc y =_ β S u ¥>_TL maKu_[FIh bÄNKF _ h LocFp_ O"H FIc h Loc u _HKL mu LˆO m fG_ cYO h FI_ ccYJ(O u _e_gf fGO _ h O"|} _ H FMu{KLob _ cYOlaKdu _H FIdH _eu cYm OgrILo_Tc4m S<†[_Tm L‡UVFIH FIcrIc b'F O"|} Xd HO"W N O"b X LQdHKLQH WcLo|ed NzLo|cgEq}DO FIHKLo| cYO dYO(N X L E L@S dcYO"FdYE O"}Dm OšL‡UVH FIF H mFIcO „ILˆO h cL`FIcYcYOgO"rIb'WFL4aKcu F*_ cYO aKu _ dYS O ÀÞy d HKh O"Lob'|} FI_HKX Xd _ HO"WLob _ b'F u _Tm_ nlXb'FIH _ EDO a O(N`NKF[d _gu FIb O(h ) O(h ) 0

i = 1, . . . , n,

n+1

6

u O"} _'m_ WLob _ yi+1 − 2yi + yi−1 =

2

0

−y 00 (x) + q(x)y(x) = r(x) h2 (qn+1 yn+1 − rn+1 + 10qn yn − rn − qn−1 yn−1 + rn−1 ), 12

i = 1, . . . , n.


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q– §  µÔ´>—/‘²?©¨ ‘Øæä K ¤ HN`XKNJ#VLM^V—NR+O?]M^V— Z KNJ

i RDR

(1 + x2 )y 00 + 2xy 0 − x2 y = 1

zb W!O œXKNMAKNRPSRVLJLKSUFVXW VLOR »eV—NOJ#V´˜NO†˜SUKNJ¡KNRT Pt—bXT

h = 1/2

¸

y1 , y 2 , y 3

y(−1) = 0

y(1) = 0

¯

y2 − 2y1 + 0 y2 − 0 + 2(−0.5) − (−0.5)2 y1 = 1 2 (0.5) 2(0.5) y3 − 2y2 + y1 y3 − y 1 (1 + (0.0)2 ) + 2(0.0) − (0.0)2 y2 = 1 (0.5)2 2(0.5) 0 − 2y3 + y2 0 − y3 − (0.5)2 y3 = 1, (1 + (0.5)2 ) + 2(0.5) 2 (0.5) 2(0.5)

(1 + (−0.5)2 )

tOCOJ#T#MAK ¤ OnUKN™

¬

¬

y1 = −0.24 y2 = −0.365 y3 = 0.24

¸ 

d FLob%O"b _%a d h™_ {KcFH _ WcF*d _gfG_ NKF} _gu cdHpS α1 f (a) + β1 f 0 (a) = 0,

in α2 f (b) + β2 f 0 (b) = 0,

a L3d _ b%O f O"b _'au S L@ScYOMETL m FIJMcLobeL u _ r }DO"beL@S _ _ _gfG_ _Tm _Tm u _ _gu u _Tm _Tm _ a _¬a anlLob'dHpF S‚u HKL™rMrIF¿c _'Lob%m O"L‡UVb FIH FIcE÷„ _ H WcFIb0d NzX E E r }TL a [ d FIb O E O"dH } LobeLoHO"b y00 =

y1 − y−1 . 2h

nm_ _ EDOec_ FIJMcYO"c}DO y_ NKF h FqcYOMETL m FIJMcYOQS Å FIH[b _ HO u X m L`E u _ r }TL a EGF h N O u L m L‡UVFIH FIc„ILˆO h cYOíFIcYOgrIWYO WLob { F*FIcYOgrIW −1

−y1 + 2y0 − y−1 y1 − y−1 − p0 + q 0 y0 = r 0 2 h 2h y−1

oL J u FI| m EGFI| FIcYOgrIW~dYO h O"|} _ LoJ h™_ rILob _ S dEGF F h N d O_T mm E O[_ NKNzF[Lob H F_ m { O u FIETL h™ _h O"u |_ } r }?_ GJ*dXHKL™d rp_O"H}TO"XgW NKFI_ b FI_} aKum HOO"d W _g_ h b Og„I_*L NKmF_Wd L _Th LmW_ W_ghcNKL{ F u L daKHKu LoL4W dh LHKŸ L }G£ FDS_ b‰Å WsFIHBFIH dYfGO[_ ETJ(L4Ob'cYO"F dYu _TO"m } LF_  m_ WLob _ { FW _gh NK{ FhdHKLoW h L Ÿ }GF*E u L aKu Lo| u _ r }DO"|c}TL3cYO aKu _ dYO(N _eu O"} _ E f H _ WL4} _gu E%ácL m F h L u ETL@S 2

ôY1Eõô2CõÔ1 ÀÂÁ ǃ=Å Á Cs;Å=k; à ÿ[Å=[?; à ÿ[Ç Á Ï ^ HKL R?K Z OR?K^T!MNR?KNJM^V—NR?KNJ ]M^V— Z KNJ=HW!MNH›KQ› TLMAK^W T NKF cF h LocFpO"HKcYO‰U©Xc}G„IL N O Loc S ž O"HO m L3cF h LocFpO"H‘ c _DaKu u _TL`mc_ F‰b _mH _ FIb _ _ Xd _ h HO"WL u L<} _ a b‰aKWu LocYOg„IL NKF mh EGFI_ |ÊJ(OgrMF u cLo|ÊdH _ W h FIb _ EsS*jld _ HO"WLob _íh O"|} _•m L‡UVFIH FIcrIc _ b'F Y O WLob cF LocFpO"HKcL L FIbãJqEGF Lo} cFIJMcYO"c}DO"beL@S f

y

y0

n[F h LocFpO"HKcFqFIcYOgrIWsF*Lob%O(N _e_ W h Lo} _

yi−1 − 2yi + yi+1 =f h2



yi+1 − yi−1 x, yi , 2h



,

i = 1, . . . , n.


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(b, β)

yi = α + i(β − α)/n

È i ZQS t É

y 00 = f (x, y, y 0 ), y(a) = α,

£ FM{KL u IF EqNKF T_ m TE L a cYO T_ m ξ QdYO"HO"b'F u FIH ξ dYO[NKFd _gu H FIWc _ LoJMWHO u L u O"} _  m OšW _ y 0 (a) = ξ.

yi−1

yi

yi+1

y(b; ξ) = β

S

β

α a

b

d}DO F u FIm H Fâ_ á} c_ghLoL`HO"b'b F _ u _Tm_ J(O'H FM{ FIEDO"cgNKFqcF h LocFpd O"_gHKucFIFšbÜFIcYL™{ OgrMrIFIWsb FDSI_¿d u O"F‰} cdHp S m Ÿ OÊF h LoW b _ _ Xd _ HO"WL u L S¬u O"jlc d f _FIcHO"u Wc Lo_ b b'_¿F h u O"_T|m} _ _  E(ξ) := y(b; ξ) − β

d _gu FIbãd _gu H FIWXgNKFIb _'u X m L E (ξ) = 0

∂y(b;ξ) ∂ξ

SB†FâácLoHO"b _

y 00 = f (x, y, y 0 ),

m_ WLob _

ξ

z :=

E(ξ) = 0

∂y ∂ξ

Loc€J _Tm EDO(N O"cgNKFIb

y(a) = α, y 0 (a) = ξ

d FH FM{KXgNKFIb _'a L aKu FIbÚÈ i ZQS t É LocËÈ i ZQS w É d _gu FIb m_ WLob _eu X m L4ETH F m c _DaKul_Tm E _Tm O‰U©Xc}G„IL NKF Yô 1Eõô 2Cõþ ; Á É Á\Ä ÿCÐÅBC(ÉsCs;>=@C = Èg9": =l?; à ÿ[Ç Á Ï z 00 = fy (x, y, y 0 )z + fy0 (x, y, y 0 )z 0 ,

¦

z(a) = 0, z 0 (a) = 1.

