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AJ R) (+ L-9&(# LbB+ I&+,- 3(+ 0*+,-.0 '=[>! J+ C)T) '&(< ([#W .7 61++ c(/S9Id ! e # J . a(7I! $I&) ^(E%0, U&I)E) C),- ./+=9- I,- `,GR Y9I! . (+R J/9&P+,- J/GH J+ a(7I! O0< $+):) .>9I+,- 3(+ 0*+,(A `,G,

C,(),- C7/7I),- UE9,- `,G J/6/ . $0M) C),- ./+=9- I,- (minimizes) +# J+ a(7IW-(error growth) ,(6 (H9/_ J+ 9&>! J R) (H&,- )K/67).

A C),- .0B+W- 5*6 ;+ a(7If, .20)I+,- e- #W- g- JG\6 c9&# C0/ (+/([/0< ^0V),- ./2/R " ([%1 ).

A U//M) K(M):- CA hI0)/ Q&&*,- $/0%),((evaluation) N8()# ^(E%, K97'(7*+ ./&&< 3(#(/6 J+ .6 07+ ./&&< => Q&&*,- $/0%),- $*>/ -GH " U0*,- J+ L-a

O+E/ QG,- D/&%,-"3(+ 0*+,- $/V:) "(information processing) 3(#(/6,- D/% ",- '(7*+(given data) .0I&+,- 3(+ 0*+,- $B+)(input information) N8()#,- "

.>9I+,- 3(+ 0*+,- $B+) .6 07+,-(output information) UE9,- `,G $B+/ (+R "C,(),- C7/7I),-:

WKš«b�« �U�uKF*«

unput information

WO�“—«u)«

Algorithm

Włd<« �U�uKF*«

output information

WKšb*« ¡UDš_«

unput errors

WO�“—«u)« ¡UDš√

Algorithm errors

Włd<« ¡UDš_«

output errors

1ÍœbF�« qOK×²�«Ë W¹œbF�« ‚dD�«

W¹œbF�« �UÐU�(« ∫‰Ë_« qBH�«

!"# (i) 1234567 $%&'()* +, #- %).#.

(ii) 854211 /%&'()* +, #- %010.

(iii) 3.1415926 / +, #- %2*3%&#45.

(iv) 4.265 /6&&#45 6&* #.

Types of Errors ¡UDš_« Ÿ«u½√

Round-off Errors V¹dI²�« ¡UDš√ ∫ÎôË√

7(6- 8# 0x: 95:'.;* 9 (exact number)

x: %&.&#<= %*& (approximation) 99)>/ 0x

? 6@%&'()*% +(,* (absolute error) A(B. CD#E)F&:

!xx "#$% 0

-*% +(,*%./01 (relative error) % 1)/,. CD#E)F&:

xr"$%$%

%2#$3!* (Rounding)

4#556 7*8 9) 556 2:$;3 (n) <#-1 )*% =9>$?% @) (significant digits /

figures / positions) . + # G;'*/H 6&*& +, #$H G&*I J)In K? + #/H ,*- L MH#7N G;'*/H + #n O9PH' O9P'. 9HQF& 6- '- #&&R= 6'9 'S ,*T U#=F& 6- ,*V A(@?(one

unit) G:=<*/H WQI/H 6,T HXV ,* !2P(truncated part) CN( 6* #.T- '- J -9P'O G;'*/H K?n . 6@? O9P' CN(/ :.;/,. M,&',2* G:=<*/H WQI/H HXS 6,T HX YH'

G;'*/H K? + #/Hn 9PH' O9P'. 9HQ& 6- 6T*& '- O- #&&R= 6'9 U#=& 6)9HQ& O9,5'.(

' %.2(/,.* 55 * 2#$3!'n <#$A *% =9>$?% @) (decimals) : 9). +, #$H 6@? + # G;'*/Hn – MH#,7N- ,Z>T J)I= K=/H'– %:,2.. CXP=.

) B9C(1-1):

)i( 1235000

(ii) 854000 (iii) 3.14159

(iv) 4.26 !4.27

"#! $%& 0x: '( )*+ ,,

x: -./)/%01 .+/ 23

: 456(+34 7(83

/ #9 :;0% &&$'x (' )"*+n #,#-. /0 ) 2/< !n =/>? @%A' B-%( 4C9#;D:

n!"# 105.0

<#;D 4C9 EFG+:

4631.2

4671.20

2104.0004.0 !"$$

!#! @: 2105.0 !"%

#H< I3;13;) x #//%A' #/+-%3 =/>?.

!J";D 4C9 ;+ x = 2.4620

12 10051.01051.00051.0 !! "$"$$

! #! @x -%3 =/>?,>4 @%A' B.

##! x.

#+ EFD #! $%&"a, b #+ # D+ ,,'n B-%.

> K%*34 :?;ab L6' E;+ +' @ 1>/2n B-%.

34 M%;8 .+N0a/b , ,>+ %/O ,,' L6' E;+ +' @ 1>/.

< ;P"Q) R%P- 4CH EFD L39 .+N034 K%*34 I1S/1" #+n I6/ ;+D ;+T3 ;"U+% V B-%:

4K%*34 ./6+W3 .)R%P0+34 .S/1"3: b ×a

4.+N034 ./6+W3 .)R%P0+34 .S/1"3: b ÷a

" #! $%&x X3 =/>? ,,'n @%A' B-%.

! ,,W34 B;-%x #+ %)D! Y,> 34 Z/> [*4 +34 :1>1 I134n!10 12320#4 536 7"89$3)\4 #! .]>F+ [+,W1 _ ./`4,1)_4 %;&?.(

<,,W34 EFG+:

& 400000.0001234.0

<FG V .>/>? ./%A' B;-%! .N+8 2/./ "W+ B;-%! .G.

),,W34 ;+"/: 000006.0001234.0 &

<#;/ "W+ #;+-% .>/>? ./%A' B;-%! .W)%! 2/.

37:*;:

'D< I(W/ .>/>?34 ./%AW34 B;-%\4 ,, %;+"/) 56(+34 7(834 BS> ! .+/- #' Y

I)N"34 7(834 .+/- #' Y%D< I(W/ ./ "W+34 B;-%\4 ,,'.

*<' ="#> '% ?2@A4 &8&7@"6B'% C2"<3$

Round-off error bounds for elementary operations

" X3 ,4,'\4 Ka%P0" ;""! $%&n "! V B-%2 ,4,'\;) (0< =N+/x 50>1 I134(%A34: 11

W¹œbF�« �UÐU�(« ∫‰Ë_« qBH�«1

<#H: nabba !"#!" 105.0

nbaba !"#!' 105.0/

< 4C9 EFG+! ;"*%<#: n = 2, a = 0.56, b = 0.65

<#H: 36.0,364.0 $"$ baab 2105.0004.0 !"%$!" abba

4="#> '% 7D" 9 7"B26* ?2@A:

+ L6' ;"+N- BG ;+ b,,' I< ."/W+ .+/- ;")%* 4C9 ;""! E;/*;/% B 6W+34 # ,,W34 2N&" EFG+< V ./6?\4 .+/034 JS1" V:

( ) 76.006.006.067.0 $'"

D3./6+' :D I< K/%0134 c4%S9 [+ # ,- [+ EFG+< V ./6?\4 .+/034 ;","' d1"1 _A' #/+-%3 K/%0134 %I6/ ;+ .0);N34 .-FW34 ;"/(W1 V #//:

(0.76 × 0.06) ÷ 0.06 = 0.05 ÷ 0.06 = 0.83

* 0.76

/ e K/%0134 #' d1;"34 7(834 # D: |Total round-off error| = 0.83 – 0.76 = 0.07

'%7"B26*'% C2"<3$'% =" # 8 ="#> :

,- ./);N>34 J;/6+W34 K/1%1 %Gf/ (order of arith. operations) - + [ K/%0134– 9./`;T"34 .S/1"34 I< %//g1 L3.

<#! E;/*;/% B 6W+34 #+ EFG+:

(a × b) × c = a × (b × c)

3 4 I< ;+D V#;<%(34 h ;N1/ _ ,- K/%0134 [+ #D :;G+334 #/+-%3 Ka%0" Z/> I3;1#//%A':

(0.56 × 0.65) × 0.54 = 0.36 × 0.54 = 0.19

0.56 × (0.65 × 0.54) = 0.56 × 0.35 = 0.20

* 0.19

E4C%#"F 3 G&. HI 7'%& HI HB69 J@A #B

Max. relative error in a function of several variables

"&#! $% f I< .34,n %/g1+: nxxx ,,, 21 '

!#! @: ( )nxxxff ,,, 21 '+

c;(8\4 #! $%&" Y,%&+34 (individual errors) L6' Ie J4%/g1+34 iCe I<

K/1%134:

nxxx ,,, ,...,, 21

< .34,34 I< I6D34 7(834 # D/f - ;T- < ;+ ./";G34 .)1%34 , ,> :;+e9 [+– e:

n,--,-,$,

....... 21

n,--,-,$,/

....... 21max 21

I)N" 7(8 L?-! # D/ I3;13;)(m.r.e. + max. relative error) e :

n,--,-,$

.,...... 21

3 K2L(2-1):

!J4%/g1+34 .3_,) #/1/3;134 #/134,34 #+ :D I< I)N" 7(8 %)D! ,S :

nxxx ,...,, 21 ;T`;(8! :nxxx ,,, ,...,, 21.

...... 21

---$ '

nxxxf ,--,-,$, .1....1.1 21max

,--,-,$

,,,,---$ ......

W¹œbF�« �UÐU�(« ∫‰Ë_« qBH�«1

37:*;:

/ .+/- I< I6W&34 7(834 #! E;+ +' ]>Ff L?-\4 ,>34 #+ %/GD) :-! # D/ 7(863 c;(8\4 #\ j3C – ./6+W34 ./>;"34 #+– E;*W) ;T*W) Ig6/ #\ 2S11

)W)34 KS + ;T*W) Z/>$ K3;N %8k4 .( l/*! 4C9 EFG+<20.000 Ka%- V,,' L39 ,,' :D4 @ ;N/ 7(863 L?-\4 ,>34 #H< V./%A' B;-%!000,20105.0 4 "" !

@ ;N/ @!1 I6D34 7(834 ;+"/) VY%/)D .+/- iCe (total error) Y,;' # D/ [- 1+34, ,> I< 0.005.

3 K2L(3-1):

!7(8 %)D! .+/- ,S .34,34 I< I)N"f Z/>:

xyzyxff

9#! B6' 4C:

004.01

0005.03

003.02

,!,-,$

105.0004.0180005.012003.09max

$"-"-"$,f

00583333.018105.0

(8./`;T" _ ./6+W) ./T1"+ ./6+' :4,)1N4 #' mA;"34 7(834 e n;(1-_4 7

(infinite process 2 finite process)V .3,;W+ B4,81N;) ./6*;&1 .3,;W+ .6+;D+D

Truncation Errors ŸUD²�ô« ¡UDš√ ∫ÎUO½UŁ

n +S+) 2)/%01) , ,>+ :+;D1 K;N> V5 %<) J;<%>"+ i;)A! J;>;N+ n +S+D, ,>+34 :+;D134 ;T6G+/ I134 .>;N+34 E;)/%01 @ ;N/.(

3 K2L(4-1):

>4 %&34 .3,;W+ :4,)1N4 #' mA;"34 n;(1-_4 7(8 KN5 ( )1.01 $,$

,- xxx

)./6*;&134 .3,;W+3;:

3 .+/- K;N>y ,"'x = 0.2 #! B6' 4C9y=1 ;+,"'x=0.

) ( !"#$%& '(&:

! 02.112

2.02.0

#"$"$"

cxyxdxdy

) ( )**+& '(&:

01.11.01.011.0

"+#"#"

#"-#"#

xyxxyyiiiii

",-!./0 1!2 3"45 6&-.&-#: Truncation error = 1.02 – 1.01 = 0.01

" 758 *.#0 9-:-5#& 6; <-!2= 6> ?@>(initial data) 6.& 9-:-5#&-4 A/ 3% BCD% -E5FG 'H(:<75F%+% 7#AI. 6; JKEI= L+# 9 .

Initial Errors WOz«b²Ðô« ¡UDš_« ∫ÎU¦�UŁ

W¹œbF�« �UÐU�(« ∫‰Ë_« qBH�«1

G9 "!2 J*G 6; 75**+& 75#-M(& 75F%+& N.. J*- .% J"!2 3O LAP: ?@> 35F%G 6> 9 "!2& 7 %MQ& 7 z = x/y.

"3O LAP:yx .., 35!"#$%& 35**+F& 3-#5AQ. -%> x, y R5.A.& SFG T5(# U3O:

#; 7!"#$%& 7%5Q& 3% B0*z :75#5AQ. 7%5/ RM( z. 5(T:

! ! 1#%

* 2#2#"

()* ##"

21,1###"

erroroffroundyx

roundedyxz

Propagation of Errors ¡UDš_« b�«uð Ë√ —UA²½«

* 2##"

* 2#%&'

O6; 1!2& 3O )z 3% J*& ".% <-!2O 3% 3"4%x , y *5*I 1!2 3%"R5AQ.& 7I5.:.

@V 758 *.#0 9-:-5#& 6; 7!5M# 9 A5W. 9*O (initial data) JA5#4 9 A5W. S&V 3V '-Q5; U758-E:& X8-.:& 6;% &'#()% *+"+, *)-./)(ill conditioned problem).

: 7& *& 7%5/ R-M( *5A: -::O LAPf(x) T5(:

:x 6Q5Q( **G.

:f Q5Q( 7& *57.

)i( :3O LAPx. : 6#M: **G(rational number) **+F& R5AQ. ">x Y:O T5(3% 68-E: 0 **+# * *GO 35K2. Z5!.M5 RM-( *I"5 0 N-/A= 75A[+& .

3O )Oxx 2. : 7%5/ 6; 68 *.#0 1!2& ">x.

3"45 6&-.&-#"01%2 34% -&5)% 7& *& 7%5/ 6; '#-Q%& f ) *& ".%& 1!2& ">" 7%5/ 6; 68 *.#0 1!2& 3G X.-:& x:(

! !xfxf 2."11

)ii( : 3O LAP1f :& *& 3% !M#O 7& * 6> 7f T5(#O R\AQ. -E:f 6!+. )O7%5Q& 75#5AQ. 7%5/ f ] 3"4. J*-G1f 7+!.Q% ]"^/ 7F^MFM.% (truncated

power series) 7& *& _"4P% 3% (expansion of f) f [.

" 3"45 6&-.&-#5 67& 84% -& '#-Q%&:

! !xfxf .2."1 12

)iii( : 3O LAP !xf .2 :> RM-(&-# 7#"M(%& 7%5Q& 6) 9-#5AQ. < AI`# )O

roundings ;75#-M(& 9-5F%+& 6.(

#" 3"45 6&-.&-&59"#: )% - ) "!2& ">1 9-#5AQ.& 3% *& ".%&:(

! !xfxf .2."1 123

"6F5 -%5; a#M -% b52F. -::4%5:

! ! ! ! ! ! ! ! ! !xfxfxfxf

xfxfxfxf

xfxfxx

.2."1$.$.

.2."1$.$

"6F4& 1!2& 3"45 6&-.&-#:

! ! 3212 1#1#"12.1" xfxf

/;7&5<% =/ *>+ 5/)% 6%'?<% @"A' ) B7C

W¹œbF�« �UÐU�(« ∫‰Ë_« qBH�«1

/ B7C(5-1):

; RM(:M -::O LA3/1e _"4P% 3% !Q; S&"= *"*( 7M%2& N *2.M-#7& *& xe .-M(& '4 3O LA; "#!Q; 75A[G N-/AO 7+#A1# ]AI.M 9- . N5/ *I"O

.2%& <-!2= FR-M( 6; 7P3/1e6F4& 1!2& *I"O ND 3%" U.

!31, "" xexf x

)i( ( '2*. 3O 3% B0*#; U 75A[G N-/AO 7+#A1# ]AI.M 9-#-M(& '4 3O T5x

&-# 9-#-M(& 6;Q7!"#$%& 7%531 75#5AQ.& 7%5Q&-# '2*..

3333.0".x

" S&V )*c5 @>5 01%2 3% -&; 7& *& 6f.

!6519600004.0

3333.03

...0000333.03333.0

...3333333.03333.0

3/13333.0

2"$1$4

) JA[+& _&@"R5AQ. N*G 6:+. 75A[G N-/AO JA[G 3O LAP# 75A[G N-/AO.(

)ii( " R-M( 3% B0*# -::O T5(xe *"*(& N5/ ,"%I% R-M(# 6P.4: U ]"/ 7FMFM.% 3% S&"= 7M%2& x 7& *& _"4P%&xe R-M(# )O U

7+!.Q%& 7FMFM.%& ,"%I%(truncated series)

!!4!3!2

11432 xxxxxf ####"

; S&V )*c5 @> 3`567& 84% -&

! ! xx exfxfe.. 2."1$.$ 121

xxxx ex.... 2###.#"41

!4!3!22

6275000003.0

!63333.0

!53333.0

89: ##2"

89: ##2" .. xx

)iii( " 3"* B-!"#$% B-#-M( 7+!.Q%& 7FMFM.%& ,"%I% R-M( 3% B0*# Y:O -%#R5AQ. )75A[G N-/AO JA[G '-%+.M-# )O(Q&-# -EG"%I% RM(: U5Q%& N7#dA:

3955.1

0005.00062.00555.03333.01

3333.01!4

3333.0!3

3333.0!2

3333.02

####".xf

" R5AQ. 3"* 7+!.Q%& 7FMFM.%& ?@> ,"%I% 3O -%#)JA[+# )O N-/AO75A[G (">:

! 9630439552.11 ".xf

; R-M( -::4%559"#: )% -&:

89: ##.# .

...1 xx $ 567

89: ##.# .

...1 xx

R5AQ.& Z% *R5AQ. 3"

9630400002.0

9630439552.13955.13

" 7%5/ e#H.%0+E)% -&5):

2425000011.0

1#1#"11

% 7/* S&V B-#5AQ. 355"-M.% 35**G 35# aA-P& R-M( )*c5 3O 34%%& 3 B *I 7$P2:% 75#M:(very poor relative accuracy) .5F%G 3O )O7 6.& fA!&

5 g)O 3% A5D4# '/O 35.5%4& 35# aAP& -E5; 3"4 7$P2:% 7/* S&V )*c. */ -%E:%7I5.:& 6; .)**+& <-W&h S%M5 @>".

;3O -:$A; @`: 21xxy 2"

Numerical Cancellation ÍœbF�« ¡UG�ù«

W¹œbF�« �UÐU�(« ∫‰Ë_« qBH�«1

!"#$%&" '() "*+ 21 xx !"# $%&'" ( )*# +,"# -./ 0#+1 0#234 56 578 9.8,y / 0#+1 0#237.

6-./ #:; 0<=>:

105.05763.0

105.05764.0

0001.00001.021

@A13B!'" C2+D>"# A>3D"# EF.83 $%&'" ( )*# +,"# -@ E.

C2F 7 A>+&B8>"# A43 "# A7.B/ C+.G; HF.,! -@ 913 IJ.,"# K:L H=> 56"# -32#+D>"# M2% E@ N.4"O# .P36 9!1B! A78.!> A'=>*# -> -37B3 .>/ Q-372.DB>

A3".B"#.

!"#(6-1):

) ( !3D3D,"# -32:1"# +.13; +32! .!!@ R2S3A3!.="# A12+"# A"+.T>" -:

022 $#%% axx

/#E++T"# N.4"O# 0.7!B1> -32:1"# -3:L +.13O -3B78.!> -3B43 9B.