E

S

È i ZQS w É

5

£ FM{ FIEDO"cgNKF[H _ WcF f OedH _ W h FIb%O −

d dx



p(x)

dy dx



+ q(x)y = f (x),

0 ≤ x ≤ b,

J­H _ WchLob%O%u d _ _gfGa _ NKFIb%O y(0) = 0 Loc y (b) + σy(b) m=_ 0 }(NK_ FIHNKF p(x)_g≥fG_ δ > 0  q(x) ≥ 0 Loc σ > 0 >_ NKF FI}TETLoEDO FIc c L }DO"cgNzX•U©Xc}G„IL NKF y ∈ C [0, b] }TL3J(O { rpOšH WcLob%Ošd NKFIb%OšLoc€beLocLobeLoJMLoHOšU©Xc}G„IL N 0

F (u) =

‚ m F›N O

Z

b

2

 p(x)u02 (x) + q(x)u2 (x) − 2f (x)u(x) dx + p(b)σu2 (b).

ä T Z NK O‚›®Õ ä OnU†bXV!™!KµJLKSUFVXWK NKF m OšcYO"b'F aKu _ 0

y(x) =

y ∈ C 2 [0, b] n X i=1

ci φi (x),

L™{ rMFIb _ dHKLoW h L Ÿ FI} _ W h Lo}GF


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q– u S L@S —^TtbSR?K œNHR!_SO `XK S ^ HKL £ OM¨ h FIL f | Î £ L u J _ ETL6b'F u _Tm L3L™{ rMFIb _ beLocLob‰Xb }(NKFIH a _

i Rg

φ1 , . . . , φ n

F (u) =

}(NKFIH`NKF

Z

b

0

 p(x)u02 (x) + q(x)u2 (x) − 2f (x)u(x) dx + p(b)σu2 (b), u(x) =

n X

ci φi (x).

À a LdYO"H „ILˆO h cL _Tm E _Tm Ld _ c b _ HO(N _ WL u L FIcYO"}TL 0 S §`O"} _‰m_ WLob ‰_ h oL cFpO"HKcL a L aKu FIb }(NKFIH`NKF Z i=1

i

b

J(O



c1 , . . . , cn

 p(x)φ0i (x)φ0j (x) + q(x)φi (x)φj (x) dx

aij =

Loc

Ax = b

0

Z

b

φi (x) dx. f (x)

… m LoJMWLoH F*WYO"JMcLo|~U©Xc}G„IL N`NKF T_ m ETL a c _ O h L4W _ea L aKu FIbãH FM{ h NzLoEíLoc€}DO"} _em_ WsFIHkW _em_ W h NKFIc~dHKLoW h L Ÿ FI}sS ëÎM^T!J=KSUFOQVXW!W R?T|KC!œNV!HRRPS!R+_SO†O `X®ÎK K Z KNJLKNR\UFV!™ NKF £ OM¨ h FIL f | £ L u JpFIEDO%b'F u _Tm Os}(NKFIH[J(O%WYO"JMcF­U©Xc}G„IL NKF‰LoJMWsFIH FIb _•u S L@S ]OÕ bi =

0

}(NKFIH`NKF

   

0, (x − xi−1 )/hi−1 , φi (x) = (xi+1 − x)/hi ,    0,

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i

i−1

0 ≤ x ≤ xi−1 , xi−1 < x ≤ xi , xi < x ≤ xi+1 , xi+1 < x ≤ b,

= hi−1

n

i

aii

=

ai,i+1

=

ai−1,i

=

bi

=

ann

=

bn

=

Z

xi

1

Z

xi+1

φ0i (x)

S

cFIcL™rMF h cYO h F'cYO€Loc u FIHKEDO h X

(xi−1 , xi+1 )

1 p(x)dx h2i Z xi+1 1 1 2 2 + (x − x ) q(x)dx + i = 1, . . . , n − 1 i−1 2 2 (xi+1 − x) q(x)dx, h h xi−1 i−1 xi i Z xi+1 Z xi+1 −1 1 i = 1, . . . , n − 1 2 p(x)dx + 2 (xi+1 − x)(x − xi )q(x)dx, h h x x i Z xi i Z ixi+1 i −1 1 p(x)dx + i = 2, . . . , n 2 2 (xi+1 − x)(x − xi−1 )q(x)dx, h h xi−1 i−1 xi i Z xi Z xi+1 1 1 (x − xi−1 )f (x)dx + (xi+1 − x)f (x)dx, i = 1, . . . , n − 1 hi xi−1 hi−1 xi Z xn Z xn 1 1 p(x)dx + (x − xn−1 )2 q(x)dx + p(b)σ 2 2 h h xn−1 n−1 xn−1 n−1 Z xn 1 (x − xn−1 )f (x)dx xn−1 hn−1 2 xi−1 hi−1 Z xi

p(x)dx +

xi

S


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i vD\

\[

Z

Àüa ÆÊ_ O uKh O"Wu XÊ_TmLob%_ O"b u _ _ J(O'H u FM_T{ mFIED_šO"hcgNKF­H _ _ WcLo|¿dH _ _ W h FI_ b _ EÇcYO%_ E h _gh N _ b'F u _Tm_  _ }TL`X_ d _ h HO"W h N O%} _gh™_ }DOt „IL N } b'F S ¥ 'b F O"|} H FM{KXgNKFIb H WcF*dH W FIb'F}TLQNzLo|€J(O"dL™{ FIb E W Lo}TL Š µ

d y(x) = f (x, y(x), p), dx

… q

g(y(a), y(b), p) = 0.

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… q

j T viu Š µ

T

Q0_`

… q]z|,T

Q0_`ba |iŠhq Q0_`baJj T v g `

gf

‚ T

a T fhgiT `:jo

µ'u

T

Q0_`

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®

áŠhq Q0_`

ju

«S

j T v g `

Šhq Q0_`

] ‚UIad‚ Š

o

®

gf

Š µ hg ` gf ‚'u

j T v g `

g `

gf Q

u

gf

Š µ hg `

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 q ^] Ia Š

] cro

ad`roa |Ug `

gf Qho

«g `

®

T fhgiT `:j

Š µ

j Af

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Š µ

u(−1) = 0,

u(1) = 0.

… q

y10 (x) = y2 (x),

Ê«T=bXT P8KSUnR+O?]MNOF— Z O £ K8´™SbXT!JLKNJ#V OR c˜SUnMAKAbSRT´!VXW T#bXT#MAK ¤ OnUKN™ž]M^V— Z KNJ#TL™zë.TU Z T—NHµ`XK ¯                    

y20 (x) = (y1 (x) − 1)(1 + y2 (x)2 )3/2 , √ √ y1 (x) = 1 − x2 y2 (x) = −x/(0.1 + 1 − x2 )

ª

Q0_` qfhgiT `ú‚ T

‚ T

µ

u

yá‚

Q0_` qfhgiT ` jŒu

S

]Jm ow

yá‚U

u

j g `

S

jŒu

v TAf]S

+

v0‹,





gf

Š µ

u

Š µ hg `

S

]Jm o0º m o0º

] Ò<4 m oo {

w

o

w

{

v0‹, g ` gf] cro

] ebn k Ñ j è0Sf] k Ñ chº m oo

gf] v g `:j

… q]z|8+ 

jnÃcba

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v0‹, S ] ‚UIad‚ Š

] k owx‚ Š] k o

+

] cbad‚ro

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y

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

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ju

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‚

+

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v0‹,Ia

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|8+

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

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a k

a m e:oa

a

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ow

w

v0‹, g `

gf

ow

¸

gf Q


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q–

ivi 

§ µÔ´>—/‘²?¨©‘˜ ¦ ¤ P8KNJ#V Z T˜SUnRV#™XMAK^W!RV˜SU ¬?]MNOE!TUKNMNO OJ#T#M^V—NR+O]M^V— Z KNJ R?K=KNR+KNOQRLP8K ]Z RVA› VV‘`8MA¬[K ¤ R!On]UKNMX™!¸ ¸LØ=KNM Z T® !¸7V­¿™šVX˜šW TV!V—NRUFVLT!!VVUfMAK™ž¤ ]OnUMAKNK™e` ]RXV!`XJ=KNJRV bA£ › OJ#K^W!VaHȞKNR]VT ZPt`N—^H+Va—NR+˜F]OJ¡MAKNJL˜8!KNT R+Z T!OJ#M`XVLKNJz™z¬C˜N]O†˜SVUUnKNMAJéKX—NHNKN`XRKNT J#PtV— ¤ Z ]¤ MN™!KQ› TaMAK^W T ¯ Ê«T=bXT P8KSUnR+O?]MNOF— Z O £ K8´™SbXT!JLKNJ#V OR ¸ c˜SUnMAKAbSRT´!VXW T#bXT#MAK ¤ OnUKN™ž]M^V— Z KNJ#TL™zë.TU Z T—NHµ`XK ¯         λ

y 00 (x) + (λ − 10 cos(2x))y(x) = 0,

y 0 (π) = 0,

y 0 (0) = 0

y(0) = 1

y10 (x) = y2 (x),

y20 (x) = (10 cos(2x) − λ)y1 (x).

y(x) = cos(4x)

λ = 15

ª

Q0_`

qfhgiT `ú‚ T

‚ T

µ

Q0_`

u

y

qfhgiT `

jŒu

S

Q0_` ‚

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‚

] m ow J

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uxx uxy uyy

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xy

xx

xy

yy

x

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q = −x sin α + y cos α,

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Aupp + Cuqq + Eup + Guq + ru = F (p, q),

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A = a cos2 α + b sin α cos α + c sin α2 C = a sin2 α − b sin α cos α + c cos α2 E = e cos α + g sin α G = g cos α − e sin α.

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0 < x < 1,

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t>0 u(0, t) = g0 (t),

u(1, t) = g1 (t).


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ui,j+1 = λui−1,j + (1 − 2λ)uij + λui+1,j ,

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λ=

1 6

i = 1, . . . , n.

S

1 2

τij



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õ ™h _ YW O h cYOšcYO"dYO"}DO[NKF

eij = uij − u(xi , tj )

S ž O"HO m L`FI} a d h L™„IL u cF a |FIb'FEGF h N O

ei,j+1 = λei−1,j + (1 − 2λ)eij + λei+1,j − ∆tτij .




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Ej = maxi |eij |

}DO"Hd _ b'FIcL E

n

≤ E0 + tn T

T = max |τij |

i vDv

S ž O"HO m L λ ≤ 1/2 EGF h N O

Ej+1 ≤ Ej + ∆tT,

Loc aKh F m L

§ _ _ Yd O f d _ b'FIcLF m m O f H FícYO"dYO"}DOÇdH _gu L o B} _Çf H F aKu O ∆x Loc ∆t dH _gu Ll\QS Å FIHeEGF h N O } c>EGFIH IF c„pOeH F O O(∆x ) S ÑB´Tָ̑QÆÐ ¨ˆ² ¶ ]?UnOJ#T Z RT#O bt—NOM^T=bXT¹K8X˜F] Z OQ_SOnUnRV#˜8®KNJ#V7`XK λ = ¸ |U (xi , tj ) − uij | ≤ E0 + tj O(∆t + ∆x2 ).

∆t = λ∆x2

NKF

2

»eV!Ttbt¸ d F*ETH F m c _DaKu L

1 6



ui,j+1 ui+1,j

Loc

ui−1,j

HO"JMETL NKFIb _ E•§`OM¨ h™_ H©NKFIE _ ETH aKu _'_ }TH _gf

uij

 m_ WLob _

∆t2 (utt )ij + O(∆t3 ) 2 ∆x2 ∆x3 ∆x4 = uij ± ∆x(ux )ij + (uxx )ij ± (uxxx )ij + (uxxxx )ij 2 6 24 5 ∆x ± (uxxxxx )ij + O(∆x6 ). 120

ui,j+1 = uij + ∆t(ut )ij + ui±1,j

£ FM{KXgNKFIb _em L‡UVFIH FIc„ILˆO h c _ FIcYOgrIW _

ut = uxx

 _Tm•u _Tm dYO aKh F m L

utt = uxxxx

 a O(N`NKF

utt = uxxt = utxx = uxxxx .

ž O h™_ }DO h c _ cYO"dYO"} _ FI} a d h L™„IL u cF a |FIb'F u _ H F›N EGF h N O~ÈÔXd _ { u FIEDO"b _ ui,j+1 − (λui−1,j

d F•LoJMWsFIH S FIb _

λ =

O(∆x4 )

1 6

É

λ = ∆t/∆x2   ∆t ∆x2 + (1 − 2λ)uij + λui+1,j ) = utt ∆t − + O(∆x6 ). 2 6

a FíU@O"} u _ HedHKL

∆x4

XcL™rILkLoc¬cYO"dYO"}DO%NKF•H F m O

^ HKL λ = m _ W Lob _ FI} a d h L™„IL u c _'a |FIb _

O(∆x6 )

1 6

1 ui,j+1 = (ui−1,j + 4uij + ui+1,j ). 6

ô\þEõÔ1EõÔ1 Ï÷?[ǃ= =VA ÅBC×Ï Á A ÃCÄ C N

65

ui,j+1 − uij ui−1,j+1 − 2ui,j+1 + ui+1,j+1 = . ∆t (∆x)2

ui,j+1

ui-1,j+1 ∆x

∆t uij

ui+1,j+1

 a L™„MFIH'dYOÇWLkWL h OÇH F m O


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q– † _ WLob _eu HKL m LˆO fG_ cYO h cL a L aKu FIbãJ(OšETH F m c _DaKu L u

i,j+1

i vD

cYOšcYO aKh F m cgNKFIbrpO a _ ETcFIb-cLoE _ NzX

−λui−1,j+1 + (1 + 2λ)ui,j+1 − λui+1,j+1 = uij ,

SG¹"´>ò÷‘QÐƨ™Ð ¦8Ji] Z OQ_SOnUnRT´˜8®KNJ#T´!V!R+™!KNMQ›!OM^TLbXT#™š˜šT ¸ »eV!Ttbt¸ x _ }DO h cYOšcYO"dYO"}DOšLobed h L™„IL u cF a |FIb'F NKF

i = 1, . . . , n.

λ



   ui,j − ui,j−1 ui−1,j − 2uij + ui+1,j − uxx (xi , tj ) − ut (xi , tj ) − ∆t ∆x2 1 1 ∆tutt (xi , ξ) + ∆2x uxxxx (η, tj ). 2 12

τij = =

õ h™_ WYO h cYOšcYO"dYO"}DO[NKF …lJMcYOgrILob _

eij = uij − u(xi , tj )

S ž O"HO m L6Lobed h L™„IL u cF a |FIb'F*EGF h N O

(1 + 2λ)ei,j = ei,j−1 + λ(ei−1,j + ei−1,j ) − ∆tτij . Ej = maxi |eij |

}DO"Hd _ b'FIcL E

Loc

T = max |τij |

 _Tm•u _Tm

S † _ WLob _

(1 + 2λ)|ei,j | ≤ Ej−1 + 2λEj + ∆tT, j

≤ Ej−1 + ∆tT

E n ≤ E0 + tn T

Loc

|U (xi , tj ) − uij | ≤ E0 + tj O(∆t + ∆x2 ).