)! ( 7%A"+.T>"# E2:1 +.13O -3B43 "# -3B.L V:

02121.111.1112 #%% xx

)@( $ # 2

FA78.!> A43 K:L.

$ % # 2

)E++T"# N.4"O# ("; E+WB +) K:L(

$ $ $ % #

12 xaa

a2 $# (without loss of significance)

JA43 "# X#+&B8# -@ Y, a2$ F@

1x$ 0.732DB E+W3("; X.)2*##A3F!T>" .P8S!.

)9( 02121.111.1112 #%% xx

10.111

545.55

2.3085

4.3086

555.55

2121.1

F78!".7A " 2&Z# 2:1'2x A3' *# A43 "# .!>+&B8# #:; Q:

$ % # 2

! ,A>3D"# ('G H: 01000.2

37FL X.)2*# -> -37>"# ++T'" %F7[>"# 2:1"# .>!: 010910.2

)@32\G -3>)2" A,3, .P3'G .!' , 5B"# A13B!"# -@ E-3 %D6(.

@>A"]+T_>"# 2&*# A43 "# .!>+&B8# #:; .:

2 xx $#

?6 ('G H ,! .!!A>3D"# :010910.02

LFX.)2@ A8>&" A,3, A>3) K:.

W¹œbF�« �UÐU�(« ∫‰Ë_« qBH�«1

!"#(7-1):

a[ E++T"# N.4"O# 23=$B XF.DB A78.!> C2F 56 A3".B"# b3 "# -> </:

(i) cos (x + ') - cos x

,3 U' G 0#+1 234 ++.

(ii) sinh x – cosh x

, U3x G 0#+1 237/ ++.

(iii) f(x + ') – f(x)

, U3' G 0#+1 234 ++.

(i) ( ) ( )2

sin2coscos ''' % # % xxx

(ii) xxe

sinhcosh11sinhcosh%

(iii) 72F'3.B A'8'8B> cF/S> X#+&B8.

( ) ( ) ( ) ( ) ....21. 2 %**%*# % ''' xfxfxfxf

@#F Q 2#2DB8J# 2.3T> I.3>d2#F&"# -37 A!2.D>'" X+&B8B 5B"# 233.T>"# +e," E: 0.%387 0#234B -@ 5!T3 fg.B!"# 56 %387 234B ("; E+W3 A3g#+B7J# I.!.37"# 56

A3g.P!"# . A3>d2#F& .P3'G V'%3 A3 .&"# K:L VD,B 5B"# A3>d2#F&"#F '()*+

(stable) 5P6 J h#F Q, '()*+ (-(unstable) . C2DB8> -F/B I.3>d2#F&"# RT7F.& A3g#+B7# I.!.37 AGF>1>" A78!".7 AGF>1> E* i3"F A.

FA8#2+" 2.\B!# -37 A)<T"# QI.3>d2#F&"# A32#2DB8# -37F 932DB"# N.%&@ F>! F@ K2+) $%& -@ R2BS!+ )-@F Q I.7.8,"# N.!=@ .> A',2> 56 2PY + ++G +T7 $%&"#n

>"# ->TK2#+D> A',2>"# K:L +T7 A3".BB>"# A37.8,"# I.3' nE . -37 A)<T"# -GF-3$%&"#nE,+ A3'>T"# A3,.!"# -> -/"F Q A"$8>"# ('G +>BTB 2F C+G c.!P6

5'3 .>36 .>P6j2T! Q0#23=/ -#2PYB -.B"., c.!L:

Stability —«dI²Ýô«

*.-(/:

;-./ #:: +CnEn,

, U3C ('G +>BT3 J I7.=n -?6 Q0123&$ 4 k!; H.D33 52(linear).

"#$% &'( $)*+: +nnkE ,

, U3k > 1 5j8_@ $%&"# F>! -; H.D36(exponential).

FGF QN.%&l" 5%&"# F>!"# 9!1B -/>3 J C+.Ge -> H/ A>3) -F/B .>+!+,C

A"F7D> 0.>F>G -F/B fg.B!"# -?6 C234 . _*# F>!"# .>@F Qk7!1B 9136 N.%&l" 58"# -* c":F, +nk X3D" (B, C237/ kB>3) m7 Bn C234 "# k!?6 5".B".7F Q.378!

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Solution of Equations

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Cubic and Quartic Equations Wł—b�« WOŽUÐd�«Ë WO³OFJ²�« �ôœUF*«

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division).

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#!#$ %&'() *''+,- &.&!/ 0'12&3!4: Horner’s algorithm for evaluating a polynomial of degree n:

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Removal of roots from algebraic equations (synthetic division)

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5$-3: 1 -3 4 2 -10 -4 2

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4 5 4 54 54 5

4 54 5nn

/01.12

M*2 '=+N-R *;*;7,!&.

!"#$(2) :

6666E*;? P&"W,6666@& C66667%*N-R 166667 166663! C6666% 6666*;-Q!& ZT"#6666$%!&(algebraic)

*%#666@,%!& ZT"#666$%!&+(transcendental) #666Y,B%#$% C+6667, '666,!& ZT"#666$%!& V!96667+ G(coefficients) 6666*"E0 #=;+96666Q +<) 6666-7;%complex .( 6666!"#$%!& 6666!#3 '6666( [66664< TF

B%#66$%!& Z&9 66*;-Q!& 66*\&",-& 66%*E- S"66E0 ;966Q .66!F 1+662+!& #66447%* T 66*E*E3!& Z *E*E3.

666E*;? D;#666E, H"666% I666J+, 666AB0 .666/0 U-666@ #666%*( #4/66623N-R . 666*;]4!&+ E*;?!& K9= D;#E,, '7 ;(&+,, C< DQ* ',!& ?+;O!& C*-, *!#,!&.

%&!'(5-2) :

666!&"!& Z6664#7 &9Ff C*,;666% 1666J#8,/! 666/-#A(twice differentiable) Z"666Q++ G C#,%*Aa, b ) );84+a < b ( *!#,!& ?+;O!& ZEE3, _*3-:

) <( R;#OFf(a) R;#OF >!#W,f(b)C< S< G:

4 5 4 5 0. 7bfaf

) D( !&"/! H;M2 *#Y4 +< .%]0 *#Y4 V#4= `*!f R;,8!& '((a,b)C< S< G:

4 5 4 5baxxf ,0 89:/

) a( f // R;#Ob& #Y!@84 #Y R;,8!& 17 '((a,b).

) "( 4 54 5

afaf 07/07/

E*;? C:(N-R R;,8!& '( *;#*,W& %*A Sc- #4<"- &9F D;#E,, ; <ba,.

- 1+5& ?;O!&)< ( "+Q+ #4! C%J*(existence) R;,8!& '( ;9Q(a,b).

- '4#666X!& ?;666O!&)D ( "666*3+ ;9666Q ;9666Q!& &9666= C< #6664! C%666J*(unique) ] ;666]4&P66A; 1766O!&1[ D+66/?%!& ;966Q!& C660 "#66$,-T& P"660 #664! C%66J* #66%7 G) .66!F #66%-;;Wd ;9Q] (PA; 17O!& ;]4&2.[

- _66!#X!& ?;66O!&)a ( 66!&"!& C< '664$*f 66-"3% #66Y4< #66%F(convex) R;66,8!& 1B66Wb) (a, R;66$E% #66Y4< +< #66Y/7(concave) &"!& C< S< GR;66,8!& K966= 1667 1&+66? 66!f

>#66?$4& 66?E4 S< #66Y! `*66!(inflection) )#="6640 '66,!& 66?E4!& '66=+ 01//f (

L#66J*< C%66J* &966=+ G R;66,8!& K966= '66(– '!#66,!& ?;66O!& N66%– C660 "#66$,-T& P"660 ;9Q!&]PA; 17O!& ;]4&3.[

- N666-&;!& ?;666O!&)" ( #6664! C%666J*– U-#666@!& ?;666O!& N666%– 666%*E!& Z6664#7 &9F [6664<66*-*;E,!& nx R;66,8!& '66( N66E,; <ba, & C:66( 66*!#,!& 66%*E!1.nx C66% 66Q,#4!&U66*-?,

R;,8!& '( L#J*< NE,@ *;*;7,!& M*2!&; <ba, ]PA; 17O!& ;]4&4.[

()& *+,)1(

3&+ ;9Q C% ;X7< R;,8!& '( "b) (a,

01/f R;,8!& '(b) (a,

()& *+,)2(

D+/?%!& ;9Q!& C0 "0#-,

01/f R;,8!& '(b) (a,

()& *+,)3( ()& *+,)4(

D+/?%!& ;9Q!& C0 "0#-, D+/?%!& ;9Q!& +34 D;#E,

01//f R;,8!& '(b) (a,

C< );84f */*/3, !&"(analytic).

C<+ & '= !"#$%!& ;9Q! ?+-J%!& %*E!&f(x) = 0.

C<+ nx ;9Q/! *-*;E, %*A '=.

' C< _*3- G *-*;E,!& %*E!& K9= '( c?W!& %*A '=:

nnx '& .1

!"#$ %& '$()* #"+!,N-R Error estimate for N-R method

C< );84+n' C< _6*3- L#6*(#7 L&;M62 R;*M62ff/ 6-#,;- ;6*M,,(monotonically) '6(

C*- R;,8!& nx,&. C<+4 5 0:/ xf R;,8!& '( #Y@84.

C< );84: 4 5 4 5

4 5xfxf

*!#,!& AB$!& ./0 1234 ?@+,%!& %*E!& *;]4 P&"W,@#-: 4 5 4 5 4 5 10; 770/100 >>'='

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&='=ff

C:( V!9-+:

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C< );84:

4 54 5

0////0

) C< );8-1. 7Kh( Kh

C< S<: Kh

hn .107'

_*3: 4 54 5nn

ffK ///1 /sup

666E*;? 666@&;" "666$- Ce&+N-R )K#666Q,T& ;666*M,%!& `#666%%!& 666E*;? +< ( #666Y-;#E, _6663-+$4 G#Y*( c?W!& ;*"E,+ +#Y-;#E, ;"4! #=;79 U-@ ',!& H;W5& U;?!& )$-! ".

*-.$(10-2) :

Z666-#X!& `#666%%!&+ GK#666Q,T& Z666-#X!& N?#666E!& '666,E*;? C666% 16667 D;#666E, C< Z666-X& '@"4= D;#E, K#Q,T&(geometric convergence) ] '6( C*c6?W!& C*6- AB$!& C< S<

R;+2!& '( C+7, C*,*!#,,% C*,*-*;E, C*,%*A:

nn A'' 3.1

_*3A Z-#X[

% 17! *;*;7,!& M*2!& NJ+ C7%*R;+2!& '( C*,;+79%!& C*,E*;?!& C: 4 5kxf

nn01.1

_*3k K#Q,T& Z-#X!& 1*%!& +=(constant slope).

C< );6668-+& #666% .666/0 E-#666@!& 666AB$!& C666% 1666234( G?+-666J%!& ;9666Q!& +666='/*:

'&'&'& 1

4 5 4 5 4 5; < .//./.012 . &'&'&'' fffk

nnnn !2

C< #%-+4 5 01&fC:( G:

c?W!& C< ]3B*+1.n' c6?W!& C6% ;*X7- 1A< C+7*@n' 16*%!& 6%*A Z64#7 &9Fk %*E!& C% -*;A4 5&f /.

/62,% 6!&" #64*?0< #44< );84f !& '6( 6( f;g$% R;6,8; <ba, C< );684+ Gf(a),

f(b) C#,8/,W% C#,;#OF #%Y! . %*A "Q+, ?@+,%!& %*E!& *;]4-(p _*3a < p < b

C< _*3-f(p) = 0 . ;9Q!& &9= %*A "#Q*F D+/?%!&+p.

;96Q!& "#6Q*b Ce& #Y3;O4@ ',!& >*24,!& E*;? C< Ph;p U6*-?,/! I/62,&9F #% !#3 '( R;6,8!& '( ;9Q C% ;X7< V#4= C#7; <ba, ?*6@-,/! );,846@ #644< TF

R;6,8!& K96= '6( "*3+ ;9Q "Q+* [4< . ;;67,%!& >*624,!& .6/0 >*624,!& 6E*;? "6%,$, R;66,8!& C66% 66*\iQ!& Z&;66,8/!; <ba, .66/0 S+66,3* S966!& >6624!& C*66*$, R+66?W 1667 '66(+

;9Q!&p.

(Bisection Method) nOBM²�« WI¹dÞ

N6J4 *&"-!& '(bbaa 1111

, C< );684+ G1p '6( ?6@+,%!& 6?E4!& '6=

R;,8!&; <ba,C< S< G:

Z6664#7 &9:666(4 5 01 1pf C:666(1pp 1 R;#666OF !"""# $ %&' !1pf (""")* +,("""-* ."""/

!1af '0 !

1bf (1"""""""234 . ("""""""5 !"""""""# !

1af , !

1pf +,("""""""-6& (""""""")17 (1"""""""234 !"""""""#

,bpp" 894 :;4<=' >: 1212

, bbpa ##

+,(""""-* ?""""4(5 * ("""")0' !1pf +,(""""-* .""""/ !

1bf (1""""234 !""""# !11, pap" :"""";4<=' >

894:1212 , pbaa ## . +,@37& ABC D<B)E7& F:/ G<HI@ J<E4 KL$ %22 ,ba.

D<7(@7& D<)M,&'N7& .# DO<,I7& F:/ P<NB@ 5)<':

!"# $%&'( $&)*'#+, -&.Bisection Algorithm

D""7J(E)B7 Q""= J(""R<6f(x)=0 +,""@37& .""#[a,b] S""<=f(a), f(b) (@,(""-* ("")17 (@3B@N).

0 T,""34 & U)(""2@7& '""/(tolerance) 0' >V""H W')""2)7&N JJ""C A""XY0 '""/,<,5@7& ?&,)7 VH W')2).

89i = 1.

0 ()7(I D<7(@7& ?&'IN7& :Z34i < N.

! 89 2abap '(#.

! (5 &:* ! 0#pf '0&)'2ab D)<Y 8HI(#p [Y'@'.

! 89i = i + 1.

! (5 &:*f(a).f(p) > 0 89a = p 89# $ %&'b = p.

JJC JEH UB3@ K7 DO<,I7& 0 J<3@ D7(2, 8HI&N [Y'@ KL ?&,<,5@7& ).

0 "5)< I,"- ") ,"L50 J"R'< V"40 \=]<' D"O<,I7& :"<34@ [("O<6 KJN@"2<) $J"HI,""-7& "") :f(p) = 0 '0&)'

2ab )D""<)M,&'N7& .""# ,'5:"")7& .( ,5:""4 .""B< ("")<#'

I',-7& F:/ TEH.

(=)(""""2@ ,""""@N&(tolerance) 0*& J""""<7'@ .""""# ,)@""""2&'nppp ,,,

21 0 A""""7*

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()J4C: ! 110 3 *) ' npfn

57'310')'npp bBI@@1000*n.[

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234)(11-2) :

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-7&I,: 4105 '3)'n

aab DYJ )9474 D<'4E) K(Y,0.

! ! 142,51 #'# ff

.7(@7& Q'JR7& K<Y ABC QX=4 [<X4@7& D<)M,&'N G<HI@H':

n na n

b np !

1 1.0 2.0 1.5 2.375

2 1.0 1.5 1.25 -1.79687

3 1.25 1.5 1.375 0.16211

4 1.25 1.375 1.3125 -0.84839

5 1.3125 1.375 1.34375 -0.35098

6 1.34375 1.375 1.359375 -0.09641

7 1.359375 1.375 1.3671875 0.03236

8 1.359375 1.3671875 1.36328125 -0.03215

9 1.36328125 1.3671875 1.365234375 0.000072

10 1.36328125 1.365234375 1.364257813 -0.01605

11 1.364257813 1.365234375 1.364746094 -0.00799

12 1.364746094 1.365234375 1.364990235 -0.00396

JEH V40 \=]412 !# ,<,5@:

364990235.112

_'(2< dIN7& 0 _0:

000488281.0121212

#'4' abpp

0 ()H':

105106.3 '' 3)35'a

QY`& ABC D<'4E) K(Y,0 DEH,` U<=X b<,O@7(# . ,:"RB7 D=<="X7& D")<O7&'p

./ D<,-C K(Y,0 D<4()L7: p=1.36523001

0 D"""\=])7(H ,<J"""R7& """)' 9p U<="""X7& ,:"""R7& A"""7* b,"""Y0p b"""<,O@7& """) .;(147&12pU<=X7& ,:R7& D)<Y ?)BcC &:* $* &:/ D#,E)7 DO<,I JR'@ $ 57' >.

."1# >+,"<H5 ("H'<C ("17 0 $* DI<"2H' D="9&' ("140 K"e, [<"X4@7& D<)M,&'N' (""""1H,(O@ .""""# &J""""R D"""";<IH) 0 _0N ,&J""""O)7& UH""""X< 0 Q""""HY &J""""R +,""""<H5 UH""""X@ J""""Y

Npp ' (<#(5 &,fX &,<fX(> (I"2'@) &J"<R ("H<,O@ !"# &:"/ ) ,L50'– 0_ D"R<@4

.;("147& ,"<,5@7& g<"7' DI2'@)7& ?&,<,5@7& J=0– V"<7* ?"h3h@iBc< 0 'J Q"h)1c< J"Y> 8").;(147& b<,O@7& ) Q9#0 V40 (4<0, ()5 >GH(27& Q(L)7& .#.

[<X4@7& DO<,I 0 ABC"1H M<)@@ D)(/ D<X(N (17 (");&J b,("O@@ ("140 ."/' (Q"""= A"""7* ("""1EH@@ D"""<J<1)@ '0 D"""<;&J@H& +'"""IN5 KJN@"""2@ (""") ("""H7(e ."""1# bH"""27& &:"""17' >

QX37& &:/ D<OH .# (19EH A7* ,<-4 .@7&' +j(35 ,L5`& G,I7&.

%&'( 5'3%!$ -&. !"#(Convergence of Bisection Method)

$&'/ :

0 T,"""34f +,"""@37& ."""# DB"""X@) D"""7&J[a,b] 0 T,"""34' > ! ! 0bf.af ) . &:*O<,I (4OHID D<7(@@) J7'@ (14!# [<X4@7&, -

np ,:R7& bZ,hOc@p D<X(N7& GO=@':

6'4' nabppnn

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l'"""Y D"""7$JH d"""IN7& U)("""2@ A"""IEc< <"""= b"""240 Df<"""X7& F:"""/ '"""5@ J"""Y'10 >AhIiEc< &:5/ +J(C'.

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V""@YJ n""BH@ Q""= A""7* Q""X4 S""<=H >[<""X4@7& D""O<,IH510'#& D""<&JHB7 DH""247(H a""7:'2,1

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20"#: J(R<* Q'(=4N I,-7& GO=@ .@7&:

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JJ""E7& "") ,""<L5H ,""H50 J""=7& &:""/ '""5< +,""<L5 (""<=0 .""#' > D"")M]7& ?&,""<,5@7& JJ""E7b'BI)7& .BE37&.

b""""BI@@ $ (""""14dH M""""<)@@' > ?$J(""""E)7& Q""""=7 D""""<,<,5@7& D""""O<,I7& F:""""/ KJN@""""2@ D"O<,I .# ()"4<H >?(O@-) _0 K<Y b(2=N-R D7J(E)7& Q=7f(x)= 0 b(2=7 q(@=4

D""7&J7& DO@""-) D"")<Yf +'""IN Q""5 J""4C. 8""94 D""@H(L7& D""IO47& D""O<,IH ("") D""7J(E) Q""=7'./' DX(N +,'X .# $'0 D7J(E)7&: !xgx # S<=g .# D7&Jx.