^ HKLLobed h L™„IL u cLlb'F u _Tm L a L u _ H F›N h O"|} _ dHKLoE _ { rILob _ EGFMr›NKF~rpO a _ ETcF€} _ HO"}GF b _ HO"b _ dYOÊJ(O u _ EËE a Ot }G_ FIb} _ HO"}TXÇH FM{KL u L h u HKL u m LˆO fG_ cYO u h _Tcm L a L aKu FIb~SaKu Å FIH[m dYOqNKF a L aK_Du aKFIub u _gHKLu m LˆO fGa _ cYh O h FIu cc f O h u O"_T|m } _ H FM{KLoh b _ E O(n) u _ m dsHO FIŸ HN OgO „IÉ L N S O"|ÊLocÊLobed L™„IL cYOíb'F OíLob%O%L L<H F J(O"|FIETc L<} FI} d L™„IL cYO%b'F OÊÈ NKF F­J(O'U@O"} H ¥TFIEGF m _ O¡} _ c>h EG_ FIH f FIc„pm OÌdHKLd _gh NzXWcFIb h λ _ cFÇd _ b'FIcLF m_ O h O"|_ } _ E¬dH_ O"} a L[Lou JMWsFIH FIa b _ d _gh NzX_ Wc _ f EGF h Lo} λ SÌn HKb%O c NKF  OÇdHKLkEGFMr›NzLo| λ O"|} dHKL™rpO"}TXgNKFIb _ WHOMETcYOMEDO"b _'h F a L u XYOg„IL N _ } _ef H F aKu O ∆x Loc ∆t _ WYOšdH _gu EGL FM0r›N S cYO"dYO"} H FM{KL EGF O(N­dHKLk} c>EGFIH FIc„IL

ô\þEõÔ1Eõþ Ô ;GCÆÅB: À = à ǃ9 à Šà ÉsCÓÏ Á A ÃCÄ C WV

65

À J(O"b'FIb _ d _ ETdH FMr›NKF*Lobed h L™„IL u cFqLoc~FI} a d h L™„IL u cF*b'F u _Tm FDP ui,j+1 − uij 1 = ∆t 2



ui−1,j − 2uij + ui+1,j ui−1,j+1 − 2ui,j+1 + ui+1,j+1 + 2 (∆x) (∆x)2

ui,j+1

ui-1,j+1 ∆x ui-1,j

ui+1,j+1

∆t uij

ui+1,j



.


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q– ¥>dsF ukm_ WLob _eu HKL m LˆO fG_ cYO h cL a L aKu FIbãJ(O­ETH F m c _DaKu L 

i D\

ui,j+1 i = 1, . . . , n

P

−λui−1,j+1 + 2(1 + λ)ui,j+1 − λui+1,j+1 = λui−1,j + 2(1 − λ)uij + λui+1,j .

`§ O"} u __Tm } _gu Lobem d h L™„IL u cYO a |FIb%a O m u X m L m  HO"c}ƒnlL™„ _gh™a _ c _ EDO€b'F u _Tm O•aKu } _ _ c>EGFIH f LoHO~J(O•E a O"} λ S ^ a H F h m c u _DaKu‰u h F b'F h Fu NKF Oku NK_TFlm cYO"dYO"u }D_eO h F O(_ NH F _ O O((∆x) cYO"b'F dHKL3FI} d L™„IL cL6O L Lobed L™„IL cL4b'F LFJ(O O"|} Xd HO"WLob _ EGFMr›Nz+L4rp(∆t) O a _ ETc)L4} _ HO"}sS O((∆x) + ∆t) 2

2

2

sC ;GC?ÿ à ǃ=÷ÅBC¬?Cs; =@CÆǍÅBC Ä = Á ; Á Å =@CÆǍÅBC Á ÅBCÆ÷ÿCÓÉÌÉ Á ÷ Ä =Ï Á Å ý = È"CX d FLob%O"b _ cdHpS u = u + u , (x, y) ∈ Ω = [0, 1] × [0, 1], t > 0 u g _ G f _ _ _gfG_ NzL u(x, y, t) = g(x, y, t) J(O (x, y) ∈ ∂Ω pN _ y, 0) = f (x, y) Loc H WcLobeLsd hJ O"J(|Og} rM_ F d c_TLob m_ Wd c _ } NK_gFIub E•u(x, m _ a h FIcL Lob'FIcJML NzL6H FM{KXgNKFIb J‰FI} d L™„IL u c _ Loc€Lobed h L™„IL u c _ b'F u _Tm_ S cdHpSFI} a d h L™„IL u cYOšb'F u _Tm OQP

ô\þEõÔ1Eõ



¦

5

t

Ã

xx

Y5

yy

ui,j,k+1 − ui,j,k ui−1,j,k − 2uijk + ui+1,j,k ui,j−1,k − 2uijk + ui,j+1,k = + . ∆t ∆x2 ∆y 2

ui,j,k+1 ∆t ∆x ui-1,j,k

ui,j+1,k ∆y ui+1,j,k

uijk ui,j-1,k

5 ½µ6½ ž O­J fDh F m

G

A>^¾‰ >A@`­±¾š?¢«> k ­±àz> M <s z¢«> k ­ ­7@ܚ zä

K Z O ]?UnOQPSR?K dYO"H „ILˆO h cF m L‡UVFIH FIc„ILˆO h cFqFIcYOgrIWsF*W _ b _ ETJpF h L ­¿V!O†˜8˜šV!RV!™V¹KNRT Pt—^V uxx + uyy = f

J(O u(x, y) cYO _ Wb _ r›NzX Ω = [a, b] × [c, d] J H _ WcLobd _gfG_ NKFIb u(x, y) = g(x, y) J(O (x, y) ∈ ∂Ω S ^ HKL m L‡UVFIH FIcrIcLkb'F u _Tm L _ Wb _ r›NKF Ω dH FI}THKL NKFIb _a dHOME _ } _gu c _ beH F Ÿp_  }(NKFIHeLoc u FIHKEDO h O [a, b] Loc [c, d] F£I}TETL m L aKu O"c _ u c _ HO"J m _ F h Lob _~a[u _ r }DO"beL x = a, x , . . . , x , x = b Loc y = c, y , . . . , y , y = d S O"JMbeLo}DO JMcYOgrILob J ∆x Loc ∆y S 0

1

n

n+1

0

1

m

m+1


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q–

ii

d

c

a

\ô þEõþEõô Á AgA à ÷G: à ɠÅBC9 Á Ï÷C d!V!F ™XRm HKV#X ˜8f ®OšKNJ#dYO"V H P „ILˆO h cYO _Tm E _Tm Ošd _ Loc

b

5

¦

x

y

O"dH _ } a LobeLoHO"b _%aa Lob'F u HKL™rIcLobeL m L‡UVFIH FIc„pO"beLF m_ WLob _

]+KSUQUFVXPSÕ

ui,j+1 ∆y

∆x

uij

ui-1,j

ui+1,j

ui,j-1

JqFIcYOgrIWYO"beL ui−1,j + uij + ui+1,j ui,j−1 + uij + ui,j+1 + = fij . 2 ∆x ∆y 2

Û _ecYOgaKurIWsFšJ(OíE a F u a F aKu OMETmLob_ _ E _ h LocFpO"HKcL a L aKu FIb3}TL<Lob%OíW h™_ rIc _€u HKL m LˆO fG_ cYO h c _ _ W h Lo} _ S­ÀædHKLob'FIHKXc } O ∆x Loc ∆y FIcYO"}DO WLob   T −I S S S    −I S T S A= S S S S −I  ,  ij

}(NKFIH`NKF

−I

SSS SSS SSS

4

−1

  −1 T = 

4

T

  . −1  −1 4

n O"dY_eO"m}DO_ dsF uKu _ _ r } _ ETcFO"dH _ } a a Lob%aKu Og„IL NKF NKF O(∆x +h ∆ym ) TFIcYO"} _ dYO u X m LYJ(O*c>Xb'FIHKL™rIc _ O"dH _ } a Lob%Og„IL N _  }TLTN WLob J*H FM{ FIEDO"cgNKFIb L FIb%O Au = b QEGF N O O[NKFcYO"dYO"}DO O(∆x + ∆y ) S 2

2

2

2


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q–

i DZ

ž O•H FM{ FIEDO"cgNKF a L aKu FIb%O€Xd _ HO"WLob _ b'F u _Tm F'J(O€HO"JMdH { FIcF%b%O u HKLo}GF a O(N*Lob%O€b%O u HKLo}DO~EÊE a O"}TLBETH aKu L™„IL cYO(NzEGFMr y cFIcL™rMF h cLo|ÇF h FIb'FIc u _ EsS

ô\þEõþEõÔ1 Ç Ã ÿCÆǍÅBCÓÅBCs?Cs:C Y

x _ }DO h cYOšcYO"dYO"}DOšdsF uKu _ r } _ ETcF‰O"dH _ } a Lob%Og„IL NKF NKF ∆h u(x, y) =

ÀBF h N O J(O

|α|, |β| < 1

1 (uxxxx (x + α∆x)∆x2 + uyyyy (x, y + β∆y)∆y 2 ) 12

|τ (x, y)| ≤

ž O Df h™_ YW O h c _ cYO"dYO"} _ d g_ u IF b-EGF h N O

uxxxx

i,j

|u(xi , yi ) − uij | ≤

Loc

uyyyy

J

Mx4

Loc

My4

 m_ WLob _

1 (M 4 ∆x2 + My4 ∆y 2 ). 12 x

max |u(xi , yi ) − uij | ≤

u _ H F›N

}(NKFIH<NKF

u(x − ∆x, y) + u(x, y) + u(x + ∆x, y) ∆x2 u(x, y − ∆y) + u(x, y) + u(x, y + ∆y) + . ∆y 2

S[d F _ MJ cYOgrILob _ J fG_ HKcgNzL4b'F›NzL4J(O τ (x, y) =

τ (x, y) := ∆u(x, y) − ∆h u(x, y)

(xn+1 − x0 )2 max |τ (xi , yj )|, i,j 2

(xn+1 − x0 )2 (Mx4 ∆x2 + My4 ∆y 2 ). 24

d‚ƒJMF•}DO LoŸ b%F O"a b F _ m Od _ h O"FI|c }_D_*aKu u OMX E m h NKL FIuc¬X}DdO(HKN Lob'XFId H _ HO"WLob _ £ L™„ |YO"H ma  _ cm__ E W_ Lob FI}_ aKucYHOÊO"d } _g_ h cOg„I„IXL N _ Sfd FLoJMHOgrIXcYO"b _ dHKLoW h L Ÿ FI} S J(OeO"cYO _gh L u u L™rIc _ _H FM{KL u FIEíE u _ r }TL FIc}THO u J*beH F Ÿp_ EGF h Lo} _DaKu L  m HKX f L™r*dYO a ácF›NK{ _ beH F Ÿp_ EGF h Lo} _DaKu L  d FIb-W |u(xi , yi ) − uij | = O(h2 )

∆x = ∆y = h

(x, y)

h

h/2

4u(x, y)h/2 − u(x, y)h 3

_ m _DaKu E u _ r }TL (x, y) S•¥TFIEGF m O u _ FI} aKu HO"d _gh Og„IL N _€h O"|} _

{ FšW _gm h NK{KL _'da HKLoW h L _ Ÿ FI}Êu H aKF u m O O(h ) J(O•dHOME ETH F c u

_ K a

u cYO"H F Lob O"b E L Lo| r }DO"|c}TL3cYO _ dYO(N _eu O"} _ EíbeH F Ÿ L h } _guku X m L4EíbeH F Ÿ L h/2 S 4

;>=QÉB=k; à ÿ à ÉB= ÀÞdHKLob'FI_HKXcY} m _'_ f Wb _ r›NKF Ω_ ca L4dHOME u _ } _T_gm u c_TLo}?m Yd _ JMcYO"b _ dYOšETH F m c _DaKu L6cYOšH _ WX JMHOMEGFIc~H WYO HKX OgrMF­O"dH } LobeLoHO L E FDS

ô\þEõþEõþ

O

∂Ω

b _ HO"b _ E u _ r }DO"|


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q–

it

2 B A 3

0

1

4

n O"b'F aKu _šu _ r } 1 Loc 2 E•FIcYOgrIWL4E u _ r }TL 0 Xd _ HO"WLob _'u _ r }TL A Loc B Q}TL aKu O­cYO­H _ WX ∂Ω S nO(N W _ HO"J m O h N Ošb'F m 0 Loc A FIcYO"}DO θ h  0 < θ < 1 SB‚ƒJ*HO"JME _ NKFIEíE•§`OM¨ h™_ H©NKFIE _ ETH aKu _eaKh F m L@P ∂u0 ∂x 2 ∂ u0 ∂x2

= =



1

1

 1 1 1 − θ1 θ uA − u3 + O(h2 ), u0 − h θ1 (1 + θ1 ) θ1 1+θ   1 2 2 2 uA − u0 + u3 + O(h). h2 θ1 (1 + θ1 ) θ1 1 + θ1

^`_Tm_ Wc _ cYO"H F m Lob _ J(O Loc S ^ O"JML u LBb _ HO"b _€u X m L cYO u _  m O a F m O(NJ(O•O"dH _ } a Lob%Og„IL N _~m HKX f F f O _Tm E _Tm OíEGF h N O m O‰NKF­cYO"dYO"}DOíH F m O _gu dHKL a Lob'F u HKL™rIcL m L‡UVFIH FIc„ILEšc _gu HO"cgNzLo| u _ r }DO"|4S ž O"HO m L u F f O h O"|} _ dHKLYH FM{ FIEDO"cgNzX O(h) LocícF O(h ) } dH _ W h FI_b a _ E m a }Th HKLoETLobä H _ W a _ b u d_ HKL™rpO"}TXgNKFIb _ _ EGFMr›NKm FÇcYO"dYO"}GFÇLožc÷cYO"m b'F u aKu f _ } _ crIcFÇcYO"dY_ O"}GFÇH _ F m O _ O(h u ) _ Lo£ b%O"b ma F_ O(_ N O"|} aKu E¡E_gh FI| r }DO"|¬cYO"dYO"} H F O O(h) S O"HO L F OÇcdHpS cF€b H FIb Xd HO"WL L L™„ |YO"H c EGF‰FI} HO"d Og„IL NKFDS ∂u0 ∂y

∂ 2 u0 ∂y 2

2

2

5 ½µ6FÚ

ã

>^¾í <ÜíkA>A@`­¡¾š?¢«> k ­±àz> M <s z¢«> k ­Ú ­7@ܚ zä

ž fDh F m J(O­|LodsFIHKW _gh L™rIc _ dYO"H „ILˆO h c _%m L‡UVFIH FIc„ILˆO h c _ FIcYOgrIW _ NKFEDO h™_ ETcYOeFIcYOgrIWYOšJ(O

u(x, t)

J J(OgrMF u cLobeLYd _gfG_ NzL u(x, 0) = f (x)  u (x, 0) = g(x) Loc H _ W cLobeLd _gfG_ NzL u(0, t) = u(1, t) = 0, t > 0 S ‚ƒc u FIHKEDO h [0,a 1] FI}TETL m L aKu O"c u c _ a _ HO"J m F h Lob _í_ au _ r }DO"beL x = 0, x , . . . , x , x = 1 }(NKFIH<NKF ∆x =  ∆t dYO[NKF dH FIb'FIb‰WYOecYOšrpO ETcFIbcLoE NzX4S utt = α2 uxx ,

0 < x < 1,

t>0

t

0

0

1

n

1

n+1

1 n+1


gihXj7kClnm8oXpqmArtsSutvwExzy!{|mAj^}n~Ar?mi{|mtpqhmz€‚qƒn„†…F‡ ˆQ‰NŠc‹ŒNŽƒ‘‘„’A‰q„“N”^”^•Q– ô\þEõ  õô ú =Ï Á AD;>=÷Å ÁÂÄ = Á ; Á Å Á d FXd _ HO"WLob _%a Lob'F u HKL™rIc _'m L‡UVFIH FIc„ _ J(O Loc  m_ WLob _ 

Ã

Y5

utt

d F _ JMcYOgrILob _

i "w

uxx

ui,j−1 + uij + ui,j+1 ui−1,j + uij + ui+1,j − α2 = 0, ∆t2 ∆x2



i = 1, . . . , n j = 1, 2, . . . λ=

S

m_ WLob _

α∆t ∆x

ui,j+1 = 2(1 − λ2 )uij + λ2 (ui+1,j + ui−1,j ) − ui,j−1 .

d F NKF

 uj+1 = 

SS

u1,j+1 un,j+1

  

Qd _gu FIb-EGF h N O 

  A= 

uj+1 = Auj − uj−1

2(1 − λ2 ) λ2

λ2

SSS

2(1 − λ2 )

Q}(NKFIH<NKF

SSS SSS λ2

 λ2 2(1 − λ2 )

  . 