D7J(E) _0 89' 5)< V40 8Y&'7&'f(x)=0 F:"/ ."# +,'5:")7& D"X(N7& +,'"X7&$ JJ""""EH D""""7J(E)7& ]L"""")# >G,""""I7& """") .;(""""14023 #'x ?$J(""""E)7& """") (""""<0 r#(""""5@D<7(@7&:

xxx ('# 23 (i)

552 3 xxx ('# (ii)

x # (iii)

,Ns Q(L)5' : D7J(E)7& !#f(x) = 0 +,'X7& .# D7J(E) _0 r#(5@:

!xfxx ?(#

S<=? &,3X _'(2< $ ?H(L _0.

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Fixed point iteration method (or Iterative method of successive approximations)

!"#$ %&!#'()* +!), -()* .!/& +012' 2 134* 531&– 67 !"89)* ,,!-)* *:!; <!( =2')* <(– +!> 4)* ?2&!>)* 6!@x=g(x) +!3A B)* +!'08)* +!012'A !"#$ C,D!1 E!1$A

– F* <:GA H1#I ,-A J28K (L– %2 03 M)N(convergence) %&!#'()* 2:O)* &$8 P ,Q A3 M)N 24R 2 134* C,D1 ,I (81A(divergence).

?2&!!>)* 6!!@ +!!), -( 81,!!) <S <T* U2!!V8x=g(x) . +A!!K 8( +!!(10A S,!!A*0xx

) +A12I+), -()* 2:O <(( +1) 33( &S +#K#K <W&L& P(sequence) +!-A 33( X !A1203 <!(+I9-)* 5*,43K A:

! " ,2,1,0;1

nxgxnn

+!1) 33()* Y:") < L *:N Z8S [/*&)* <( ,,, 210 xxx +!1 "8 (limit) $ <G!@ P$ +), -(#) \*2:O <&L1x = g(x) ] <] ^):&! "$$ g .[

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+), -(#) 6010$)* 2:O)* , O1N %&#'()*) \ 0A K ?2&L:()*:( 023 %x +!012'A+3A B)* +'08)*.

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xxx #% 23 (i)

552 3 xxx #% (ii)

+17*,3A_* +(10)* 2 134 A&2.10 x

]$%&'#: 2:O#) +'&A/()* +(10)* <S 5&#-()* 6;:

259921.123 $.[

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349.16

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25992.1

2599.1

2596.1

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+!18 B)* +a1!>)* Y:!; =1A'3A "1#Q H>$8 63)* +1) 33()* %2 03 %AK (S&)ii(

M!!)&]* +a1!!>)* +!!1) 33( ,!!Q A3&)i( N 21!!b8K@) +!!012' %2 !!03 +!!K*2, ,!!-A F* <:G!!A Z!!1+3A B)* +'08)*.

c'4)* 6'-3 +I9Q , O1d1#n& c'4)* +)_,An&:

<S U2V8$ 2:O#) +'&A/()* +(10)* 6;

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M#Q H>$8:

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<S (A&$ +), -()* =0$3 "8S CS P2:O#) +'&A/()* +(10)* 6;:

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nn%# '

Convergence of the fixed point method W²ÐU¦�« WDIM�« WI¹dÞ »—UIð

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ggg (((&& ))) '#

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P!!!>:>$-8 '(!!)*-8 +!!!, &6!!L*– @8=!!9R3$-8 '9!!!] 6(!!%91 5&321 ,, xxx - '8643!!!/(L J=94Y8 -6(7$-8) >-(>-8 5& ( $9] PB73 P3-8#3x: 303 &x

J=94Y8 +L] -6(7$-8 '>) 9*(>-8 5& ( $9] 0:" +S,* (U*$#2x:

363242 ,&,,&x

0-#Y8 -6(7$-8 C8=94&# $9] PB731x: 918271 &,&x

X/I* KL(/-8 +,-8 0:" (*:S, 8F?G#.

&a +o6!!!qL 5& M@A6(!!!7$-8 T!!!93=3 =!!!o9_(permute) !!!-6(7$-8iLS9 W9,L =4& -6(7$ a$ 0-#Y8011 *a.[

PQ L#=H$ 0-#Y8 -6(7$-8 EFG ZH&:

13121 ,,, naaa ,,,

']= -6(7$-8 0-1: 2 , 3 , ... , n

ZF!!!!,- [!!!!-F# M T!!!!93=3-8 0!!!!:"1x 0!!!!-1 !!!!9*(>-8 <!!!!$ @A6(!!!!7$-8 +!!!!? <!!!!$J=94Y8.

696%-8 +$(7$-8 0:" 9*(>-8 -6(7$-8 '/]822a+ . PQ 6#%#$-8 =S*7-8 8FG# C(96!3# #& C(!92?3=$ C8=!S*" 0$/9 X9:" -6(7$-8 '/N* 5F-8# P/9;=-8 =BN-8

(pivot element)] .@A6(7$-8 T93=3 =o9RQ C8=IS 5#(/9 <(? <1 C(H9&#.[

Gauss Elimination Algorithm ”ËU' ·c(« WI¹dÞ WO�“—«uš

0,0 ** yA

<939/9;= <93#B4 <$ 9$2=8#4-8 <#?33:

C&5D' 5)E&' :FG2H2% I#GJ: CG&K CG)L%&' I#GJ:&' B+5@- (Triangular System)

<9L#:/& 6,.L 8FG '39 <& <?$9#:

)0( !++L-&' M% >?@&' (Elimination with Normalization):

0!!:" 0!!-#Y8 !!-6(7$-8 '!!/]811a . #& C(!!92?3=$ C8=!!S*" 0$!!/9 =!!S*7-8 8F!!G# C(963#(pivot element).

]<(!!!? 8F1011

!!!!"#$" !!!! %$ &' (!!!!)*"+,ix -!!!!./ (!!!!01#$"02 3!!!!4i /!!!!))$502 (!!!!)*"+ 6"!!!!75 81!!!!9,02 :,#!!!!7)

(normalization) . !!!"#$" !!!$'0 ;<1#!!$"02 )1!!!=5 (!!)*"+,ix -!!!./ (!!!01#$"02 3!!4i 2>

3!?@'02 @#A5/<2 ()*"+ 6"75 (B*C" ("). /=AD(partial pivoting). ED EF2 G/!HI J#!"#+ J#!"#KI #I)1!0. Ax = y L!)9 M !NH02 2>!O P!*C" 3!4 Q!H)/$5 R=!7 :>!02,A 3!O

, ;S"#$"02 (4,HN"x , T1#!')U V,!*C"02 Q!'5"02 ,Oy 2 Q!'5" ,!O (!",*$"02 ;!=2,W0)X2 Y/C02 34E" ED G/HI, :

!"#2x $%% &'()*+, -.*')/0# 1'23 &'42$42# 5'% 678$'9%2# 5'% :; 5% &'<4

!" &()*+21x=> &,?*@% .8)8A2# &)B$42# &28$9%2# &>$@C, DE -:

24232 ,,, naaa ! ! !

FG* &28$9%2# 123: 3, 4, ..., n

H)I*I2# 1<J)&K"L% :ija

21 ,,, aa

aa n!!!

F'G* &'28$9%2# 1'23 2, 3, ..., n !'"2 M'2!? -H')I*I2# 1'<J1x 678$'9%2# 5'% :'; 5'%

42# 5%.*)/0# 123 &)B$.

!"#2x &()*+2 &<4$%% &()*+, .*)/0# 123 &42$42# 5% 678$9%2# 5% :; 5%

!"1x=> &,?*@% .8)8A2# &)B$42# &28$9%2# &>$@C, DE -

32 ,,,aa

FG* &28$9%2# 123 :3, 4, ..., n H)I*I2# 1<J.

*%IN#*O$B92# !" &)<%J =>143 ,,, !nxxx H)I*I2# 1<J.

.*?O2# => =4<4% F$KB 1<J :O"B &)$PB2# =>:

nnnnnnn

zxcxcxc

1,111,1

22323222

11212111

&2$"2# =>)E) (*'))9I2# Q'% !'"2# (&')*+(2# 6L%$'9%2#iic DE .8'"?2# D?$'NI

nicii ,,2,1;1 ""

81)1'02 ;<1#$"02 34 ;S"#$"02 3O.(

/!!!N#I$02 Y>!!!9 (!!!)*"+ 3!!!4 /"5!!!72143 ,,, n

xxx /1!!!. MV!!!)5/502 6!!!*+(+#C57<2.

) ( ! "#$ %&'()*+,, (Elimination without Normalization):

34 (=,/Z" 60,X2 (01#$"02 YZD

%&'()* +,-.)* :/0-,)* 1.2.$)* 3-!,)* 4") 152()* 6+&70)*

(Back Substitution)

#!';R !"#$%&'( )#*+,-.-+ /(.#,0-&'( 1,2 345 6789 :;4<'( =,"%-')+: ) >#&?.,<@( A'B)%&'((

)?.,<@( 6+2 A'B)%&'( >&(

! #! 1

68'( A,&C.("< D,<4- )99$&," :E4E&'( 1)F94' :;4<'( =,"%-')+ Cx = z (*)

G###,8C A###,"45 A###,E4E& AH";###7&$ %nn& I(.;###7 J")###K- L A###,.MN'( )O.###7)95 6###$A,')-'( /)2P%')+:

)&B95 Q9R AF8P& S&i = n T"#&U&'( >VH"#!

W+#7,"#!

B"B#8 B#U"- L Q#9R JR

I(.;7 :M%, :')-')+" T"&U&'( (XO :H) I)8PM7(.(

!"#$ %&'$( )*+,- ./ )+0"1'$( 2"+345$( 667 Number of Operations in Gauss Elimination Method

>#& >"$& :M< 1)F9 68' !")U' YX8'( AN,.M )9&B<-K( (XZn :#H A#'B)%&n )+EZ >$&, Q9VH 6"*U&>R /:

[.\'( /),4&5 BB5 /A&KN'(: 3

3 23 nnn #

S&U'( /),4&5 BB5" /].M'(: 6

532 23 nnn #

A&######KN'(" [.######\'( /)######,4&5 BB######5 >R JR)].######M'(" S######&U'( ^'X######$" ( J")######K,)+,.N-3/3n.

.89#$( 9":;,<"0 %&'3$ !"# )*+,- (Gauss Elimination with Partial Pivoting)

L B#2 A#,&C.("<'( _XO /("M< >& ?"M< JR :H JB-"'( "R JC$-.&'( .79%'(;7 J")K,Q#,45 A&#KN'( >#$&, :')-')+" I(. B#2 :')#-')+" I(B#U I(.,0#7 >"#$, B#2 Q#9R LZ

?.####,+$ `)####M<R 3####'Z JBa####, .$ b####c9, ?B)####5 I(B####U .,0####7'( 6####&)%&'(">,BB####5 >,####+ d.)####;

I)+,.N- >,,")K-& /LB)#%&'( 6,B#+-+ `)M<@( _XO 6E& :HP- >$&," (permutations)

)###*+,-.- .###,,0-+ JR JC###$-.&'( .###79%'( W+###7, G###,8+ [###,-.-'( (X###O 6###&5 6###\;,H –

Q###,45 1###KN9K JX###'(– .###79%'( >"###$, G###,8+ /LB)###%&'( [###,-.- B###,%9 ?B)###5" I(.###,+$A##N4M& A##&,2 .##+$R "X 6##&)%&'( "##O JC##$-.&'( . S##+-9 :##-'( !")##U A##N,.M 3e&## fKg- ?B)##5"

`(.Uh( (XO )*,H : :iCU'( C)$-.L( S& YX84' !")U AN,.M(partial pivoting).

A,')-'( AU,-9'( 3'Z JBa- I)N+)K A8".c&'( :;4<'( =,"%-'( AN,.M:

)+,=>:

/##9)$ (XZC A##,2"H A##,E4E& AH";##7&(upper triangular matrix) 6##$ /##9)$")O.#####7)95 AH";#####7&'( >V#####H I(.;#####7 J")#####K- L A#####,.MN'(C !)#####$%9P' A#####4+)2 >"#####$-

(invertible) !"$%& )*' JR1 C.

?"@,0$(:

:E4E&'( 1)F9'( >R >,+- ?."$X&'( :;4<'( =,"%-'( AN,.M >R S2("'(

(*)zCx !

) I)N+)#K 6,#7;-')+ ["-$&'( (' B#8(" 6#8 .#E$@( 3#45 Q#' Q#U-& 6#$z 3#fM%g& . >V#H :')#-')+"C A,')-'( A&"4%&'( A,.F9'( 345 I )9+ !)$%9P' A4+)2 >"$- >R [U,:

)+,=>:

>R )9\.H (XZA A%+.& AH";7&nn&AiH)$-& )*4$ A,')-'( /(.)+%'( >VH :

) R( !9)U-&'( 1)F9'(Ax=0 (homogeneous system) QH)-'( 68'( LZ Q' !,' x= 0.

) ( !"#$ %&'y ()*+', -./ 0-$12, 345', 6/ 758$Ax = y %9 !'.

) :( ;/<=>$',A ?)&8+@' ;AB)C.

!"#$', -./ D'E'<x 7F$GH1 ;IB)G', 6=AJ', K1<8#', ;1$L4,<JB 7M58$', : %9 6NAN$', ()*+',[the solution of (*)] (*).

?1'< : 6NAN$', ()*+A' O@9[a solution of (*)] (*).

) ( !"#$ %&'()– *+,-. /01("$

Gauss-Jordan (G-J) Elimination Method

51PGB %1Q8# R$ 3E9A' ?<)" ;I145 )SG=+ 6T ;I145', UET Q8# . 6P/ O@NP$/ VLPP&#4$', 4PP>+8', 7PPAW )PP$ ;PP'Q)8$ (PPGI+ -X QPP8B 4PP118#', RPP$ 3EPP9A' ?<)PP" ;PPI145

1P8$ 4P1Y#$ 3EP9' ;P'Q)8$', UEPT (,QJ#G)B (<I+ )++./ %=PGX 6P#', Z[Q)P8$', %P& -P$ -;PP'Q)8$', UEPPT . ?<)PP" ;PPI145 6PP/ )PP$X– ,EPPT 3EPP9 ;PP1A$W \,4".PPB (<PPI+ )PP++./ -,Q4<PP"

;'Q)8$', UET %=GX 6#', 7'] ;/)^_)B </ 6#', Z[Q)8$', %& -$ 41Y#$',.

?<)" ;I145 ;1$L4,<J/ D'E'<– )SPG=+ ?<)" ;I145 ;1$L4,<J 6T -,Q4<"1PGB', %1QP8#', ,EPT R$ 5) 4P>+8', %=PGX 6P#',< <P/ 6P#', Z[Q)P8$', %P& -P$ 3EP9',

VQPP#<', .( ;PP145C Z@$)PP8$ ;/<=PP>$ 7PPAW O,4PP1JX %PP>9+ ;PP'Q8$', ;PPI145', UEPPT 6PP/<(diagonal coefficient matrix) aQ9<', ;/<=>$ 6T %B(unit matrix) . )P++./ D'EPB<

G9 VX -<QB Z[Q)8$', ;W<$"$' 6b)S+', %9', 7AW a4c)B$ %>9+ @/ 0d4JX Z)B)6=AJ', K1<8#', ;I145 `1B5#' :)#9+.

2 3#4(2-3):

?<)PPP" ;PPPI145 (,QJ#PPPG)B– , ;PPPW<$"$ %PPP9 QPPP"<X 3EPPP9A' -,Q4<PPP" Z[Q)PPP8$'%)N$ 6/ a)58$', ;15J',1-3 .

%)PN$ %P9 6P/ )P$& %9', XQB+1-3 3EP9+ eP19 01x !"#$%&' (")&*$!+,&' (!, -!. (!, !!%&$%&'/. .01!!2 !!"3,4 5!!&6 -!!7# $,*!!#4/2x (!!.&/ 8!!9: !!%&$%&' !!&*$+,&' (!!, 01!!2;" <!!:

=$!!>"? 5!!&/@' (!!,) =$!!>"? !!"#$%&' !!&*$+,&' A'*B)!!C$D E!!&1/ .( FG*$!!+,&' (!!, =G*!!D: E&1!!D/ "&$)&')-$%, -2 H$#%? $I"34 $#372 J)&' 1-3.(

ÍœbF�« qOK×²�«Ë W¹œbF�« ‚dD�« 3ÍœbF�« qOK×²�«Ë W¹œbF�« ‚dD�«

!"#$%& '() *+,- ./%"0%& 123%& 4)567, 1/,2x 897:; <'7%=>& .7%!"#$%& ?7$ @A ./-"3%&3/4 - '%=>& '() "B#$C=.(

@777A D9777/E>& .777%!"#$%& 89777:- F7773(-15) *777$"#$ *777#C%3x F!E0777G-= <D!777,=%& H="777G/56777,% .777C0"-%& .777%!"#$%&3x 89777:; I777%6= <.777/-"3%&= '777%=>& ?/0%!"777#$%& ?777$ *777J ?777$

.773%"3%&).77C0"-%& ( @77A1/5 @77A .773%"3%& 8977:= <'77%=>& '77() "77B#$C=-16/15 "77B#$C=&6JK <./-"3%& '():

.77C",%& ?=! <'77L#$%& F"77M-(% 8=77(L$%& *77,%& @77L#0 12377%& 4 !"77#$%& N677K=HO P&9CQ '%Q R9EO 4";"G,.

"7$J 4 !"7#$%& ?7$ !7; 4"A=S7+$%& .7% !; "7B(J *7,%& 4&=LE .;"0J "--J$/= @777(/) 56777,%& .777T/9L F&!E0777G"; *777,%& !777-) N6777K .777;"0J%& .777T/9L U"777;0& "777:/O "777--J$/=

V="C%:(

121520

151610

100151610

?O "-777777:9A &6Qn *777777/K"C$%& !!777777) W777777GS- =777777K= 4 !"777777#$%& !!777777) *7777773$/) HO4&977/X0$%& ( 4"77/($#%& !!77) ?Y77A(number of operations) V="77C .77T/9L @77A– "77$J

Z"T;"777G "777-/O9– =777,-; 9!777T[/3/3n V="777C .777T/9L @777A 4"777/($#%& !!777) 9!777T/ "777$-/; <–

=77,-; ?&!9=77C2/3n 9L ?Y77A I%677%= <8;77G%& &677B% Z"77/($) *77:S0 V="77C .77T/ ?77J$/= <@(/ "$ 4";3Q:

%&'( )*+, -. */'0"12 3'*4$512 667– 826+&(:

89:%& 4"/($) !!) /.$GT%&: 22

23 nnn "

\$C%& 4"/($) !!) /]9L%&: 22

77) ?O HO .$77GT%&= 8977:%& 4"77/($) !!)]977L%&= \77$C%& I%677J= ( Z"77;/9T0 H="77G/2/3n.