ddH F'F›NKX{Kcgd Nz_ LoH| O"Wm hEGN FIO"|Êb rp_ÊO a a Lo_ b'ETcF Lou |ÊHKL™rIccLoEF _ m NKFIL‡UVE FIH FIc„MLoFc  d _gu FISlb¢§6J(F ŸO€OMLoEDJMO HOga rIFqXdcÌ_ ETN OMH ETF mL6c cY_DO'aKu J(L OgrMF u }TXcd }_gu_ H FId W_ XgJMNKcYFIO"b b _ _%ETh H F F m c Y_DaKcu F L dYO u X m L u J(O­LoJMHOgrIXc u S nlLoE _ u LoJMHOgrIXcYO"b _%a d _ b _ r›N _ J(OgrMF u cF f Ošd _gfG_ N O uj

1

ui,1 = (1 − λ2 )fi +

}(NKFIH`NKF f = f (x ) Loc g = g(x ) S ‚ƒJMdsF h N OMEDOQP u(x , t ) = u(x , 0) + u (x , 0) + i

i

i



i

1

ut (xi , 0) = g(xi ) utt (xi , 0) = α2 uxx (xi , 0)

S _ HKb‰X h ONKF

ut (x, 0) = g(x) Y*

λ2 (fi+1 + fi−1 ) + ∆tgi , 2



i

i

t

dYO'O"dH _ } a LobeLoHO"b _ J

i

RZ

x _ }DO h cYOšcYO"dYO"}DO­U _ HKb‰X h F NKF

+

∆t3 6 uttt (xi , µ)

S

S

Y5

O(∆x2 + ∆t2 )

τij =

∆t2 2 utt (xi , 0)

fi−1 −2fi +fi+1 ∆x2

ô\þEõ  õÔ1 ú ADC?ÿ =ÇÅ à 9(A : à ÅÉ Á ; Á Å QC 

u0

2

1

uxx (xi , 0)

uj+1

uj−1

_ JMLoH _ b%OecYO u O"crIcF›NKF

 1 ∆t2 utttt (xi , µ)) − α2 ∆t2 uxxxx (η, tj ) . 12

Û } a d h L™„IL u cYOšb'F u _Tm O[NKF aKu O"WL h cYOšdHKL3d _gfG_ NzX λ ≤ 1 S ^`_Tm_ Wc _ } _gu dHKLsdYO"HO"W _gh L™rIcLo| ^ †Û u X m LsJ(O‰|LodsFIHKW _gh L™rIcF ^ †Û _ W aKu O(N O(N _ Lobed h L™„IL u cFb'F u _Tm FT}TL a _ WH FIJMd _gfG_ Nzc _'aKu O"WL h cFDS


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iy

V

È ipt S ] É

}(NKFIH a _ a, b, c, f U©Xc}G„IL NKF x, y, u, u oL c u  u FIHkEGF h N O b − 4ac > 0 S d FdH F m d _DaKu OMETLob _  m OlNKF u cYO _ Wb _ r›NzX Ω JMEGFIJMcYO­U©Xc}G„IL N O x Loc y Qd _gu FIb-J(O m L‡UVFIH FIc„ILˆO h EGF h N O ∂u ∂u È ipt S R É du = dx + dy. auxx + buxy + cuyy = f,

x

2

y

∂x

∂y

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a

in



dy dx

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+c =0

m_gh™_ rpO u O m EGF m HKX Ÿ LocL

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a dy d(ux ) − f dx dy + c dx d(uy ) = 0. C1

Loc

C2

S<À J m_ghoŸ }DO"HO"} u FIHKL aKu Lo}

C1

Loc

a λ d(ux ) − f dy + c d(uy ) = 0,

a µ d(ux ) − f dy + c d(uy ) = 0.

C2

EGF h N O

È ipt S v É È ipt S  É

C1 S C2

Q P

Γ

ž OgrIcFIb _ J‰ETH F m c _DaKu L u  u Loc u cYO'}THKLoETX h NzL Γ Y}TL6cL`}DO"HO"} u FIHKL aKu Lo}DOQS[nO'cgNKF›NkLoJMWsFIH FIb _•m EGF u _ r }TL _ m _DaKu aKh m u _ _em_ _ ipt ipt ipt P Loc Q S ‚ƒJ*FIcYOgrIWËÈ S R É 6È S v É Loc¡È S  É LoJMHOgrIXcYO"b _} gu dH F a FMrIL™{ rMF*}DO"HO"} u FIHKL aKu Lo} C Loc C Q}TL f H F aKu O a } _ JML PETH F Loc c Q S L3E%cYO F cgNzL r }TL S Q}TLTN WLob x

y

1

2


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i S ž gO rIcFIb _ cYOÇ}THKLoETX h NzL Γ B}TL cLk}DO"HO"} u FIHKL aKu Lo}DOÇLoccYOÇ}DO u FIHKLd _ JMcYO"b _ ETH F m c _DaKu L u  u Loc u ÈÔW T_ m L a L[LoJÇ u m J(OgrMF u cLo|½d _gfG_ NKFIE÷O h L[dYO¡LoJÇdH F›NK{KcgNzLo|÷HOgrIXc _ E É S nO}THKLoETX h NzL Γ LoJMWsFIH FIb _Ëu _ r }GF P, Q, R, L S ZQSk‚ƒJ'È ipt S R É d _ u L™{ rMFIaKbu _ } _T_ H m LocYO u a F x_ Loc yaKu cYO aKh F m cgNzLo| u u _ _ r } S(x , y )  T (x , y )  . . . T}TL a _ dH F a FN rIL™{ rpO­}DO"HO"} FIHKL Lo} C Loc C } JML4X H FIJMcF*dYO"H F r }ícYO Γ S t Sk‚ƒJ¿È ipt S v É Loc/È ipt S  É LoJMHOgrIXcYO"b _ ETH F m c _DaKu L u Loc u E u _ r }DO"| S, T, . . .  d _gu FIbÜdYO LoJÈ ipt S R É LoJMHOgrIXcYO"b _ ETH F m c _DaKu L u S wS ^`_ c _ ETLob _ dH F›NK{KcgNKF} _ HO"}GFqJ*c _ ETLobeL u _ r }DO"beL4Loc~c _ E _ }THKLoETX h N _ Γ Q}TL f H F a } _ JMcgNKFDS x

S

1

S

T

y

T

2

x

y

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S

S

1

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Z

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µdx.

Q

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λ

S

S

S

P

S

Q

x S

y S

S

P

S

Q

S

S

P

S

Q

S

S

S

S

S

S


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žO

(ux )S

Loc

(uy )S

EGF h N O

Z

S

a λ d(ux ) −

P

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a µ d(ux ) −

Z

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f dy +

P

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S

f dy +

Z

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Q

c d(uy ) = 0.

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Q

(aS λS + aP λP ) [(ux )S − (ux )P ] + (cS + cP ) [(uy )S − (uy )P ] − (fS + fP )(yS − yP ) = 0,

(aS µS + aQ µQ ) [(ux )S − (ux )Q ] + (cS + cQ ) [(uy )S − (uy )Q ] − (fS + fQ )(yS − yQ ) = 0.

µY–Y¹g¶ˆÕp´TÖÖ´GÕp–3—`´ òQ¸¹"¸òsÕp´>¹g¶ÔÓMÕ(¶ÔòA\‰òY–Y¹"¸òл ž Loc u F f HKLoHO"cgNKFIb È ipt S R É ETJ m_ghoŸ'^ ¥ m_ WLob _ R du = R dHOMETL h™_  m_ WLob _ [

S P

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uS − u P = uS



(ux )S + (ux )P 2



(xS − xP ) +

S P



ux dx +

RS P

uy dy.

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h O"|} _'m_ WLob _eu X m L4J*Loc u F f HKLoHO"cgNKFIbETJ m_ghoŸ

QS



È ipt S i Z É È ipt S ipt É

d F­Xd _ HO"WLob _•u HO"dsFIJMc _

}(NKFIH m_ WLob _

(yS − yP ).

È ipt S i w É

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(ux )S + (ux )Q 2



(xS − xQ ) +



(uy )S + (uy )Q 2



(yS − yQ ).

]

S

S

S

S

S

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δΩ

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S C1 P

Γ

S

x S

y S


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2

C1

1

S

x S

5 ½µ6qÝ

L

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k

1

k i=1 i

1

k i=1 i i

k

i

n

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ci L(ui )(xj ) = f (xj ),

j = 1, . . . , k.