9:"12 )*+, ;47 3')*/,< Applications on Elimination Method

)= ( 3266"$12 >'0" (Evaluating Determinants)

5677,%& .77/$ 9&=E 4&=77LE ?77$ D=77LE !77C=0 W77-& F77(#- 4&!!77,$%& _&=77E ?77$77C% V=") V="77C =O– ?&!9=77C ( P"-3077G"; 42$"77#$%& .A=S77+$ D!!77,$ .77$/` ?77$ 977/X0

?/S77+ */!77;0 .77/($)= <H 77J09$ 977+-) *77J '77() .$77GT%& .77/($))?/!=77$) =O .( 177/,= H="77G0 .77C0"-%& .A=S77+$%& D!!77,$ .77$/` ?O1 ) M77, – 5677,%& @770T/9L ?77$ HO @77A–

H9S+ @(SG 1(3$ "B% .C0"-%& .A=S+$%& ?O@A !",a= < @G/b9%& 9LT%&( .7$/` ?=7J0A < .777/ J09$%& 9777+"-#%& 89777: *777+",% ./="777G$ I%6777; ./(777+>& D!!777,$%&(product of

pivots)< */!777;0 4"777/($) ?777$ H!9777A !!777) H9777C[O !777` ?"777J &6Q c0"777-%& D9"777dQ 9777//X0 \777$D!$)>&= 5=S+%&.

% & % & n

kpppA $$$ ! 21det

1/, :k ?/!=$) =O ?/S+ */!;0 4&9$ !!) *3$/.

npp ,,1 :./ J09$%& 9+"-#%&.

.A=S+$ D!!,$ .$/` 8"G,nn$ !!7#% e"70,/ 567,(% V="C .T/9L U";0";=,-; 9!T/ 4"/($#%& =O 4&=LE%& ?$3/3n !=!, '%Q !!#%& &6K \S09/ "$-/; <n! !-)

%& 8"G,% D!"0#$%& f9L%& U";0&4&!!,$.

"3$ 6Eg- ?O *; =567,%& .T/9L; 4&!!,$%& 8"G,% Z @-"73%& f7/;L0%& V9!7- <()4"A=S+$%& VJ) =K= .T/9L%& N6K ' Z"#$ ?/T/;L0%& '() Z "3$ 9J6- F3 <.

)> ( .&?@$ %&A5$ 6'(*B (Matrix Inversion)

?$ Z2J ?O h9S-A, Z, I .#;9$ .A=S+$nn$ ?O= <A.Z = I

?O HOZ .A=S+$%& V=J#$ @KA: 1 ! AZ

.A=S+$%& ?O HOA D!9S-$ 9/i(non singular).

) ?O h9S-A "BG=J#$ !"C/Q 8=(L$%& .A=S+$%& @K.(

.`2#%&AZ = I ./%"0%& D9=+%"; */+S0%"; 80J0 ?O ?J$/:

6K ?YA 4"A=S+$%& _&=E ?$=./%"0%& 4"`2#%& .)=$C$ "-/L#0 .`2#%& N:

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4"777`2#%& N6777K ?777$ .777`2) *777J=) "K!!777)=n ( 4 !"777#$%& ?777$ .777)=$C$ *7773$0 ./LE%&) "K!!)=n .%!"#$.(

"-#:= &6Q "--O HO1 ! AZ ?YA <AZ = I

=O njeAz jj ,,2,1; !!

1/,jz : F 9 !=$#%& *3$0j .A=S+$%& @AZ.

je : F 9 !=$#%& *3$0j .A=S+$%& @AI.

?O HO:

) F 9 5+%& j........ (-.

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D!77777$)>& ?Y7777A &67777JK=.A=S7777+$%& @7777AZA ! 1 7777(, @77777K ?77777$ 4"7777)=$C$% *=.7777/LE%& 4 !"7777#$%& D!7777$)O H="7777G0 4"7777)=$C$%& N67777K @7777A '7777-$/%& 5&97777L>& 17777/, <

D!777,=%& .A=S777+$I 4"777)=$C$%& N6777K ?777$ .777)=$C$ *777J @777A 42$"777#$%& .A=S777+$= < .A=S+$%& H="G0A )"BG=J#$ !"C/Q 8=(L$%&.(

V="C% 56,%& .T/9L @A=)"C =O V=– ?&!9=C ( D!7) \7$ *$"#0- ?O "--J$/WGS- 4 =%& @A '-$/ 5&9LO . F"7M-%& '7() V="7C% 567,%& .7T/9L "-T;L &6YAAZ =

I D!777,=%& .A=S777+$ D!777$)O *777$"#- 1777/,; <I – W777GS- 4777 =%& @777A– D!777) "777B-O '777() 4"77BC0$y )'77-$/%& 5&977L>& @77A 4"77BC0$ D!77) HO ( .A=S77+$%& D!77$)O=Z "77B-O '77()

7(,%& 4"7BC0$ .7(;"T$%& *=(corresponding solution vectors) x I%67; *7+,- "7--YA <– N677777777777K *=77777777777(,%& 4"77777777777BC0$ ?77777777777$– 8=77777777777(L$%& .77777777777G=J#$%& .A=S77777777777+$%& '77777777777()

"K!"C/Q1 / AZ.A=S+$%& I(0 D!$)O @K 4"BC0$%& N6K ?=J0 1/, <.

$ C'D(3-3):

?O h9S- '''

)i( F&!E077G"; !77C=O V="77C .77T/9L– @77LE%& F"77M-%& *77, ?&!9=77CAx = y ) ?=!4";"G,%& @A ./!"/0) & 9=GJ%& F&!E0G";= < D!$)O =O 5=S+ HO */!;0.(

)ii( .7777G=J#$%& .A=S7777+$%& 97777+"-) 87777G,&1 A F"7777M-%& *7777,;AZ=I F&!E07777G";= <@b C%& ^"J09 &.

)iii( .$/` !C=O|A|.

(i) <(ii) =S7777+$ ?> Z&97777M- 42$"7777#$%& .AA *7777, ?7777$ *7777J !"7777C/j F!E07777G0G @7LE%& F"7M-%&Ax = y .7G=J#$%& .A=S7+$%&= <1 A ?/b 7C%& *7,- I%67(A <(i) <(ii)

Z"#$ *&kG%& ?$.

594257

251610

47/547/147/7

235/16235/22235/13

235/6235/67235/93

162213

)iii( ?/!=77$) HO =O ?/S77+ HO */!77;0 F077/ F77% W77-> Z&977M-(k = 0)?=77J0A < : .77$/` D!!,$%& =./ J09$%& 9+"-#%& 89: *+",.

% & 2355

472252 !$$!/ AADet

.&?@$ !"#$ %&'() *+!,-$./ *(+&01./ 2&(,$#./ %%3

!"# $%&'( ")*+,- .+"/0- 1"230- 45– 67*3 8"6#&9 :;6<0- 465 =-8'!# ! $5!>?30-A @86<!0- $5!>6?3 ABC $&"D)0- 45 '/&E- :'(0- 45 ")B?< =F+ G0;! H

I$66/!7*30- $5!>66?30- !66I =663&E- :'66(0- =J K,66)&5 H1 A . L"66+,- =667330- =663 ="667! !"66# $66%&'( =663 MN866+ :;66<B0 !"66# $66%&'(– :'66(0- 4665 166?<) =F66+ =-8'!66#

!>666?3 A666BC $666&"D)0- 46665 '666/&E- $666&O!5 $666&2B23 $5U P&!666*,0- $666&3Q'-!R .666+() S6662 88C 4>BR0-n T-'30- =3.

U)J T"+2V =73&!:

) J( !"# $%&'( ")*+,- -;V– !7*30- 8"#&9 =-8'!#1 A:

W'X0- T"&B3C 88C =Y5 /$3/%0-:

Z3#0- T"&B3C 88C! /['(0-: nnn212

23 23 !

) W( !7*30- 8"#&9 :;<B0 !"# $%&'( ")*+,- -;V "3)&+1 A:

W'X0- T"&B3C 88C =Y5 /$3/%0-:

Z3#0- T"&B3C 88C! /['(0-:

34 23 !

!667*30- 8"66#&9 =-8'!66# !"66# $66%&'( 4665 U66)J \J1 A 866C =!667& T"66&B3C 8 $366/%0-! W'66X0-)['66(0-! Z663#0- T"66&B3C 8866C G0;667! ( M"66+&'%, M"&!"66/33

23 n "663)&+ H

!I :;<B0 !"# $%&'( 45 1+"%30- 88*0-3

)4( 5(,16)5&$37 !8 (*9!:;$ :Factorization of a matrix

$5!>66?3 166&B<, A660V ]"66,<) ="66&<E- P66*+ 4665 $66*+'3A W'66X 166?"< A660V=&,5!>?3 : $&B>6/ $6&2B23 $5!>6?3 "3I-8<V(lower triangular matrix) L '6RE-! H

$&O!5 $&2B23 $5!>?3(upper triangular matrix) U.

=J \J: A = LU

_;66I 166&B<,0- $66&B3C A366/, M"66)"&<J! : 4662B230- .66&'>,0- !J 166&B<,0-(triangular

decomposition) . $5!>?30- '?")C ABC (!'` G")I =!7, M")"&<J!L '6?")C !J $5!>?30-U.

")&860 =J P'6>) H16&B<,0- -;6I 163C 45 :;<0- $%&'( =3 @8">,/N- $&>&7 $5'*30!n

45 $08"*3n 1!D#3: Ax = y

a&<A $5!>?3nn" . $5!>?30- b+?, H:;<0- $&B3C =3 @8<-! @!(R 8*+A:

Tc&8+, \J 8#!, N U)J P'>+ G0;! . $5!>?30- ABC ")B?< 8%5 SB*) "37!1A =3 $5!>6666?30-A 466665 M"+!'6666X3 1!E- :66666?0- ['6666(+ H+ ,111 / aai S6666O' :6666?0- =66663i .

$5!66>?30- @8"*,66/- "6))73&!A $5!>6?30- =6631A 4665 M"+!'66X3 1!E- :6?0- $5"66XY++ ,111 / aai SO' :?0- A0Vi a&<ni ,,2,1 !* W'X0- $&B3C d5"7& -;I!:

11ALA *

######

&&&&&&

niaaii ,,3,2,/1 1111 !**

8&8#0- =3&E- :'(0- =Y5 1230"+!1y $3&8%0- U,3&%+ (+,'&y $Oc*0"+:

11yLy *

466I :;66<0- $66&B3C 4665 $66&)"20- @!66(R0- =Y665 Tc&866+, \J 866#!, N U66)J P'66>+!$5!>?30- =&!7,

######

&&&&&&

4)"6620- :66?0- ['66(++ ,2212 / aa )) S66O' :66?0- =663i S&66%B0 G660;!ni ...,,4,3* . @'663! @8"*,/- "))73& 'RJ1A $&/7C $&B3*+: 221 ALA *

+ , niaa

...,,4,3,/1

########

&&&&&&&&

=Y5 -;7I!: 22111 ALLALA **

1230"+!: 22111 yLLyLy **

a&<2y =&,!(R 8*+ =3&E- :'(0- !I.

N U)J P'>+!W,7) =J "))73& .+/ "3 '&'7,+5 H Tc&8+, \J 8#!, :

U: $&O!5 $&2B23 $5!>?3 'Re.

z : =3&J :'( 'Re.

jL:$&B>/0- $&2B230- $5!>?30-.

###########

&&&&&&&&&&&

ijl: :fC"666X,3(multiple) 666O' :666?0- Sj S666O' :666?0- =6663 ['666(& \;6660-!i

1-Q,RN- $&B3C g")2J.

=Y5 G0;+!:

######

&&&&&&

) Mc23 (= L

a&<L $&B>/ $&2B23 $5!>?3.

121520

+ ,+ ,

121520

12/12/30

!"#$%& "'( )*!+,&-. )%+/ .01 2%3 !4% .0/56(non singular) 78#9% :#- .0 ;.6)<'!=8 >?@AB C:

A = LU ,

y = Lz

D'9L E F'GHG& FB6$I&!+9J +KH/ F'"LM-. +5"I+%N F'H$.

6U F'O6B F'GHG& FB6$I&.

FB6$I&-. @1 P6M% 0Q%'96A )HRH S9 !O))HT& UNS3 63 (@''GHG& @'H&+N V-1 L , U.

!"#(4-3):

& P##9 @##& P+##G& W##B F##'LX-. )*!+##,&-. F##N6&41-3 Y6+##4- Z0##9-. F##M'"L= )*!+###,&H- )<'!###=8 >3 @6!###=)-. [+###/8"*. @6!###= >3W###Q[4 ( FB6$###I& P###'H98 !###463 @''GHG& @'H&+N V-1 )<&+,&-.L , U.

WQ[4-. [+/8"*. @6!= Z09-. FM'"L= P9-. @& V-6?. 6LX-. \'XH8 +%%/&'WH' +&'B:

=J 8#) "D&0V ")B?! TN8"*3 $C!3#3 'Re =3!:

121520

=J =3 .%<,0- 1D/0- =3!LU = Aa&< H

121520

=Y5 G0;7!: yLz *###

5(,16./LU !"#$ %&'$( )*+,-(!. /(0.12"3 )4!56+$

!" #$%&'( )&*!+ ,-#&*$!+ . &/%* #&$$01 #&2 %1#3 4&!" 56(7&'1 8& %9*! #&2L, U

5 !#*!+ 5 :;$!+ )6 <=9%! >(#? 5 1@:+(/ A+B/*C#D:

)*-78:

A#&&&;$!+ 4&&&%3 >(#&&&?! <=&&&9!+ 5&&& 1@:+(/ =&&& 7$* E&&&011!+ E&&&1 E#&&&0 +="yAx

<(7&'%! 8 BD* FG E(BD 56(7&'1!+ 8& %9* E&011!+ E&1 H&$I6 JA K:&L 8&'#9 4&!"E M%M1 E %1#3 :A = LUN 9 J

! "ijL 1# 5 %7C 5 M%M1 56(7'1.

! "ijuU # 5 (%3 5 M%M1 56(7'1.

5 !#*!+ O#PQR!#D E *6(7'1!+ E *#S :'#$3 4TURV*(:

EG W:7$A 5RD:1 56(7'1nn$ . A P E1 51 P 80%6nj ,,2,1 :

ij,,2,1

1,,2,1

ija A&&&P: <&&&'!+ )&&&6 :&&&'$R!+ (&&&Si A&&&P: B(&&&1R!+(j )&&&*!+ X:&&& /Y+ 56(7&&&'1!+ )&&&6<=9%! >(#? 5Z :UD #2 %3 8'9$.

ijm K:#&&&&&&L!+ (&&&&&&S(multiplier) <=&&&&&&9!+ X(&&&&&&U/ )&&&&&&6 AB/*&&&&&&C+ F=&&&&&&!+ BB&&&&&&R!+ FG! " ijiji EEmE '$& 5&&!B#R1!+ )&&6 K:&& VL F=&&!+ BB&&R!+ FG J)<&&'!+ ( A&&P:j

5&&&!B#R1!+ E&&&1 ,*#&&&$!+ [:&&&UV !)<&&&'!+ ( A&&&P:i 8&&&91 5&&&?*#$!+ 5&&&!B#R1!+ 8&&&9* A&&&M J AP: 5!B#R1!+i.

))7'9+: @1:!#&D K:#&L%! #&$S #$:\Gijm E&1 ] B&Dij1 4&%3 5&!^B%! #&$S AB/*&C+ F=&!+B9!+ 56(7'1!+ )6 A#R!+L.(

)6 ]QM16:

+ :"; (4-3))_D#C!+ 8#M1!+ (S(:(

Q B&&D* FG O&&1* +=" H&&$G ;&&9Q ( 8+@&&*/^+ 5&& %13 `#&&$MG O)<=&&9!+( 8B&&D #&&$$I6 J 56(7&&&'1%! E &&& M%M1!+ E %1#&&&R!+ 4&&&%3 8&&&'9$ EGA E &&& M%M1!+ E %1#&&&R!+ 4&&&%3 8&&&'9$ J

56(7&&&'1%!A*&&&'9$ )&&&*!+ )&&&S( J <(7&&&' 4&&&%3 5&&&%D#Z1!+ OQ B&&&D*!+ 8&&&1RD #&&&2 %3 8 56(7'1!+ XB13G(A.

<*14!56+$( =$> )4!56+ :*?'1 )*+@A L , U

)6 4$1 <+:UG XB3 a1 >(#?! <=9!+ 5Z :U )6 81#R*$ EG E01 H$G A%R$ H&&&&C7$ O&&&&P(!+) OQ1#&&&&R1!+ 56(7&&&&'1 #&&&&2! O^B#&&&&R1!+ E&&&&1 O#&&&&3(1?1 XB&&&&3 8&&&&9$ FG

#2C7$ .( M0 E# 9G )6 E0!(5& +BD!+ =&$1 #$ B&! X:6(&*1 4$1 !+ <+:UY+ 80 E(0* ^ X: .

E *3(1?1!+ 89$ EG ]QM1 B :$ BZ6:

2211 , bAxbAx

N 92b )6 #1 5!+B1x .+ E1 (BD BP #$S(5 +BD!+ E1 <=9!+ B R$ EG F:(:L! #&11 J1R!+ E1 ]+: D0 ]+BB3 #$7%0 O# % . +=S K$?*$ < 0 Eb+ c:$C(): D0!+ BBR!+.(

56(7'1!+ 8 %9* #$RU*C+ +="A E * M%M1!+ E *6(7'1!+ 4!"L , U: A = LU

A#;$!+ EI6: Ax = y

A#;$%! ]#-6#01 E(0 : LUx = y

E U CD!+ E M%M1!+ E 1#;$!+ 4!" _:7* +=S(: Lt = y , Ux = t

A#&&&;$!+ EG d&&&L+((Lt = y H&&&%9 E&&&01 )1#&&&1Y+ W (R*!#&&&D(forward

substitution) A#;$!+( JUx = t )7%/!+ W (R*!#D H%9 E01 .

#&&&&$1%3 +=" +=&&&0S(L, U A#&&&;$!+ 8&&&&9$ EG #&&&&$$01GAx=y O#&&&& %1R!+ E&&&&1 BB&&&&RDF(#&&C 222/12 nn $ E&&1 ] B&&D3/3n 5&&Z :U =&& 7$*! K(&&%U1!+ BB&&R!+ (&&S( J5&& %13

(#?! <=9!+ 56(7'1!+ 4%3 >A.

#2&&C7$ OQ1#&&R1!+ 56(7&&'1 #&&2! )&&*!+ A;$&&!+ E&&1 :&& D0 BB&&3 8&&9 #&&$$01 e!=&&D(O# %1R!+ E1 8PG BBRD( 5!(2CD.

E && M%M1!+ #&&2 %1#3 4&&!" 56(7&&'1 8&& %9* )&&6 5 &&C#CG 5&& :;$ 4&&!" Eb+ : &&\$( .

`+:Z*C^#D 5 :;$!+ 5$S:D E01 ((induction).

)*-78 LU:

EG W:&&&7$A &&&RD:1 56(7&&&'15nn$ .EG W:&&&7$(kA 56(7&&&'1!+ )&&&Skk $

8(G aU#Z* E1 5$(01!+k )6 B(13( <'A .E#0 +=":

! " 1,,2,1,0det & . nkAk

XB 9( 5 %7C 5 M%M1 56(7'1 B?(* H$I6 ! "ijL 1

niii ,,2,1,11 N 9D

XB 9( 5 P(6 5 M%M1 56(7'1( ! "ijuU

H$G N 9D LU = A

<*14!56+?$ )5?1.+ -!6 L, U:

EG #$L:6 +="A 56(7'1nn$ 5!B#R1!+ EI6: A = LU

8 %9*! 51B/*C1!+A E M%M1 E %1#3 4!"

! "! " njiuU

8&&&M1*2n 8&&& S#?1!+ )&&&6 5&&&!B#R1ijij u,1 #SBB&&&3(nn %2 . ]#U(:&&&\ a&&&L$ EG E&&&01 (:&&&/b+ WRD&&&!+ E &&&R$( 8&&& S#?1!+ f=&&&S W&&&RD 4&&&%3 5&&&'#/ . NQ&&&M )&&&% #&&&1 6 :0=&&&$(

5&&&7%*/1 _:&&&U NQ&&&M 4&&&!" FBg&&&* U(:&&&\!+ f=&&&S E&&&1 O#&&&3(1?1– 5&&& 1SY+ W&&&RD #&&&2! 5 %1R!+– 8 S#?1!+ f=S E R*!:

)A ( )B*-C):1*$!0 ((Doolittle's Method)

U:*\$ #2 6(: niij ,,2,1;11

56(7'1 )6 5 :UZ!+ :'#$R!+ EG FGL B#9h #2%0) . 5&Z :U E1 H %3 8'9$ #1 +=S( ]#ZD#C #$ G: #10 <=9%! >(#?.(

)ii( )B*-C)D!-E ((Crout's Method)

56(7'1 )6 5 :UZ!+ :'#$R!+ E(0* EG U:*\* f=S(U %0 ]+B#9h #2.

niuii ,,2,1;1

)iii( )B*-C)FE2$!G1 ((Choleski's Method)

E *6(7'1!+ F:UP :'#$3 _DU$* EG #2U(:\(L, UEG FG J:

niuiiii ,,2,1;1

D"4!56+?$ -G"3+$( :*?'1$())*;?;+$( "H?+(!I =$>(

Direct Factorization of Matrices

4&&&&&%3 8(&&&&&'9!+ #&&&&&$$01 <&&&&& 0 #&&&&&$ G: E &&&&& M%M1!+ E %1#&&&&&R!+ UL, 56(7&&&&&'1!A

>(#&&&?! <=&&9!+ 5&& 1@:+(/ A+B/*&&C#D . E %1#&&&R!+ E =&&S 4&&%3 8(&&&'9!+ e!=&&0 #&&$$01 ( 5PQR%! 5%D#Z1!+ 5U CD!+ O^B#R1%! :\#D1!+ 89!#DA = LU

)!#*!+ 8#M1!+ E1 e!= E D* #10:

+ :";(5-3):

56(7'1!+ 8%9:

E M%M1!+ #2 %1#3 4!"L, U N 9

3,2,1,11

)8* !(B 5Z :UD 8 %9*!+ FG.(

5PQR!+A = LU i6#0* X#UR1!+ U(:\!+ O9*:

131211

8 S#?1 5RC* )6 5 !#*!+ aC*!+ O^B#R1!+ #$ UR* f=S:

! "1911

3333191

3323321331

3232219

22321231

31311131

232327

231321

222227

221221

21211121

113112

'%&$%$&

'%&$& %

& '%$ %

'%&$ % &

#&2% %9* 5&Z :U e&!= B&RD :0=&$ A&M J O#6(7&'1!+ E&1 ]#&'#/ ]#3($ <j:R$ Eb+(5 M%M1!+ #2%1+(3 4!".

%*-J1:

8#Z )4!56+$A #2$"0*0'1$"3 )3#!+ (positive definite matrix) O$#0 +="

) G( 5%M#1*1 56(7'1 (symmetric matrix).

) K( 51 PAxx t F:7' : k H?*1 FY 5D?(1x.

5&&&&Z :U _&&&& DU* B&&&&$3))0&&&&C!(\* ( EG ,*&&&&$ B&&&& B9*!#D 5&&&&D?(1 56(7&&&&'1 4&&&&%3 5 :UZ!+ :'#$R!+) )&6L (GU (?(1 #&2%05&D 5& (%R!+ 5& M%M1!+ 56(7&'1!+ EG( JU )&S

:l(B1(transpose) 5 %7C!+ 5 M%M1!+ 56(7'1!+LEG FG J:

tLLLUA

5 D#&&C9!+ O+(&&&U/!+ BB&&3 8&&&Z e!=&&D(– <&&&'$!+ 4&&!" ]#&&&D :Z*– 56(7&&&'1 8&&& %9* B&&$3 5&Z :U!+ f=S . /%* E01 ( J )0C!(\* 5Z :UD #2 %1#3 4!" B B9*!#D 5D?(1 8& %9*!

)% #1 6 O#6(7'1!+ E1 m($!+ +=S:

tLLLUA

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Factorization of Positive Definite Matrices (Choleski's Algorithm)

5 %7&&C!+ 5&& M%M1!+ 56(7&&'1!+ :&&'#$3 K#&&C9 E&&01 L K&& *:*!#D ]+B(&&13 ]+B(&&135 !#*!+ n '!+ A+B/*C#D:

11111 a

45 & 6

kjkikij

jjij a )5 :UZ!+ : k :'#$R%!(

kikiiii a )#$R%! +B3 5 :UZ!+ :'111(

+ :";(6-3):

5&&& !#*!+ 56(7&&&'1!+ 8&&&%9 )0&&&C!(\* 5&&&Z :U A+B/*&&&C#DA )&&&6 B&&& B9*!#D 5&&&D?(1!+X:('!+ :tLLA .

5.375.21

75.225.41

312111

333231

5.375.21

75.225.41

( ) ( )( )

( ) ( ) 15.15.05.311

5.15.05.075.22111

25.025.41

322313333

21313222

22212222

&''&''&

&''&'&

5.05.02

15.15.0

$%!&':

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L)!1+"tLLA &

@J3+" 7.M=+" A:= 7-

yxLLt &

455-,-&+" 45&.M=+" A: @=*5 N#+"!:

O%J*&+" @%J3+" 7.M=+" A:+ 'P5)J+" E# !Lx = y 4%& 99%D Q")%;> O%+> N9R%/%%5,&*+" 99%%D S%%1= T.%%25)P/ N!.%%F5 '52.%%F:+" <.%%5,&*+" S#%%:+" '%%P5)J @%%( '%%2!,J&+" <.

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29 23 nnn **

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76 23 nnn '*

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@%J3 A%&.D(linear factor) @%(n '%5!=*&+" '%&5P+" '%5:.= 4%& ^Y.=/5%F _%=B( ?(in

significance) L9.5H Z&n.

!" #!$% &'()*+,+-%.&/ 0/1,! 2 34 , 54,6-%.7 7849 :%4, ;$ <(= >- 0/ !" 0,?,/@%.7 @!8 .7 5A7+ @,+&B".7 5$ #!$% #=) :C@ 0.DE".7 @>(727-3.(

!"#$%& '()*+,$- .,/-&"01 2-#, .034#, 56

;4444FA.7 :&4444>(.7 54444- 3!44446F".7 #= G@4444H(Ax=y I4444,-A 04444/ !" 0*!H4444J"+44444,+-%.&/. $44444E.!K% 044444,"L@7!A :+A%44444E( #= 544444MH, 044444.&-.7 N8444449 ;44444* 544444,6-%. OP!= ;

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()*+: @,0F+" @-,-&+" 7.M=+" A:Lz = y.

#$% '&

(,-.,/: N!,*+" @-,-&+" 7.M=+" A:zxLt &.

1,,2,1

#$% '&

ijjjii

$%!&':

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#$% '&

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$-/0/'"# ,10'#*2 3"4 5,6*78'"# -0!9 *! :;,9.

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.& '(!01& 4!$/ 48 4$&&+" 4&A L9)0=&(singular) A%5,:/ .C+!L-U !8 ? L9)0=& )5e 4!$/(non singular) A5,:/ .C+ V5+!L-U.

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<PP: "#>A T.5*JY '5)JP+" L)J5F+" J)G(strictly diagonally dominant).

ijii ,,2,1;1

!!!!"#$ %&A '!!!!(')*+#, -!!!!,.%/(positive definite) -!!!!0(12 3!!!!(,2* 4!!!!$/( 5!!!!"67 8 9#:"+; <=> ?%#.+ @A)+;Ax = y %',-7%B!C/+; @%B!C+ D('!,* E& F!/> 4 80*!G/ -!(=/H+; IAJ 4%$*% K1(stable) #!,(10*+; L#!2MN, 1ON!** P E& 5!"67 Q+#!*+#,% 8

F(=)* 4$/(A 4((O=O/+; #R(=/#> <+SL , U.

#7%B!C/+; 4!/ T#!M U%" -(120+; -(ODO #7%BC/+;; F/*!V* 5!(7 -7%B!C/+1#B!!!CW; 4!!!/ 1!!!(O$ <!!!=> -!!!,%=2/+; -(,#!!!G)+; #!!!(=/H+; ''!!!> F!!!(=0* 4!!!$/( Q+#!!!*+#,% 8

#R(=/#> <+S -7%BC/+; F(=)*+.

QJ -(120+; -(ODO -7%BC/=+ -/#H+; K1%C+;:

#########

&&&&&&&&&

nnnnnn

,11,12,1

343332

232221

Q!!!7 2!!07 '!!!.%* 4& 4!!$/( -7%B!!C/+; IA!!!J Q!!7 -(1B!!!C+; 1!!(X 1!!C#"H+; 4& E&+; 1!!20+;K1!!V#,/ I1#!!G( <!!=> %& K1!!V#,/ 5!!"(/( <!!=> %& Q!!G(Y1 . K1%!!C+; 2(!!G,* 4!!$/(%

Q+#*+; F$V+#, #R*,#*$, -7%BC/+; IAR+ -/#H+;:

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(Crout) Factorization of Tridiagonal Matrices

########

&&&&&&&&

,*+,*+

F(=)* 4$/( 5"& Z1B"A -(+#*+; K1%C+#,) %1$ -0(12:( A = L.U

#########

&&&&&&&&&

############

&&&&&&&&&&&&

2H( ;A!!!!!!J% E%#!!!!!!G( P'#!!!!!!H/+; 4!!!!!!/ \;''!!!!!!> #!!!!!!"(3n-2 4!!!!!!/ 5!!!!!!GB" ''!!!!!!H+; Q!!!!!!7F(J#./+;- .iii ud ,,1 .QJ P'#H/+; IAJ%:

11 d"* (1)

niii ,,3,2;1 ""+ (2)

nidu iiii ,,3,2;1 1 "/" )* (3)

1,,2,1; )"" niud iii , (4)

#!H/+; IA!J 9;'M*!GP \P#!O/ ]1" 4& F, % 4(*7%B!C/+; 1!C#"> 4(!(H* Q!7 P'L,U E& U%!"+; ;A!J 4!/ D/#H/+; -7%BC/ 4%$* 4() Q2M+; 9#:"+; F) <+S 1(V"

-(120+; -(ODO -7%BC/(tridiagonal).

4& Z1B"A (=)* 4$/( 5"&% -(120+; -(ODO -7%BC/ 4((O=O/+; #R(=/#> <+S #R=L , U %J 5=) _%=2/+; Q2M+; 9#:"+; 4&%:

Ax = y

0 LUx = y1

4& Z1B": Ux = z (1)

0 Lz = y1 (2)

9#!!:"+; \P%& F!!)"7(2) 9#!!:"+; 9!!O(1) _%!!=2/+; <!!=> F!!C)"+x 4!!/ F!!^& ''!!H, ?%#.+ @A)+; -0(12, F)+; ;AJ <=> F%C)=+ -,%=2/+; #(=/H+;) 4%', E& F!(=)*

-7%BC/+;A #R(=/#> <+S.(

!"#$%& '()%& *"!#%& +,- ./ (0%& ./121 .3-45$%& +/6,7 ./$8 &-)

-!(120+; -!(ODO -7%B!C/+; F!(=)*+A 4(!(O=O/+; #!R(=/#> <!+S \#0,#!G -!71H/+;L , U

\#0,#G 4(71H/+; Q2M+; 9#:"+; F) 9O 8Ax = y -(+#*+; ;%2M+; a,*":

9:-; :%& +/6,7.3-45$ A:

FH.; 11 *"d

_G); 111 / du ,"

9(0=+i = 2, 3, ... , n-1 _G);:

FH.; 11 ))"

9"/#"1 :'()%& *"!#%& +,:

) &( 9#:"+; F) Lz = y

W¹dDI�« wŁöŁ wD)« ÂUEM�« qŠ

Solution of Tridiagonal Linear System

FH.; 111 /dyz "

9(0=+i = 2, 3, ... , n _G);: - .11 1 ))" iiiidi zyz

) _( 9#:"+; F) Ux = z

FH.; nn zx "

9(0=+i = n-1, n-2, ... , 1 _G);:

- .- .

")))")"

1 !" iiii xuzx

!"#(7-3): ! !"#:

$%&"'()* +,-A ./01,(.2 3)4L, U ) $/51!67&!/8(9 :./;#)* +/- </=&>&Ax = y

!"#: :./;#)* +/-& $/1!65)* $/1@A@ $%&"/'()* +/1,-B $/1(C!*&D 7*&6D E.FBGF

H,1 .( <=# $1!65)* H@A@ H6D)*:

$%&' : ()&*+,"# -.!/A:

"###"#"

A = LU

!"#!$ :%&'() *!+#() ,-:

! ! ! ! ! ! ! ! ! ! ! ! ! ! !

4/5111

3/4111

2/3111

"##"#"

! ! ! 1

"##"#"

- ! "#$%&' (): 14321 """" xxxx

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nidi ,,2,1,0 "./

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) C( G/G)9&#@ .@H 7 "#$%&' *4 I37#J7&' .4 567.

) K( F#/J,L ./0,1&' :0,/M&' ,0A D1)9 "#$%&' *4 I37#J7&' .4 567.

N<++++! D++++/@,9 .++++/%#;7E O++++11)9 >7++++P/ F#/4#++++PE F#,0++++A *++++,J9 .++++/&#9&' .++++/0$%&' ./7=0' -&'.

."1+#:

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.+/,-&' "$%+&' Q+J@ *+4Ax = y I3-G+7&' *+4 .+5/5, I'0+/W9 BGX+9) 0+6#%Y I37#++J7&' .4 5++67A O++H97&' 0++6#%Y Cy #++7?/8; C ( O++H97 *++4 :0++/@; I'0++/W9 Z++&E

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0 ,!$(8-3):

*,-&' "#$%&'Ax = y >/9&G#J7&#@ [R0\J]7&':

0001.320001.1

>C BC:

0001.3

20001.1

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◊dA�« WKOKF�« rEM�«Ë ◊dA�« WM�(« rEM�«

Well-Conditioned and Ill-Conditioned Systems

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0001.329999.0

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9999.220001.1

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#?%Y :G/J@ .7/G1&' .,1%&' >Y F#7#79 .589-7 :G/GH&' V,#19&' .,1% > ;9.

(;++++++A&' >++++++7 b0++++++% `&<++++++;) >/7/19++++++M7&' *9&G#++++++J7 >++++++7 C21 1,1 ( .++++++,1% >C

(++27 !0,33ax >/7/19++M7&' G++)C Z++8Y V++19) ++! 11 (0++-c' >++7 .++@/0L !21 Z++%J7 T

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>=0#-%(3, 0) E F#C- 7 ).)/0, 785 1*9,6Ax = y G 65. !"#$%& '(#$%&(residual

vector) )A.6%/-% HI,3-%.(

0002.0

0001.3

20001.1

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/ )+3 1JB K-&.*(norm) :* 6) HI,3-% %&'

0002.0max21

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L/c+M) )56748$%& /0412$(matrix norm) >!)0)05-% ] B*S!P3-% >!?*3I3 Q!C? >+./3-%nn5 >!)0)05 >!-%2 H#d.. 1),+./!3 1),B*S!P3 :g >!)- ,-% =*/!^-% 4!05,

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/)) +3-% 1). 13*)>)+).=-% (E%2T,6% /`9) ] B*SP3C-:

3-% / )+)1: )&))

& xmax1AA

/ )+3-%21: 212

2xmax AA

]# 9 %&\ H#$ ] .`\ 193)*3 4ijaA & >+./3 >B*SP3nn5 1JB:

La/i+M# U`3- .*:

111max

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A >B*SP3C-

11 &',,&0

12 &',,&0

13 &,',&0

1 2 77,4,4max &&6)

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xmax1AA

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7 Y)$ / )+3-% %&' Q63)*;<= >+?$%& /012$%& (induced norm).

)1/*@:

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1ea .rxx '(' A (i)

..xxx 1

ea '('

AA (ii)

) 1$ =/^.0y,0xe 77.(

O)5r A0.,3-% HI,3-% *'(residual vector) A!.)/0,-% U5C-ax >.!6#- . E !F#C-Ax

A0B/"%&: >)- ,-% >)2)"3,-% ] .`f 7[*$ b ,5#:

)1=1C$#:

H!!!!I,3 :gx :* !!!!6) 2!!!!+M. :&n >B*S!!!!P3 :$*A >!!!!+./3nn5 ) / !!!!)+3 ]%& A+).=A:(

x.x AA (

)1=1C$#%& A0B/":

) 1gx.x // &.(

1$ #Y/B %&JB xxu &

1xxu &&

1JB K-&.*:

umax.x

)A+).=-% / )+3-% L)/+, 13.(

>- 5-% AB >5Y%* >)2)"3,-% 1,3 >5P*x = 0.

31/2#: 1 3I!6#3 !3"#\ >B*S!P3 / !)+3* H!I,3 / )+3- U 0)(compatible) !005 %&\ >N8+-% x.x AA (

] "I,3-% <)3I- K-&*x ABnR >+./3-% ] B*SP3-% <)3I-*nn5 0)05-%)>.

)1/*@%& A0B/" D"0#:

)$( >#) .,3-% ] .`\(i):

3 4rxx

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>06,3 @/*P. 7 3j%2 H3%2T,6% UF) 1$ QC?.

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3 4 1A.AAK '&

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behaved).

(Condition Number) ◊dA�« œbŽ n¹dFð

31/2#:

>B*SP3- U 0)A !"#\ @2/S#3 /)W:/4E%& )@4F+ )56748$ (well-conditioned

matrix) =/!!^-% 22!!? 1 !!9 %&\K(A) 1!!3 !!.)/N >B*S!!P3-% _&!!"-1 . !!"#\ U !!0)* )44G1GH

:/E%& (ill-conditioned) 9$ =/^-% 22? 1 9 %&\ 13 /)`9. /.1.

E !F#-% 1\ U !0)*Ax = y :/4E%& A4F+ I04*@ (well-conditioned system) %&\ >B*S!!!P3-% ]!!!# 9A H!!!#\ U !!!0)* ; =/!!!^-% >#!!!65:/444E%& J4441GH >B*S!!!P3-% ]!!!# 9 %&\A

=/^-% >C)C?.

$ J0K(9-3):

E F#-% 1$ ].`%Ax = y U `3 AB Q=+3-%k-l =/^-% U)C? E F#.

20001.1

121max

0001.320001.1

1 2 0001.30001.3,3max &&8)

50005.5000

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3 41002,60000,200001.3

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! " ax/z.10tAK #

ax: +$Q'( @,O2T' +.;%1-'( PB'( Ax = y

z: +$Q'( @,O2T' +.;%1-'( PB'( Az = r

r: +1.-C'( FA-C'( axyr A $

6 ,2""""X%# (48ax F""""A-C'( @()Q-""""?( ,""""22=C;# +""""$Q @,""""O2' +"""".;%1- P""""B /""""5 +1.-C'(r 9 @,"O2T' $/."XC'( P"B'( /B2 K%,1-- 0D.,--C 0;.;%1- P/TB YT* P/NBT'

0'/.1C 0<)' PN2 ;B 0;%;%=-'( 0;TCD'( 345 ></2/.