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5 ½µ6ó

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∂ + ∂y



∂u q(x, y) ∂y



+ r(x, y)u(x, y) = f (x, y),


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2

}(NKFIH Ka u O

1

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Loc

ϕ1

ϕ2

∂u ∂u (x, y) cos ϕ1 + q(x, y) (x, y) cos ϕ2 + g1 (x, y)u(x, y) = g2 (x, y), ∂x ∂x

} _gu OšJMXcYO"cgNKF*c _ HKb%O h F*E u _ r }TL (x, y) ∈ S S ϕ2

S

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ϕ1

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p, q q(x, y) > 0 r(x, y) ≤ 0 I(w) =

1

ZZ

ZZ

" #     1 ∂w 2 ∂w 2 2 p(x, y) + q(x, y) − r(x, y)ω dxdy 2 ∂x ∂x Z



 1 2 −g2 (x, y)w + g1 (x, y)w dS 2

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f (x, y)wdxdy +

S2

1

y S

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x∗

RGS = −(U ∗ + D)−1 U

Loc m _ W Lob _

λ=−

x∗ U x . x∗ U ∗ x + x∗ Dx

σ := x∗ Dx =

n X i=1

x∗ U ∗ x = α − βi

λ=−

Loc u _ Yd Ošd _ b'FIcLF m OlNKF |λ| < 1 S § µÔ´>—/‘T×c¨©‘ Ê«T´˜NO†˜SUKNJ

aii |xi |2 > 0

α + βi := x∗ U x,

S † _ WLob _

Å FIH<NKF A d _ JML u LoETc _em FâácL u cYONKF

RGS

−U x = (U ∗ + D)λx.

¥TF m O(N m FâácLoHO"b _ Loc u _ H F›N

(λ, x)

|λ|2 =

α + βi σ + α − βi

α2 + β 2 . (σ + α)2 + β 2

x∗ Ax = x∗ U ∗ x + x∗ Dx + x∗ U x = 2α + σ > 0

Loc

(σ + α)2 = σ(σ + 2α) + α2 > α2 ,

12x1 − 3x2 + x3 = 10 −x1 + 9x2 + 2x3 = 10 x1 − x2 + 10x3 = 10

­[O R=MNObXT P8T KS_AUnVR+—NOO `X]KNMN™XOFOf— Z JLO £ K8KS UFVXW!O[W V—NOJ#V O bSM^T PSHRT!J#VzW!™T!V!M^T!T¿]V T _AV—NO `XKN™XOcOR T!H˜8˜NÕ NK OQW Z V!™XOÆJLKSUFVXW!OF¸ x(0) = [1 0 1]T

Ê

Ê

]MNO T!H˜8˜NÕ KNOQW Z V!™XOCJLKSUFVXW!O?]T ¬

x(1)

)

Ë\VXP8KNR.MAKAbSH Z UFTU`XK

x(1) x b = [1 1 1]T

¸

 0.75 =  1 , 0.9 

 0.75 =  0.9722..  , 1.022..

x(2)

 1.00833.. =  0.99444..  , 1.025

x(2)

 0.991203.. =  0.994084..  , 1.0002881

¬

)


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Z iDi

½LßZ

^ HKL3b'F u _Tm L ¥… £ HOgrIXcYO"b _ (r+1)SOR

  (r) (r+1)GS (r) = x k + ω xk − xk ,

}(NKFIH£ NKF ω H F h O"} a aa Og„IL N amQ}Th™_L dYO"HO"b'u _TF u m FIH 4J(Oí}DO u FIH F f O a F­LoJM}DO Ÿ F m Oíb _ HO•WL u L ¥… }DO"H[õ*O"X z¥TFIL EDOšb'F OQS † _ WLob _ xk

(r+1) xk

=

(r) xk

1 +ω akk

bk −

k−1 X

(r+1) aki xi

i=1

n X

(r) aki xi

i=k

!

0<ω<2

,

S ^ HKL

ω=1

NKF

k = 1, . . . , n.

[õ H F*J(Oed _Da dsFM{Ku L u FIE õ*O"X a a z¥TFILh mQŸ h™_ EGF‰b'F u _Tm FYL m hF›N Ÿ OedYONKFu  m Oe_gW u _ rMFNKFqJ(O"d _ H F m NKFqb __ c _gm u _ c _ hou J(OšdHKL b'_ FIHKc_gLsh dYO"HO"b'F h ŸFIH ω > 1 dHKLoW L FI} x u h W L NKF[u H FMŸ {KL _'ETL_ } xu QrMFldYO‰J(O"d H F NKF*O FIHKcLoHO  W W NK{KLsdHKLoW L FI}•dHKL 0 < ω < 1 S…ld Lob%O cL ω NKF F } „MFIcL L@S À/b%O u HKL™rIcL _ W h Lo}TL4EGF h N O (r+1)SOR

(r+1)

xk

_ JMLoH _ b%O

1 (r) = xk + ω akk

bk −

k−1 X i=1

(r+1)GS

(r+1)

aki xi

n X

(r)

aki xi

i=k

!

,

k = 1, . . . , n,

  x(r+1) = x(r) + ωD −1 b − Lx(r+1) − (U + D)x(r)

Loc }DO"Hd _ b'FIcL

(ωL + D)x(r+1) = (D − ω(D + U )) x(r) + ωb, RSOR = (ωL + D)−1 ((1 − ω)D − ωU ).

¸ SG¹"´>ò÷‘T×c¨ˆØ ¶ ä R?KJ#V!MAK!V!R+™!KNMQ›!OM^TUnO bXTI]V Z `NH+—NR+OEbXT P8KSUnR+O ™!K8UFV!Me™7]MNOJLKNMNH »eV!m Ttbt¸fG_ h h u S NKF a fGd _ _Tm cgNKF u u HKLo} _g_gu u cYO­b%O u u HKLo}DOT}TLLob%O cYO m LˆO fG_ cYO h L F h FIb'FIc u F  cYO LˆO cYO L`F FIb'FIc F   S… m€u _Tm€aKh F dYm LOqNKFqJ HKcgN O HKLo} cYO%b%O HKLo}DOs }TL6Lob%O )

ω 6∈ (0, 2)

RSOR = (ωL + D)−1 ((1 − ω)D − ωU ) (ωL + D)−1 a−1 i = 1, . . . , n (1 − ω)D − ωU ii (1 − ω)aii i = 1, . . . , n

Å IF HNKF m F u FIHKbeLocYO"c u O¡FIcYO"}DOÇdH _Tm X} u X h O aKu cLo|¬ETH F m c _DaKu LFEGF h N O ρ(R d _gGf _ N J(O­} _ c>EGFIH f FIc„ _ NKF ω ∈ (0, 2) S SG¹"´>ò÷‘T×c¨™çº© KI`XK A ®KNMNJ=OnU˜8!T|]VtbSOnUnO™XRVÂWKê R+OnUnRTÎJ#TUnMNO†!T¬7]VUKNJ det RSOR = (1 − ω)n .

! V!R+™!KNMQ›!OM^T#bXTI]V Z `NH+—NR+OEbXT P8KSUnR+O ™!K8UFV!MX¸ »eV!Ttbt¸ ^`_Tm_ WsFIc m_ }DO"JMX€J(O%õ*O"X a a z¥TFIL m F h™_ E _ b'F u _Tm_ S

SOR )

≥ |1 − ω|

¶ ä ™#]MNOJLKNMNH )

Loc¬d _gu H FIWsFIc ω ∈ (0, 2)


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A = L+D+U

1 C(α) = −D −1 ( L + αU ) α

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0/

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A11 A21

A12 , A22

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A22

A

T

  −I A=  

  , −I  T

T

−I

4

−1

  −1 T = 

A

−I

  , −1  4

4

−1

A

µ = ρ(RJ )

ρ(RGS ) = µ2 ωopt =

¢

2 p 1 + 1 − µ2

ρ(RSOR (ω)) =

§ µÔ´>—/‘T×c¨™Ð Ê«T#J#TUnMNO†!V ¬

ρ(RSOR (ωopt )) = ωopt − 1

(

ω − 1, q

1 − ω + 12 ω 2 µ2 + ωµ 

4  −1   A=  −1  

¬

−1 4 −1 −1

−1 4

−1

−1 4 −1

ρ(RJ ) = 0.6036 ρ(RGS ) = 0.3634 ωopt = 1.1128

1 − ω + 14 ω 2 µ2 ,

−1 −1 4 −1

¬C›!M^T‘œ

  −1     −1 

za ωopt ≤ ω < 2

za 0 < ω ≤ ωopt .