/""5 +""$Q'( @,""O2'( 6 J%""M2Ax = y$/.""XC'( P"""B'( 6/ 9 /""5ex 6/ 9+.;%1-'( PB'(/ $/.XC'( PB'( ;. L%,M'(ae xx x $

/5 +.;%1-'( PB'( (4E' +1.-C'( FA-C'(:

! "! "

©Íd¹dJ²�« 5�×²�« Ë√® Iterative Refinement W¹d¹dJ²�« WOHB²�«

(or Iterative Improvement)

F""""A-C'( 6 Z6 x +""""$Q'( @,""""O2'( P""""B /""""5rx $A . @,""""O2'( (4""""5 P""""B./'(/YT* P/NB x ,22=C;) :,;%O2 ( $/.XC'( PB'( YT* P/NB'(ex W;B:

xaxex %$

='/– 0";TCD'( 0";B,2'( "C– K";%1-'( S,"$Q6 [,"25 !"#(rounding errors)

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,E-1.,? Cax . %;%=-'( ,22=C;/(iteration) 0";TCD'( 345 %(%=-.– K,"?B 0";TC* x

0""C;< K,""?B/ P""BT' 0"";.;%1-ex - 0""'/.1C 0""<) I(4 0""C;< Y""'8 @;""1'( K%,""1-- 6 Y""'8 .

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K0?($% E?$% C$D: x = Tx + c

*W%K7,% Q0'/7 =X, 20WK7,! ! nR00x (: @

! 11T2

%!&'()* +',-: ! !

! ! cccx.

! !cTTTIT

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..........

1'V$% QB170 &K"F Q'"/7$%(:

0lim "3.

kT (1)

2(-0 *- *+"-$%( ='('6$% 1'V$% Y"60< (: %9: 2-$(:

! ! 12lim%

3.%"$$$$ TITTTI k

*MM+ '(-9MM!$% 1'MMV$% UMM/?7 %9ZMM+ %9MM-:((1) MM0"T3$% 2ZMM+ @ !kk

xlim3.

"MMT$ 2(MM-0

K(.() =("P7(x YI[!( 4I#$% CBF E)?3( @:

»—UI²K� WOÝUÝ_« W¹dEM�«

;.7!$% 2< =<x YI? E[!0.. M4I#$% CBF K!7#0 E? C$D Q'"/7$% 2Z+ *$"7$",((1) @(: 4I#$% 89: U/?7$ *+"-$%( ='('6$% 1'V$% 2< \(B#!(:

ii 41 ;15

*$"7$",(: (T) < 1

./012:

]3"- "!B- 'P< /0'1$% Q'"/7 2"- @'H)<.

3!$2(14-3):

*1O$% \"_3$% E?$ *,(-". /0'1 \KO7P3P "33< 'A3

;MM//?7 2< QMM.0 =9MM$% 1'MMV$% KMM.(< /,"MMP$% MM0'_3$% \%KO7MMP",6 89MM: Q'"MM/77 CMM7? @ *W%K7,% ;.7! =a /0'1$% !0x.

30)*: (: /0'1$% Q'"/7 1'V:

*: &L0!!$% $K"#!$%:

Q'"/7$% 1'V 2(-0 b$9,(

2%16 (<286

% 'MM1/$% cMM)3 Q"MMP? Y"MM!(!F ETMMP$% 2MM! J0MM$ *MMA01$2 ]"+(AMM)!B$ ,MMP3$", &'MM0,-$% !39n MM0+"- Y"1('MMV 2(MM-7 >MM0?, "MM:'",7O% ETMMP0 1('MMV ]MM[ d?e, KMM/+ %9MMT$( @

(sufficient conditions) MMM0'0'-7$% U'MMM1$% 89MMM: Q'"MMM/7 "MMMT3! =< 2!MMM60 . 89MMM: 2MMM! ('V$% +(A)!$% '"0#! Q"P?, UB#70 "! 13!( @!B$ 0'1/$% &'10P$% 1'V "T +(A) @

70 "! "T3!(K0K?7$", ,.(! +(A)!$% 2(- QB1 0$"7$% ]"0'_3$% b$9 R6(7 "!- @.

."'/#: 0'0'-7$% H0)$%:

! ! ,2,1;cx.x 1 "$"$ kT kk

K0?($% E?$% C$D Q'"/77 x = T.x + c

*W%K7,% E? =X, 20WK7,!nR0)0(x1'V$% U/?7 %9D @:

0? >T '0'-7$% +(A)!$ '"0#! =< (:T.

%!&'()*:

Y%'MMMM_3 ;MMMM3a– /,"MMMMP MMMM0'_3 2MMMM!– MMMM4I#$% UMMMM/?77 ]"+(AMMMM)!B$ '"MMMM0#! =a 0$"7$%: ! TT :2

1'V$% U/?7 %9Z+ 11T

2Z+ ! 11T2

'"/77 b$9,(Q'"/7B$ 0P"Pa% 0'_3$% CBF Yf"3, 0'0'-7$% /0'1$% Q.

E"[! *+ YI[!g-hi:

1'V$% U/?7 %9D: 14 16

=<: 14

=<: 486 (<4%16

77 *,(-". /0'1 2Z+*W%K7,% *,0'/7 E? =X, 20WK7,! Q'"/.

X1OB$ K(K? f"1FD 5! /,"P$% 0'_3$% E!V7 0$"7$% 0'_3$%(error bounds).

."'/#: +(A)! '"0#! =a *$"7$% 1'V$% U/?7 %9D 11/ T;

]"T.7!$% #,"77! 2Z+ ! ,2,1,0;cxx 1 "$"$ kT kk

Q'"MM/77– *W%KMM7,% ;MM.7! =a ! nR00x - ;MM.7!$% CMM$DnR0x UMM/?77 ;MM3< "MM!- @ 0$"7$% X1O$% K(K?:

! ! !kkk

;xx.xx

MM/0'1B$ MMB["!! MM/0'1( /,"MMP$% MM0'_3$% \%KO7MMP", MM0'_3$% 89MM: MM3:', 2MM-!0 MM/0'1, MM$K"#! '9MM. Q"MMP? *MM+ XMM1O$% '0KMM/7 *MM+ U,"MMP$% EMM)A$% *MM+ ]MM#,7% *MM7$%

3$% MM7,"[$% MM1/) MM/B1!$% MM!0/$", '"MM0#!$% E%K,7MMP% 5MM! .( 20MM, 'MM0,-$% ;,"MMV7$% _MM?I0( /,"MP$% &'M0Oa% M30",7!$% 20,( 7,"[$% 1/3$% /0'1 *+ X1O$% 30",7! . M30",7!$% 89M:(

1'V$% U/?70 20? '0'-7$% c"/0D 2! "33-!7 &'0Oa%. ! ! <:%$ kk xx 1

>0?< !$% R!"P7$% E[!0;, j(!P . c"M/0k *$"M7$% 1'MV$% \%KO7MP% "3e3d- l!e0 "!-'0'-7$%:

) 1'V, !0x

1ki &$.(

./012)1(:

+(A)!$% K".0D ETP$% 2!T *,(-". /0'1$ ,P3$",)K".0D =<jT U,P "!-"MMMTA0'#7(!$% K"MMM.0D Y"MMM!(!F Q#MMM)0 2MMM-$( @ +(AMMM)SGT % +(AMMM)!$% *MMM:(T MMM/0'1$ J(".– EK0L . M$mK, 207/0'M1$% 2M! EM- Q'"M/7$ "M0+"- Y"1'V *1#7 0$"7$% 0'_3$%(

0B)a% +(A)!$%A ) \"_3$% E?$yx "A ( '0'-7$% +(A)! $mK, J0$(T.

.444/012: 2(MMM-7 MMM0'1/$% &'10MMMP$% 1'MMMV UMMM/?7 +(AMMM)! =< 2< ]"MMM,[D 2MMM-!0K0K?7$", ,.(!.

.4"'/#: +(AM)!$% ]M3"- %9DA *M1O$% \"M_3$% *M+Ax = y !KM0K?7$", M,.( @ /0'1 2Z+G-S *W%K7,% ;.7! =< '"07O% 5! Q'"/77 0'0'-7$%.

3!$2(15-3):

) <( /0'1 2! YI- 2< 2n0,J /0'1(G-S *1O$% \"_3$% E? K3F Q'"/77

K0?($% E?$% C$Dx *W%K7,% *,0'/7 E? =a ! nR00x.

) Q( 1 U,13MMMMP "MMMM33< 'MMMMA3 MMMM/0'J \"MMMM_3$% %9MMMM: EMMMM?$ *MMMM,0'/7$% EMMMM?$", 20WKMMMM7,! @ ! !t000x 0 " . 'MMMM0'-7$% ]%'MMMM! KKMMMMF 'nKMMMM4(k) CMMMMBF E(MMMM)?B$ MMMM!LI$%

2% =< @ 0'VF \"4'< 7P$ Y"?0?) E?$%: ! 6105.0xx %=:% k

) <( ]I!"#!$% +(A)!: (((

MM0'1/$% &'10MMP$% MM0#14 +(AMM)! @ 2MM! YIMM- 2ZMM+ /,"MMP$% MM0'_3$% CMMBF f"MM3,( /0'1J /0'1(G-S *W%K7,% *,0'/7 E? =< '"07O% 5! Q'"/77.

) Q( /0'1, E?B$J 0$"7$% &'()$% *+ ]mK"#!$% 563:

!"#$)2:(

!"!#$ % !&'#( )*+, -#,-*'. / -#+ !!#0 -,!#1' 2#3 45165#( 7-"*8#+. 9#:&.(k)

;##<,=+ ;##>! ?##.@ A"##B"C. ;##6"C:+. +. )##+ D##E', 5##+* F A##,C> !##=6 G-*8<##( H8##. A5##I) A5I+15-3.(

%%%&'!(: ;3"J###B+. /###11$ 8@A 5K-"!###+ "LtA ;###,=:1. ;###,-:1. 7-:,###(. :-###M

(strictly diagonally dominant) N###3 ;###1,-: )###+ 4O###* )J ;###1,-:"G-S %5###P<. A###$.yx A !,$". A$. ?.@ Q-51''x 2R !'6 26,-1' A$ HS! " n

! ! ! 4/6

!"#$!%&: ! !t000x 0 $

'(): ! ! ! ! ! ! ! 5.1xx

'* +, -.&: ! ! !01

1xx.xx #&# #'

/0"1 !.2%3 ,4()J 556 /" " 7#8,k 9%1" :#; <, =8, '=

! ! (*)105.0xx 601

## (&#'

'> ?#."): ! 6105.0xx #(&# k

@AA%" 2#8, BAA;8, '&AA7" C> !kx /" AAD6 E!AAF > /#AAG8 +!;";AA1 HAA"8I !.JAA1& C4AA8,.

/."!%#=8, '78&(*) – !"K=8, 4$L. M"; @N!O.P8,– @.K#:

! )(*105.05.1. 6

1)(& #

M"; %$ jT'

2351653.22

105.05.15.01

5.0)(*

(&(#0)

! "# $!%& '()* +,%-.% /,0,()1* 2*3450 6721, xx *.89(,06 ! "#$%1,91* :3;1*:

" !"#$ %&'( )*+,-.

%&'( )*+,-– ./0,&(.

12&3'( )*+,-)4 )+,56$/ 7+*$'2 '8092: ;/,+,39.(

%&'( )*+,-– <0+=)4 *$'2 '8092: ;/,+,39+)+,56$/ 7.(

( )*+,-2 >$!3 0(&? %&'– .: @3 ./0,&(: )(det,1 AA

;A0'B:$/ )C&:(: <" 0(&? )+$'9$/ D: %&'($ !"$/ )*+,- 7/0E9F'2 G'-E? H? IJ(9$ >$!& K )*+L0$/ )+0'+9CA/ )+,F3$/ )M+6$'2 ;'2'F"$/ )N'3 G/,(O

I+,*9$/ )(+9J.

,52 )*2'F$/ )$PF:$/ <" 0C? 1$Q IF'" 'R9'2'F" G/,(S2 7&*+F H!$/ .? TU2'F$/ <"$/ D: .,'L& K;L& H? 1N -*N .++&JB: .+:L,2 V'59"A/ D+-9F+ . W/,VJ&

(1-3) s¹d9

(2-3) s¹d9

(3-3) s¹d9

;'N&56: X,&6 1N )+$'9$/ )+JY/ ;A0'B:$/ )C&:(: I93/yx !A:

Y�U¦�« qBH�« �UM¹d9

Z![ .: 0&6*:$'N K 7'L,\/ .: 0&0": 00B2 V'59"A/ D+-9F+ IF'" H? .\ ] &9 .? )$PF:$/– ': 0" _$O– ;'2'F"$/ Z![ <`: 1N a0"+ ':.

1#+ ':+N.+9+JQ .+9+-E .+9$0'B: .: )J&3: )C&:(::

)?( )i( )+N&56:$/ X,&6$/ 1N )C&:(:$/ D^Ax = y.

)ii( )*+,-2 )C&:(:$/ <"G-J ;'2'F"$/ )N'3 G/,(O D: K !"#$I+,*9$/ G'-E? IJ(9$ )-&2^:$/ )+,F3$/ )M+6$'2.

)iii( .+:L,2 V'59"A/ D: )C&:(:$/ <" 0C? 00C H? 1N -*N .++&JB:<"$/ ;/&-E )N'3 <@E. U2'F$/ <"$/ D: .,'L P-E$/ IF"/& K

.: <3 1N 12FJ$/21, xx.

)I( )i( ;@:'B:$/ )N&56:$ )+9/!$/ 7+*$/ 0(&?A.

)ii( )*+,- H0b9 .? .&:^:$/ .: <[G-S )C&:(:$/ <" _$O )+,+,39$/ c)*2'F$/)$O I,'*99F <[ H?c)+8/092/ 7+L HP2 .+8092: <"$/ _.(

)iii( .+9:+*$'2 W'8092:: # $ # $5,5 0

1 ! ! xx

-2/,+,39 )B2,? U; )*+,- .:G-S d,9 /!': ,VJ/& K:

cA 7? -&2^:$/ <"$/ &"J )B2'99:$/ 7+*$/ I,'*99 <[

)2'(e/ ;J'3 . f/& : .+9+8'RJ$/ .+9:+*$/ .: <3 1N P-E$/ IF"'N K7BJ .+,+M9:#$21, xxg8'9J .: h+#C <6"9 ': _#C U#C& K.

;A0'B:$/ )C&:(: <"$ 70E9F9 19$/ %&'($ !"$/ )*+,- .? 7&#B:$/ .: .&39 .? <"$/ ;/&-E G'J`? 1C&, /!O U0? g8'9J 7&:B$/ _#C 1-B9 )+JY/ )+-E$/

(4-3) s¹d9

(5-3) s¹d9

Y�U¦�« qBH�« �UM¹d9 ∫Y�U¦�« qBH�«Y�U¦�« qBH�« �UM¹d9 ∫Y�U¦�« qBH�«3

,9 U+,- .C >$!& K.3:+ ': ,23? H=39,:$/ ,6JB#$ )*#-:$/ ):+*$/ ;A0'B:$/ I+9;/,+M9:$/& . _$O <&6&#$ ;A0'B:$/ I+9,9 )+:[? .'+2 )$PF:$/ Z![ .: 0&6*:$/

g8'9J$/ 1N )L0$/ Z![.

)+-E$/ ;A0'B:$/ .: )+$'9$/ )C&:(:$/ <" I&#-:$/ .? T,5J: 0.50 x + 1.1 y + 3.1 z = 6.0

2.0 x + 4.5 y + 0.36 z = 0.020

5.0 x + 0.96 y + 6.5 z = 0.96

.+:L, i7[\ X'-B: ;@:'B:$/ a+" . g8'9J .? T,5J I+,*9$/ G'-E? <+#*9$&7'L,? )`@` 7[\ I939F ;@:'B: H?& )+$'9$/ ;'2'F"$/.

)?( ,++M9 H? .&0 X'-B:$/ ;A0'B:$/ )C&:(: <" !"$/ )*+,- 7/0E9F'2 0(&?'R2+9,9 1N.

)I( (: <" !"$/ U+,- 7/0E9F'2 d,E? X,: 0(&? D: .3$& ;A0'B:$/ )C&: .3:+ ': ,23? H=39,:$/ ,6JB#$ )*#-:$/ ):+*$/ .&39$ 'R2+9,9 ,++M9) W'8092:

)`$'`$/& _$&\/ .+9$0'B:$/ <+0292.(

)j( .+*2'F$/ .+#"$/ .+2 .,'L.

)+JY/ )+-E$/ ;A0'B:$/ .: )+$'9$/ )C&:(:$/ <" I&#-:$/ .? T,5J:

2 10 13

& &*+ A IF'"$/ .? .: ,`3? .+=E92 7kl )+,mC 7'L,? <:'B: .? _JB:2 Kx

W/,56 Z,29B+F _$&\/ )$0'B:$/ 1N.

) ?( <nnnB( 0nnnB2 X'nnn-B:$/ ;A0'nnnB:$/ )nnnC&:(:$ ]+"nnn6$/ <nnn"$/ )nnn*+,- HPnnn2 0nnn(&? <:'B:x W/,56 _$&\/ )$0'B:$/ 1N.

) I( 3 X'nn-B:$/ ;A0'nnB:$/ )nnC&:(: <nn" %&'nn($ !nn"$/ )nn*+,- 7/0E9nnF'2 0nn(&? 'nn: In+,*9$/ D: .3$& K <"$/ G'J`? ;A0'B:#$ <+029 H? .&0& ,++M9 H? .&0 1[

(6-3) s¹d9

)rounding( D:($'nnnnn3 )+2'nnnnnF"$/ ;'nnnnn+#:B$/ G/,nnnnn(O 0nnnnnJC <nnnnn"$/ ;/&nnnnn-E G'nnnnnJ`? W/0( X,+23$/ 7'L,\/ IJ'(2 X,+M6$/ 7'L,\/ <':[S2 >$!& o,-$/&.

) j( :$/ ;A0'B:$/ )C&:(: <" !"$/ )*+,- 7/0E9F'2 d,E? X,: 0(&? 'n:3 X'-B )nn*#-:$/ )nn:+*$/ .&nn39$ ;A0'nnB:$/ Inn+9,9 <+0nn29 Dnn: .nn3$& ,nn++M9 H? .&0 1nn[

.nn3:+ 'nn: ,nn23? H=nn39,:$/ ,nn6JB#$)nn292 W'80nn92:)nn+J'`$/& _nn$&\/ .+9$0'nnB:$/ <+0( KI+,*9$/ D: >$!3&.

) 0( .: <3 1N <"$/ .,'L)I ( K)j ( 1N <"$/ D:)?.(

%&'( )*+,- .: ;/&-E a@` U2-– "$ <0+=.+9+JY/ .+9$0'B:$/ <:

.+,+M9:$/ .: <3 ):+L 1N P-E$/ IF"/ 7`21, xx . .+9+8/092A/ .+9:+*$/ !E0,0 21 !! xx.

)+JY/ ;A0'B:$/ .: )C&:(: 1#+ ':+N:

)*+,- 7/0E9F'2G-S Z![ <" IF"/ )M+6$/ U+2-92 >$!& )C&:(:$/ _$O <69 )L0 0&0" 1N )+#:B$/ I,'*99 .? _$O W'+,/,390.1 . ):+*$/ .? T,N/

.: <3$ )+8/092A/32 , xx W/,56 H&'F9.

(7-3) s¹d9

(8-3) s¹d9

Y�U¦�« qBH�« �UM¹d9 ∫Y�U¦�« qBH�«3

)*+,- 7/0E9F'2 )+$'9$/ )+JY/ )+-E$/ ;A0'B:$/ )C&:(: <" I&#-:$/ %&'(– <0+=.

I2F$/ .'+2 D: ;A0'B:$/ I+9,9 WA&? 0CPN X,&,^ >'J[ ;J'3 /!O . U2- 7`)+2+,*9$/ 7+*$'2 W'8092: X,&3!:$/ )*+,-$/ .: .+,+,390!ix 7+L D+:($i.

.? T,5J:

# $ AA :det.

X0:C? &? &56 H? <+0292 7*9 A . )+,F3$/ )M+6$'2 ;'2'F"$/ ,(? )+0'+9CA/)mB$/ A)+, (I+,*9$/ )(+9J G'-E? H? IJ(99$ <6"9$ 1$'9$'2& K

)-&2^:$/ <&#"$/ _#C.

(9-3) s¹d9

(10-3) s¹d9

) ( !" #$%& '()&* +,-./*#0 1 2– 3$%,4 5*3/#%:

-1 +,!0&* 673$8-&* +9#-%- :)yx

(txxx 321x &

-2 +;#<8-&* +=#>?-&*1'A.

-3 +=#>?-&* @33)-

) ( !"#$%&'( !)#*+&'( ,(-./"011 A !2'0/'( !23.'( 405#&6&'( 7#89 -6#::

.vx,ux,zx !!! AAA

!2;<( 4=-0%&'( >& >2/2'0/'( >2/5#&6&'( >& 7$':

) :( ?) !5#&6&'( /$( !2)#*+&'( !@2+'(yx !A.

) ( A2B ,: C-A*;& !5#&6&'( 4;0$ (DE 0& -F-9 G#06' HD9'( !I2A3 ,(-./"01C-A*;&.

) J( G#06' HD9'( !I2A31 -6#: C-A*;& A2B !5#&6&'( 4;0$ (DE : 79'(x K 4L&0%&'( !)#*+& C--9&#A M:det (A).

)1( 6 4 2

)2( 42

!23.'( A2B !2;<( 4=-0%&'( >& !5#&6& ?82 0&2):

4.21.040

0.11.03.0

1.005.01.0

) :( !5#&6& 79' 4(A& NLO !%10//&'( 4012AI/8' !2A2A$/'( !I2A3'( !@2+ P13!2Q(-/1=( !212AI/'( ,2I'01 R0Q-/1& !I10"'( 4=-0%&'(:

x = 0, y = 0, z = 0

$&2 7S ,$;– 0T285 ,/8+9 ?/'( UQ0/;'( V85 (RW0;1– 3#1X&'( 79'( >2&./(exact solution) Y4=-0%&'( >& !5#&6&'( ZDT'

(11-3) s¹d9

(12-3) s¹d9

Y�U¦�« qBH�« �UM¹d9 ∫Y�U¦�« qBH�«3

) ( G#06 !I2A3 ,2&%/ >$&&'( >&– !5#&6&'( 79 ?) ,-./"/' !2A2A$/'( 7-2[!23.'( A2B 4=-0%&'( >& !I10"'(.

2AI/'( ,2I'01 R0Q-/1& !I2A3'( ZDS >& >2A2A$/ P13!I10"'( !2Q(-/1=( !21 0T"*;.

3#1X&'( 79'( >2&./ ,$;$&2 7S \A.: CA&.

?) !I10"'( !I2A3'(# !I2A3'( ZDS >21 >A0]):.(

) :( !^I2A3' !^2A2A$/'( !@2+'( >& 4(A2A$/ !OLO P13G-S !^ _&&_%a&'(] W[^6'( A^b;() (!I10^"'( !'c^"&'( >^& [!^2'0/'( !^5#&6&'( 7^9' <( 4=-0^%&'( >^& A^2B !^2;

!23.'(!2Q(-/1=( 4012AI/'01 R0Q-/1& K :

x = 0, y = 0, z = 0

) ( 4^;0$ (D d(# Y0T285 ,/8+9 ?/'( UQ0/;'( >& 3#1X&'( 79'( >2&./ ,$;$&2 7S!^106e( : Y?;0^O'( A^2A$/'( UQ0^/; ?^) !^]-'( !^6A- ?^S 0^&) K ,^%;) --^5 ,^$ M:

,2^] ?^) !929^+'( !2A^f%'( ,0^]Ag( x, y, z A^2A$/'( -^%1 0^T285 ,/8^+9 ?^/'(Y?;0O'(.(

!"#$%&' !"&(!)*&': !

01.101.2

$+,-*&' !*./0( )tx 21 &'.

)1 ( 2#+3)* 4!561 2787& 9!,!:0&' ;'6<= >* ?@0&' 2A#6$ 9A,$ '@= B31 9,7'C&= DE" F%G" #"&(!)*&' HI J $AI:

& 00498.0

00298.00

01.101.2

+K L+:0*&' M0&' HI N&!"&!,+: ( )tx 67.1834.0 &'

)L ( L+:0*&' N,#6A"&' M0&' '@K #:0"& 2#6#6O"&' 2#PQ"&' 2A#6$ 4(%":'.

(13-3) s¹d9

(14-3) s¹d9

" - #:

) 1( 2#&!"&' 9!I+PQ*&' * MO& 2/,!A*&' 2#"'@&' 9!.<"*&'+ 2#"'@&' 4#A&' L:0':

) L( 2A,!:&' 9!I+PQ*&' * MO& NP#$&' 6$A&' ?Q3 2*#5 (<+1.

) R( S2#,6!A" 2A,!:&' 9!I+PQ*&' T1)6#6O" 9!I+PQ* 6,")G" #0 T1.(

" - $:

#:0"&'+ U+!<& ?@0&' 2A#6$ 4'(%":!, N&!"&' N$%&' 4!V3&' M0 (<+1T6#6O"&' .$6W&' ((X 2*#5 6Y(Z5 [&@O+.

8.3778.321550.5

12.6809.58003.1

) L!:0&'4!561 2),6\.(

" - %:

N&!"&' N$%&' 4!V3&')Ax = y(: !

0001.3

20001.1

$+,-*&' B/0( )t11x ' .2I+PQ*&' 1 ]6P3A ^#0, _!P#P$ _'6#`" 96#`" (5

90,Q1 !

29999.0

(#(<&' N$%&' 4!V3&' M0 (<+1 !

0001.3

29999.0

,!:0&' ;'6<H,4!561 2:*%& 9! . 2I+PQ*&' MKA S$6W&' 2/#/X

(15-3) s¹d9

(16-3) s¹d9

(17-3) s¹d9

Y�U¦�« qBH�« �UM¹d9 ∫Y�U¦�« qBH�«3

" - &:

7/7*&' !.#/*!X C&= #"#&!"&' #"I+PQ*&' * 8O M/0# #L, U +O# ^#0,ilii +'1 ) 6W!,*&' M#/0"&'– 2A#6$M"#&+(.(

" - ':

, 2I+PQ*&' M/0 NO:&+W" 2A#6$ 4'(%":!A L6- MQ!0 C&= 2#&!"&' a6+Q&' NI #"I+PQ*tLLA .' ^#0L 2#/P: 2#7/7* 2I+PQ*.

" -():

,! 2#*E6'+% b!,")9+6O ( * MO M0 (<+1 2#6$A&' 2#787&' 2I+PQ*&' M#/0"& ##&!"&' ##$%&' #*!V3&':

(18-3) s¹d9

(19-3) s¹d9

(20-3) s¹d9

" -( :

,! 2#*E6'+% b!,")9+6O ( M0 (<+1 2#6$A&' N787&' N$%&' 4!V3&' M0& N$%&' 4!V3&'Ax = y ^#0:

:A 2#6$A&' 2#787 2I+PQ*10 ×10 2#&!"&' 9!58)&!, ?c6)G":

9,,3,2;1

9,102,1

:y B<"* 110, Z)G*2#&!"&' 9!58)&!, ?c6:

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6#6O"&' 2I+PQ* 1 ]6P3T 2#6#6O"&' 2`#Q&' NI

( ) ( ) ( )*cx.x 1 *'* kk T

2I+PQ*&' NK !

6#6O"&' L6!A"& NI!O&'+ T6+6-&' $6W&' 1 9,7'(*) M0 C&= x = T.x + c

N,#6A" M0 T\ Nd'(",'( ) nR-0x +K: -1/3 < a < 1/3

2#6#6O"&' 2`#Q&' L6!A"& 2#:!:\' 2#6V3&' e,$ ( ) ( ) cxx 1 ! kk T

!"#$ %#&!' ()#*+G-S ,- #"#.'+) /0123 4"5:" # ULDT SG

1$$ $!

,$6+) 7&89+) :5+ !"#$+) ;<- =>$' ?"5 @+<0Ax = y 4"5:

+ $! ,

(21-3) s¹d9

(22-3) s¹d9

(23-3) s¹d9

Y�U¦�« qBH�« �UM¹d9 ∫Y�U¦�« qBH�«3

!"#$ ?3 A. ?B C>D)J !"#$0G-S ,E)*'>) ,>"#!' :5 FG %#&!'' *9H I

,$6+) 7&89+) :5: &&&

!"#$ ?3 :. %#&!' J#*)J !"#$0G-S ,$6+) 7&89+) :5+ &3K!">$' *9HAx = y 4"5:

)B( &'

)%( 0;1

)L( 0,0;2

!"#$ &9!>$ )<MG-S ,$6+) 7&89+) :5+

,>"#!' ,E)*'>) :5 FG !"#$+) %#&!'' :K/" #0x 7N9> 7'>OB ? P)0 Q)<&3+0 QR 7B 7./ I #"#.'+) C)#3 **H(k) +) Q "#SH 7&T#B N>#G U&5"52 :5+) VWH :025W+ 3XA

9"&>'3+) ="!5'+ 3XA+) FB: " # 4105.0xx $./$ k

"$6+) CR*&N3+) 7&89Ax = y Y"/: &&&

(24-3) s¹d9

(25-3) s¹d9

(26-3) s¹d9

(27-3) s¹d9

)B ( 7"!+ Z*3 FGa /0123+) [>2'A Q*"*5'+&> >O03

)% ( \#]]]]19a=-0.25 . ]]]]!"#$ ?]]]]3 ?"#]]]]"#.' =]]]]>$G-S >3 ,E)*]]]]'>R) :]]]]5+&> U&E*]]]]'F#12+) .Q)<&3+0 QR 7B !"#$+) ;<- %#&!'' :-

DAD+ C&>&(5+) %"#!' ^3 ,.(+0S' !"#$> ,+&'+) ,$6+) 7&89+) :5 *O0B7&T#B: Ax = y

,$6+) 7&89+) &9"*+ ?B \#19Ax = y 4"5:

!"#$ %#&!'' :-J ?3 C)#"#.' DAD =>$ Q)<&3+0 Q7&89+) )<- :5+ &-&9!>$ )<M F#12+) ,E)*'>R) :5+&> U&E*'>3 !"#$+) ;<-" # 0x 0 ! . >0W$3+) C)#"#.'+) **H 7.

Q7&T#B '( V+M :2' T* VWH :025W+

,$6+) 7&89+) ?B \#19Ax = y Y"/:

:"W5'+) 7)*6'(&> 7&89+) )<- :5 *O0BLU *N> H0$!3 C&>&(5+) _)#OM ^37&T#B DAD.

(28-3) s¹d9

(29-3) s¹d9

(30-3) s¹d9

Y�U¦�« qBH�« �UM¹d9 ∫Y�U¦�« qBH�«3

$#S+) **H 3"T #*T0 I "#"#.'+) "12'+) !"#$ ?3 a*5)0 a0$6 =>$K(A). #&"N3+) 7*6'()11.

?B \#19:

C&>&(5+) _)#Ob> 7&T#B DAD V+M >#!3:

)B ( /0123+) :W5A a#02+) ,/A = LU ,$6+) 7&89+) :5 *O0B0 IAx = y.

)% ( .'+) ?"(5'+) !"#$ ?3 U)*5)0 U)#"#.' =>$ CW25 F<+) :5+) ?"(5'+ F#"#Y"WH $#S+) **H 3"T #*T0 IK(A) . 3"!+) ^3 3"!+) ;<- ?#&T 7D

$0>c3+) .#&"N3+) 7*6'()01.

?B \#19A, B, C N>#3 C&/0123nn. ?3 A. ?B0 IA, C #"d /0123 ?B0 I a*#193y ?3 ?0.3 YO'3n #29H.

YO'3+) #2&9H a_&1.> %(5' e". f#S)x4"5 I:

yx 11 $$! BCA

H :025W+ !"#$+) ;<- ,/ >0W$3+) ">&(5+) C&"W3N+) **H 7.0 VWxQ

,$6+) 7&89+)Ax = y Y"/:

(31-3) s¹d9

(32-3) s¹d9

(33-3) s¹d9

!"#$ ?3 C)0$6 4AD <19G-S :5+&> U&E*'>3 I,$6+) 7&89+) )<- :5+ ,E)*'>R) ,>"#!'+)" # 0x 0 !. #"#.'+) /0123 *O0BSGT $ #"#.'+) C)#3 **H #*T0 I

VWH :025W+ >0W$3+)x 7&T#B "9&3D+ U&5"52 .#&"N3+) 7*6'()01.

?B \#19A: /0123nn. a*#193 #"d.

x : YO'31.n.

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>&(5+) C&"W3N+) **H 7. Q "+&H a_&1.> %&(5+ &KO&'5' ,'+) "QQ

?B \#19:

,$6+) 7&89+) :5+ ,.(+0S' !"#$ 7*6'()Ax = y 3"T %(5) 7D ?30 I#)*!3+): yz 1$At

4"5: &&&

&O ?09&T =">$'+ >(&93 a#02 ,/ ,+&'+) ,$6+) 7&89+) %'.),>0.:

(34-3) s¹d9

(35-3) s¹d9

(36-3) s¹d9

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?09&!+) )<- ?3 C)#"#.' DAD =>$ F#]12+) :5+&> U&E*'>3 F#"#.'+) #*T0 I7&T#B (36 V+M :2' T* VWH :025W+ >0W$3+) C)#"#.'+) **H.

:"W5'+) *O0BLU e012W+ CA"*>' FB :3H ?0*> "+&'+) /0123W+:

7D ?30 ,$6+) 7&89+) :5 *O0BAx = y ?3 :.+:

)B( &&&

y )%( &&&

=>&(+) :)g(+) :5 *HBh-hi /0123W+ >(9+&>:

?"KO'3+)0:

)B( &&&&

y )%( &&&&

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(37-3) s¹d9

(38-3) s¹d9

(39-3) s¹d9

!"# $%&' ()#

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>.8? 13@AB' *+',-.C' /0-!B4. D4+,-.! E'3"37-.

F3@G #HIB4. J<8IK- E4L0-!B' 9! M%M% $4%& $6M8!B' 9::

kiyCBAx ii ,,2,1;11 !!

N"&A, B, C: I.3! E4O6@A!nn$ = !6MI!

9! P76A, B: =Q,3@G! 3"R O6@A!

6kiyi ,,2,1;: ! E4L0-! M%M%

! *B'6-B' JMS

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(40-3) s¹d9

1 n !"#$!.

%"&'()*!+, -#. /"01#+ 2'*3456 7#4 '!4i

!"#$ %&'()!$:

-1 891,1!

-3 891,G = ECF.

-4 :43#+ki ,,2,1

%'*+,!$ %&'()!$:

-1 /#1A 24#!'. -+; 244<#<!AA

-2 /#1B 244<#<! 24#!'. -+;BB

-3 :43#+ki ,,2,1

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,@ B0C(A 45.0 ,1 D;0C(1 0-E 4:501 436CF1n . 4-1G0$6"#$ ,@ 95H$ I62:1#$ J8*. 4-#&.#$1!A:

! )7*A D06F#$ 3LU.

! !"#$ %&'(#$ )*:

nieLUx ii ,,2,1;

K-*ie %L0 ;61:#$ 6Mi N.&(#$ )*#$ OP.1 ,62-3 QD;*6#$ 436CF1 3ix

%L0 ;61:#$ 6Mi 436CF1#$ 31!A.

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) @( )-7*.#$ ;P6@LU 436CF17#A.

) J( !"#$ %&'(#$ )*Ax = y )-7*.#$ =&1;".81LU 3)@.(

(41-3) s¹d9

(42-3) s¹d9

) U( I62:1#$ ;P6@1!A !0V#$ ;;< J8*$ %H QK(A) 36CF17# 4A =&1;".81 0&-:1#$(1.

!"#$ %&'(#$ )* ;P6@Ax = y K-*:

) @( 28#6V. 4/-0! %$;".8&5.

) J( 4/-0! ,1 )2 %$;".8&5J 4/-0!6G-S OP.1#&5 @;5$ ,-./-0!#$ ,1 )2 36 Q>0CF#$ 0x )0( 4W-F#$ T-5!. 3 01.8$6 Q X#Y )F. ,@ X#Y 4-0-02.#$

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(43-3) s¹d9

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‰ULJ²Ýô« »U�Š

Interpolation

WO�ULJ²Ýô« !«dłô W¹œËbŠ º

WO�U�_« ‚ËdHK� sðuO½ WGO� º

WOHK)« ‚ËdHK� sðuO½ WGO� º �dH¹≈ WGO� º

v�Ë_« ”ËUł WGO� º

WO½U¦�« ”ËUł WGO� º

QD)« —UA²½« º w�JF�« ‰ULJ²Ýô« º

�U¹œËb(UÐ w¾¹e−²�« V¹dI²�« º

WO³OFJ²�« W×¹dA�UÐ ‰ULJ²Ýô« º W²]³¦*« W×¹dA�«Ë WOFO³D�« W×¹dA�« º

WOFO³D�« WO³OFJ²�« W×¹dA�« WO�“—«uš º

W²]³¦*« WO³OFJ²�« W×¹dA�« WO�“—«uš º

lÐ«d�« qBH�« �UM¹d9 º

‰ULJ²Ýô« »U�ŠInterpolation

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y = f(x) 0 2 12 21

&%/R .4@!f(3) Q!#V24)+:

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e- E24B! *)1:-! Q!#V24)+ 8?/+02-! )%2, 4, 5.

f- 8+/?E2-! )%E24B!(cubic interpolation).

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Lagrange interpolating polynomial WO�ULJ²Ýô« !«dłô W¹œËbŠ

‰ULJ²Ýô« »U�Š ∫lÐ«d�« qBH�«4

2,3*4!%:

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[+!!(%= [+!!(=D 2(*(!!U$% 2!!0*,$% ;!!0 O!!7-#- 5!!L*78 567!!U( G!!-$% 2!!0*,$% .? R?4 ",4;D%)$% R)WA$% ;<=0$% G.

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=2n + 1 ;0' @5*708 /i)".

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J!!$9 MV!!0k– C=!!D,0 B!!*),- Q!!$:)(xpk 2!!$%+7$f(x). $% N9!!F C!!^0 G!!4= 2!!*70A– 2!!*708

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Interpolation Formulas by Use of Differences (Interpolation at

Equally Spaced Points)

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2&4/502 657." (5%89 :;54) <$=0)> ,"==0?2 ,"@A?>B2&4/50$= 6$!3/5012 C"5D B

E>;.?23/012 F$*% ,>3/ G"+ H$A%"= $!"9 I$9$0!?2 ">$0/! 6$! J;K/0% LM" L9> H .M/4!?2 E>;.M? I$.";K/?2 JK= LFK% J2;K/012 2@N 6=#> H C"D?2 O@N JK=.