™!K Z `šT

4

ρ(RSOR (ω))

]Ti`XK


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Z ipt

1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

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0.2

0.4

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ë ` à­ í¾ àe¾­sk  s D [

Ï

0.8

1

1.2

1.4

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2

?>   D 

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0/

Kk (A, b) = Lin(b, Ab, . . . , Ak−1 b).

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k

k

T

k 2

k

k

k

k A−1

T

k

T

ô  õÔ5Eõô ;Å Ã Ç Ä = È Á ÉÿCÆÇ Ã ;>=VA Á Ï u _gfG_ h _Tm_ _DaKu m ö a a fG_ _•_ h _ fG…l_ H cYO h cYOcY O cYO%dYdOšJ fG_ WHKcgc N O ö cYO%F a aH FIF c>WsX}GFIH „IfGL N _ O EDO _ cYW Oh Lo}DOQFS FIc>WsFIH E W Lo} NKF U

J

Ó

A

H

£ O m 3L WL3LoJMHOgrIXcYO h L h FdHKETLo| k Ka u _gh sd „MFIE Q Loc H S ‚ƒJ ¥ a }DO h O"HKcLob-bec _gŸ FIcgNKFIb a



k

T

k

k

k

1

k

k

k

k

Aqj =

AQ = HQ

j+1 X

 m_ WLob _

hij qi .

hij = qiT Aqj

hj+1,j qj+1 = Aqj −

j X i=1

}(NKFIHBNKF

m_ WLob _

i=1

qi i = 1, . . . , j

A = QT HQ

hij qi .

S<ÀBFI} u _ H

qj+1

m_ WLob _ LoJ

Q

_ H u _ 


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q1 = b/kbk2 j = 1, 2, . . . , k z = Aqj i = 1, . . . , j hij = qiT z z = z − hij qi hj+1,j = kzk2 hj+1,j = 0 qj+1 = z/hj+1,j

^ H FI}TLocLFYrMF NKF S ÀÂO hofG_ HKL u b‰X ha F a }THKLoED_ Oíb _Tm L‡áY„ILoHO"cYOÇõ[HO"b=z¥T„ |beL mQu _ EDO _ H u _gfG_ cYO h LoJ(Og„IL N OQSíÝ hofG_ HKL u FIb a Fš} _ crpO€dHKL LoJMWHO"cFIb k O L3dYO} NKF h = 0 S ¹"—<¶ˆÕp´G”à‘Tc× ¨©‘Y‘ = MNRV Z W!O `XKN™"T Z › V!MNOnUKNJ ˜XK Z T® !VÎO bS™T‘`šTÂW V j = k ¬µ`XKNMI`XK k = dim K (A, b) ¸(Ê«T j+1,j

å

!™ K Z `šT `XKNMC`XK ™VXW!O Z RT

n

j = 1, . . . , k

]VXW!J#TUnMNO†!T C¬ ˜SUFV Z ]_SO ]T´˜šV ¶ Gª—^TtbXT´bXT ¸ ž O"| u FIETc _DaKu ÝlHKc _gh™m L NKFIEGF f OO hofG_ HKL u b%O NKF bec _gŸ FIcgN6Jb%O u HKLo} _ Loc J(O _ _DaKu O h FBJ(O m FIEGFDÆ c _gŸ FIcgNKF u _ g _ h _ u _ _ h e _ u m a J*b%O HKLo} NKFJ(Ošd c b%O HKLo}  J(O­HO"JMdH { FIc dYOlNKF O"|} X L O"b S AQj = Qj Hj + hj+1,j [0 · · · 0 qj+1 ],

Hj

j×j

H

Qj = [q1 · · · qj ]

O(k 2 n)

k

O(n2 )

^`_ k } _ HO"}TLo| ÝlHKc _gh™m L NKFIEGF*b'F u _Tm F*LoJ



QTk AQk H = Q AQ = QTu AQk T

Kj (A, b)

QTk AQu QTu AQu





Hk = Huk

 Hku , Hu

d _ JMcYO"b _ H Loc H Q}TL3Lob%O a O"b'FcL™r h F*Loc~J fG_ HO(N m F a c _ h S ÀBF h N O AQ = Q He }(NKFIH He m_ WLob _eu O"} _  m O H m_Tm O"b _ { FETH Ka u ™L „ _ k

uk

j

j+1

O(n)

k+1,k

j

j

j

[0 · · · 0 hj+1,j ]

ô  õÔ5EõÔ1 ÆC Å ý à 9 Á sÉ CÏ Á A ÃCÄ C dd _ FBFINKc F _DaKu OMa ETLob'L@S F ^ u HKL™L™{ rIFIcYb O_ " NKF u X m L Loc a Lob'F u HKL™rIcYO u _ H F›N u HKL m LˆO fG_ cYO h cYOšLoc€ÝlHKc _gh™m L NKFIE•O hofG_ HKL u FIb U

’

Y5

A

H

H=T

¥TF m O(N LoJ

AQ = QT

m_ WLob _

α1  β1  T =  

β1 α2

S S S βS S S

2

βn−2

SSS αn−1 βn−1

βn−1 an

Aqj = βj−1 qj−1 + αj qj + βj qj−1 ,

}(NKFIH`NKF α = q Aq S À a F a }TXdYO(N h O"|} _ J(O"dL™{ FIb _ E _ W h Lo}TL4x6O"c„IJ _Da FIEGFqb'F u _Tm F j

S

T j

j

  .  

a F _ WsrIX u c _


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q1 = b/kbk2 β0 = 0 q0 = 0 j = 1, 2, . . . , k z = Aqj αj = qjT z z = z − αj qj − βj−1 qj−1 βj = kzk2 βj = 0 qj+1 = z/βj

^ H FI}TLocLFYrMF NKF

Z ipy

S

>¥ dsF u rMF NKF dim K (A, b) = k  m_ WLob _ ^`_ k } _ HO"}TLo| x6O"c„IJ _Da FIEGFqb'F u _Tm F*LoJ

βk = 0

n

S 

Tk T = Q AQ = Tuk T

T Tuk Tu



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d _ JMcYO"b _

Tk

Loc

Y—V

U

Tuk

βk

V

xk Ax = b krk k2 = kb − Axk k2 .

x0 + Q k yk ek Qk H

y ∈ Rk

x0

xk

xk =

q1 = r0 /kr0 k2

e k yk k2 = kkr0 k2 e1 − H e k yk k2 . kb − A(x0 + Qk yk )k2 = kr0 − AQk yk k2 = kr0 − Qk+1 H

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0

1

0

0

0 2

k

k+1

0 2 1

k

0

k

k k 2

k k

5CÚe6ó Û  à­ ­A>z  <s : s

‚ u hoFIf HO u LoETc _ H aKFMh™{ _ FIEDO"cgNKF h LocaKFpu O"HKcLo| a _gL aKh u _ FIb _ E f NKF _ y WHOMETcYOMEDO"c _la } _ HO(N3E[E a FI|*XrIa WsFIcLo}TLo|qc>Xu _Tb'm FIHKL™_TrIcm F h Lo_DcaKFpu _O"HKc_ F ÅO HKL FIh™W_ EDH FDOQS`S À À u XgNzEGL FIh cL u { FIrIHLocO u L XHKL Ola cYF O[Z E XN }TED}TO"cgH©N NzO‰L LLoJM} h NzXLoc rIc v _šDh F[dHKJ*LLrMu FIFIb'HO FIu HBLoETdcHKLoEDbeOlL4cb'FFE u _TFImWO"XgbeNKF L3b'J(O F h LocFpd O"HKcF dH a L aKu FIb'H FE  a d h OgrpO­dYO a Fd _gfDh F m O u L u X m L4E ]  R   Loc i w S  ¸

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