!"#$%&'( !)*+&'( !%,%-' .($)&' /!$01

Forward, backward, and central differences

$!3/012 ,P2 Q/+ $%0;& 5?2&? =50%?$= 6f &&5' &5%' $5A!"# B>5MK!1!n ,5! 555.M/4!?2 555";$"/412 F$555*%?2 . 555=/;! F$555*%?2 555'>!R! ,) J;/.%5550 ,P2 ,555! ST2&555/=2>

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nihixx

i ,,2,1,0;.

W;55KX" 22!%,%-' .($22)&' 322%,0% (forward difference operator) - ;VY55" U@55?2> ?2&?2 QM'f – "?$/?2 #ZK?$=: #f(x) = f(x + h) – f(x)

[5555!;?2 $%!&4/555502 2@855559if Q5555M' 6&5555"?)( ixf I$5555#ZK?2 <5555/3% ,) $5555%%3!"9 H "?$/?2:

$ % $ %121

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16 50 24

81 110 24

175 84

256 194 24

369 108

625 302

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#ZK?$=: )()()( hxfxfxf "'

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)2()(2)()(2 hxfhxfxfxf ! "'

% 55'>!R!? 55".M4?2 E>;55.?2 6>&55R $55% ]>b3 2@d (55%) e55+Z"> ,>3/5509 -$55FK! F$55* $A0.% LN 6>&R?2 2@N ;D$%'– "&&K?2 !"*?2 G"+ ,!– E>;5.?2 6>&5R ;D$%'

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,5!> HF$*% `=;) ,! -$FK! '>!R!? ".M4?2 E>;.?2> "!$!\2 E>;.?2 L?>&R LFK" 63c?2,"=/% ">$0/!?2 M=$*/!?2 ;D$%K?2.

0x 0f 0x 0f

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2 f# 1x 1f 2

3 f# 2f' 3

2x 2f 1

2 f# 2x 2f 3

2f# 3f'

3x 3f 3x 3f

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$ % $ %hxfhxfxf21

21)( !"(

6D+% $A%!>QM':

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22))(()(2

hxfxfhxf

hxfhxfxfxf

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IZ!$K!?2.

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hxfxEf

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hhDf(x)f(x)

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'+ U+h .59:; O=1W S, + J- + ,ba, $

S!(G1"D*8 !#$ %%& O%!5:0 P Y?7 I?&[8 %C78 .(5Q =5\V1 'G(5 ]$+ ^)8, .(5Q =5\V1,h =DG$(78 Y78 '+ *Z(broken line) ."78%78 S4(5 UM78g %"/_1 UM"78,I7,[8 ]1#1N(g! .?V1( =5` .78%(discontinuous) ]1D3D S#1, L=!DG$8 %8%:5D.

S!(G1"D*8 @"5= '"& .57!(G1"D*8 ."78%78 .D3D, .5?)!>1 .?GN( SC 'G(5,.505/G178 .C5=N7!0 S!(G1D*!0 I(D5 !( ,+ K.505/G178 <!5%,%C7!0.

'51 #$ SG '50 .505/G1 <!5%,%C a%Y1D5 UM78 <!5%,%C7!0 J95:;178 R5=#178 .(%Y1""D(78 @="" 78 S"")W+ '""( %""/5, K .""505/G178 .C5=""N7!0 H*!(G1""D8 I(""D5 '51/0!""11(

a,578 . I?& S(1N1 .(!/78 .505/G178 .5%,%C78 '[ H8=b$,c .5W!G .$,=( !$5%?W K<08,47!0 S!(G1D*8 .#5= JW .?0!Q .57!(G1D*8 .5%,%C78 ',G1 '+ '()$7 .505/G178 .C5=N

'+ d""7M I""7Z .W!"")e!0, K.""5%,%C78 !""673Y %"";,1 J""178 O=""1>78 I""?& =8=(1""D!0 S"")!>1?7O="1>78 LM"- I"?& .?"V1( d7MG .5$!478 .#1N(78 ',G1 . ."505/G178 .C5="N78 '5,"G1 '+ *Z.57!(G1"""D*8 ."""5%,%C78 <!#1"""N( @""">11 '+ T="""1>5 * %"""$& I"""1C K."""78%78 <!#1"""N( f"""(

%B#_/78 I(D1 J178, O! /(78 !#$78(nodes).

!"#$%&'() !*!+%&") ,!-.") /!-+& (Cubic Spline Interpolant):

.78% !$5 &_+ 8MZf O=1>78 I?& .Wg=B/_([a, b]%8%&+ .&,(;(, K

bxxxxa n "###" 210

Cubic Spline Interpolation WO³OFJ²�« W×¹dA�UÐ ‰ULJ²Ýô«

""505/G178 .C5=""N78 'h"W K H8%""B#_& I("D1 .57!(G1"D*8 .S .""78%?7f @""#C1 .""78% !"6$P0 ig=""B/_1.57!178 ,=N78:

S .505/G1 .5%,%C– :(=7!0 !67 :(=$jS - .59:; O=1W SG I?&

1,,2,1,0;],[ 1 &"* njxx jj

S .78%78 a5Q f( !6(5Q @0! 11f '+ U+ K %#/78 SG %$&:

njxfxS jj ,,2,1,0;)()( ""

O%#& U+ %$& '5159:; '51=1W I#1?( J-:

+( '+ U+ K'51=1>78 '51!- I?& '51505/G178 '515%,%C78 !1(5Q @0! 11:

2,,2,1,0;)()( 111 &"" *** njxSxS jjjj

R( '+ U+ K'51505/G178 '515%,%C78 '51!- J1#1N( !1(5Q @0! 11 !(G:

2,,2,1,0;)()( 111 &"!"! *** njxSxS jjjj

j( '515%,%""C78 '51!""- '""( S""G7 .""5$!478 .#1""N(78 !""1(5Q @0!"" 11 d7M""G, U+ K'+: 2,,2,1,0;)()( 111 &"!!"!! *** njxSxS jjjj

%,%C78 ,=N '( '5157!178 '51&,(;(78 '( O%C8, @#C11(boundary conditions):

+( .C5=N?7 .5$!478 .#1N(78 a%/$1S %$& O=5Y[8, I7,[8 '51%#/78 '( SG K'+ U+: 0)()( 0 "!!"!!

=N78 8M- I(D5, : =C78 %C78(free boundary).

R( .C5=N78 .#1N( !1(5Q k,!D11S ."78%78 .#1"N(,f '51%"#/78 '"( S"G %"$&'+ U+ KO=5Y[8, I7,[8:

)()(;)()( 00 nn xfxSxfxS !"!!"!

=N78 8M- I(D5, : <g0B4_(78 %C78(clamped boundary).

,=""""N0 .""""505/G178 ^98=""""N78 i""""5=/1 '""""G(5 ]""""$+ b""""C35, '+ *Z k=""""Y+ %,%""""CJ>G1 H!#0!D O=,GM(78 ,=N78 !$- ];!1C$ !(7 .0D$7!0.

/!-+& : +!*0") ,!-.") (Natural Spline)

U+ K="""C78 %""C78 ="""N @""#C1 J"""178 .""505/G178 .C5="""N78 J""- ."""5/50 78 .C5=""N78 =N78)4 (– .505/G178 .C5=N78 i5=/1 JW H!#0!D =,GM(78 +.

/!-+& : &1*234$") ,!-.") (Clamped Spline)

J""- .""1g0B4_(78 .C5=""N78 U+ K <""04(78 %""C78 =""N @""#C1 J""178 .""505/G178 .C5=""N78 =N78)4 (– .505/G178 .C5=N78 i5=/1 JW H!#0!D =,GM(78 R.

S"5, R5")Q LM"YP5 UM"78 SG"N78 ]0"N5 ."5/50 78 .C5=N78 I$C$( '+ bC35, '=() .(?G ]5$/1 !( ,- 8M-,spline ( '"( ." #$ S"G0 L= l="(_$ 2"5C0 ]"5W aGC1"$ !(%"$&

078 !#$<!$!5:

( ) ( ) ( ))(,,,)(,,)(, 1100 nn xfxxfxxfx

H!""(,(&, S(1""N1 !""6$[ d""7M, @%+ <!""05=#1 I""7Z U%m""1 <""g0B4_(78 %""C78 ,=""N 'h""W.78%78 '& =4G+ <!(,?/( I?& . '+ R";5 %,%"C78 ,="N '"( n,$78 8M- @#C17 ]$+ *Z

'51(5#78 '51!67 H!#5Q% H!05=#1 ,+ .5!6$78 J1 #$ %$& .#1N(78 J(5Q !(Z i=/$.

"VC?7, O!"" /( .""78%7 .57!(G1"D*8 .""505/G178 .C5=""N78 I""?& S,f @""50 1 !""$$G(5 K78 ."""""5%,%C78 I"""""?& i"""""5=/178 J"""""W O=,GM"""""(78 ,="""""N78 <"""""08,478 '5"""""5/17 ."""""57!178 ."""""505/G1

jjjj dcba ,,,:

1,,1,0

;)()()()((*) 32

&*&*&*"

xxdxxcxxbaxS jjjjjjjj

(56: =N78 '+ ^)8,)1( .\5V7 !$=!51Y!0 @#C1(jS .505/G1 .5%,%C !6$+ 25C.

#!7#3: =N78 @50 1)2( I7Z U%m5:

1,,1,0;)()( &""" njxfaxS jjjj

-,<08,478 %%C1 .Q3/78 LMja.

#3"#3:

) +( =N78 @50 1)3( – + ] .;51$ a8%Y1D8 f() H!5$!4([ I7Z U%m5:

1,,1,0;)1( 321 &"***"* njhdhchbaa jjjjjjjj

) H!>5=/1( jjj xxh &' *1

) H!>5=/1( )( nn xfa '

) R( .?)!>(0)(xS j - .Q3/78 JW .W=/(78 (*)– I7Z .0D$7!0x I?& SVC$:

1,,1,0;)(

)(3)(2)( 2

&*&*"!

xxdxxcbxS

jjjjjj

=N78 @50 10,)3( – I?& SVC$ R:

1,,2,1,0;32)2( 21 &"**"* njhdhcbb jjjjjj

) H!>5=/1( )( nn xSb !'

) j( .?)!>(0)(xS j! I7Z .0D$7!0x I?& SVC$:

xxdcxS

)(62)(

=N78 @50 10,)3( – I?& SVC$ j:

1,,1,0;3)3(

njhdcc

) H!>5=/1( )(2

1nn xSc !!'

<!""Q3/78(1), (2), (3) <""08,478 '5""0 0=""1jjjj dcba ,,, '+ 2""5C, <""08,478

ja <08,478 .5#0 %!;5Z S,!C$W .(,?/(.

<"""08,478 H*,+ %;,$"""Djc <"""08,478 0="""1 ."""Q3& %!""";5Z @"""5= '"""&jjjj dcba ,,,

'( SG iMC0 d7M,jj db , '( 23478 <!Q3/78(1), (2), (3) .

)3()()3( 13

1 !&". * jjjhj ccd

JW .;51$78 LM- T5,/10,(1) I?& SVC$:

4&&&&". ** )(

3)(1)1( 1

'+ U+:

)1()2(3

!*&&" ** jjj

<""08,4?7 '5""1Q3/78 '51!""- T5,""/10,jj db , .""Q3/78 J""W(2) ) f""), %""/0j-1

'( H*%0j (I?& SVC$:

"*&& ** )2(3

)(111 jj

***&&&

,I7Z =V1Y1 .Q3/78 LM-:

)2(1,,2,1;)(3)(3

!&"&&&"

chchhch

jjjjjjj

'( J Y a!b$ 8M-,(n – 1) JW .7%!/((n + 1) S,6;(:

njc j ,,1,0; "

<08,478 a5Q '55/1, a!b$78 8M- SC %/0,jc<08,478 I?& SVC$ K:

1,,1,0; &" njb j

.Q3/78 '()1( !<08,478, K:

1,,1,0; &" njd j

.Q3/78 '()3( ! ..505/G178 <!5%,%C78 '5/$ d7M0,:

1,,1,0; &" njS j

i5=/178 a8%Y1D!0.(*)

(;( '( U+ <##C1 8MZ ]$+ I7Z =5N1 .57!178 .5=b$78, , %,%C78 JB o=BN J1&)+ (

,+)R ( =""""N78 J""""W '51=,GM""""(78)4 (78 'h""""W K.""""505/G178 .C5=""""N78 i""""5=/1 J""""W a!""""b$ S5-!;(78 JW =,GM(78 J Y78jc %5C, SC ]7 ',G5.

<$!G 8MZf O=1>78 I?& .Wg=B/_( .78%[a, b].78%78 LM- 'hW K:

9(56: O%""5C, .""5/50 .57!(G1""D8 .C5=""N !""67 ',""G5(unique natural spline

interpolant)=C78 %C78 JB o=BN @#C1 O%5C, .57!(G1D8 .C5=N U+ K:

0)()( "!!"!! bSaS

! !" #$%&$'()*+,-". /0". 1234 5607 8/&0$ 9&" -%7:. 90&34 ;":

)()(;)()( bfbSafaS ! !

!"#$%&:

#&7"/ <<=- <<>&2=7 <<-!&"? 3 <<4-". /$/<<0". 12@3<<*4 17A$<<-B- #<<- C <<&D #D ')+><<: E&F B-". 1G #&7&G H?jc ;"I) >&/" J)K&G L(n + 1) 1G 9"/ =-(n + 1) E$!B-

)( jc /&0$ E0 M" N7 >". 12O". P Q>". .IF #D$ L.

#D R3> 9&3Q>". E)S 83$%I-". NT 7>".$ U$-3". P./O7: ):

30". /0". 12@3*4 5&)27)9&=&)2". 90&34".:(

) W&3=7 #-nc(

00)(62

0)(0)(

)(62)(

)()()()(

2000000

"! "! %

#$#$#$!

xxdcxS

xxdxxcxxbaxS

#D XB& 9&=&)2". 90&34". #D YD#&234". 5607:

0,00 !! ncc

'V/ <<<=-". 9<<<A$-B- Z<<<- # <<<7"/ =-". # <<<7 F)2( C <<<&2O C <<<- Q> /<<<"$7 bAx !

#######

1)12(2200

2)21(210

1)10(20

nhhnnhnh

######

nnnhnnnh

!"#$%&'A ()*+#,% *+-. /+0! /&1+2&13" +-456&' +-*56&' (*5-7&' 5*8 96:2 /5;&' <1=,&'"Ax = b )-:" >: ?&.

!"#!$: @3AB%&' ):&' /5C*D8 9-352E:

()')()') )()()()( 0 afxSafaS

0)( baf ')

()')()') )()()()( bfxSbfbS n

) F-*42 G%nb( nbbf ') )(

)(1)( 010

baf *&&')+)

)()(3)(32 010

1000 Iafaah

chch )&&'*(

H&IJ": 2

)2(32)( &&&&& **')

nnnnnn hdhcbbbf

)3(&)1(

)()(3)(32 11

111 IIaah

bfchch nnn

nnnn &&

&&& &&)'*(

G12&)14%&'),( III 4%&' K"%L% M% EN)1)2( ) O1-5; O1%1=, )&"2Ax = b P-::

#######

1)12(2200

2)21(210

1)10(20

nhhnnhnh

######

nnnhnnnh

!"#$$$$%&'(A )*$$$$+,'( (-$$$$.! /'*$$$$0'*1" $$$$2345'( $$$$2645'( 7642$$$$8'( 46$$$$9 :$$$$5;0 )25' <2;" =; >' /4?'(nccc ,,, 10 !.

! @$$?A,2'*&B0$$8C( $$2123B0'( ;26$$9'( D$$& =$$B E$$AF ="$$%;'( G("$$4? /$$A2 *$$&2 &"A3& '(<' 7<2;"'( 23214'($2;"'( 2'*&B08C( 2123B0'( ;269'(" H /4I6$J9 :$5;0 /$0'( 7<

G$$1K&'( <$$;'( /$$4?'( )*$$+,'( DL $$+;M& N$$& H– =$$2O*P&'( /$$!jc - >$$2AF *,A$$%; Q-$$'('( D$$$& =$$$B /$$$! $$$2645'( /$$$KMK )*$$$+, D20'*$$$; =$$$2A;0 $$$52641 >A;,$$$8" H(factorization) (-$$$O

:1*8'( =%#'( /! *O*,6B- /0'(" H )+,'( D& R",'()S'*K'( =%#'(.(

$$'(<'( DL T6$$#,f <(<$$FU( <$$,F $$! V6J3W&nxxx ))) !10 <*$$P2X Y"$$A4&'(" H 2123B0'( ;269'(S D2469'( :5;0 /0'(:

0)()( 0 ('(''nxSxS

nixfa ii ,,1,0;)( !((

1,,1,0;1 &(&( * nixxh iii !

1,,2,1

hhhaxxaha

iiiiiiii

Natural Cubic Spline Algorithm WOFO³D�« WO³OFJ²�« W×¹dA�« WO�“—«uš

) 2645'( /KMK [*24? [*&*+, =;0 2'*0'( G("4?'((

0,0,11 000 ((( z.

Y8;()25A' /A2 *&: 1,,2,1 &( ni !

,)(21 1111 &&&* &&( iiiii hxx .

iii h 1/(.

iiiii zhz 1/)( 11 &&&( -

nnnn zcz ((( ,0,11

)25A' /A2 *& Y8;(: 0,,2,1 !&&( nnj

1*&( jjjj czc .

3/)2(/)( 11 jjjjjjj cchhaab *&&( **

)3/()( 1 jjjj hccd &( *

9'( 2123B0'( ;26 23214'(/O 1"A4&'(:

1,,1,0

;)()()(

],[)()(

&*&*&*(

xxdxxcxxba

xxxxSxS

jjjjjjj

!"#)9-4(:

/'*0'( ="<P'( G*,*21 :\!"0 /0'( 23214'( 2123B0'( ;269'( D\"B:

x 1 2 3

f(x) 0 1 4

2,3,2,1

6)]13(14[13

)(31002102

*&&( haxxahahh

0,0,01 000 ((( z.

4/11/ 111 h!

2/34/)06(1/)( 10011 " " zhz #

0,0,11 222 cz

2/32111 " czc !

23/)30.(11/)14(

3/)2(/)( 1211121

""" cchhaab

2/13/)2/30()3/()(

2/13/)02/3.(11/)01(

3/)2(/)( 0100010

$"" cchhaab

2/13/)02/3()3/()( 0010 " " hccd

2000000

)1(21)1(

)()()()(

"$"$"$

xxdxxcxxbaxS

2111111

)2(21)2(

23)2(21

)()()()(

"""$"$

"$"$"$

xxdxxcxxbaxS

]3,2[)2(21)2(

23)2(21

]2,1[])1()1[(21

%"""$"$

%"$" &

!!!!"#"$%&'( )"*!!!!+'( ,!!!!-..)& /0 1!!!!2" 3!!!!&'( 45*!!!+'( !!!!6,% /!!!!7 8!!!!.)&'( 9-!!!!:'( /!!!75 "$"#4'( :/7 9%610 , SS ;'<%5 = "#"$%& ">5>):

S(1) = 0 = f(1)

S(2) = 1 = f(2)

S(3) = 4 = f(3)

?,@"05:

3)2()2(

2)2()2(

1)2()2(

?(*"A05:

0)3(,0)1( !! ! SS

4)13(2)(21 00021 " "" #hxx

Clamped Cubic Spline Algorithm W²³¦*« WO³OFJ²�« W×¹dA�« WO�“«—uš

!!!!'(>'( /0 B*!!!!CDf >(>!!!!EF( >!!!!DE !!!!6 G*H$I7nxxx $$$ 10 !!!!J47'(5 >,!!!!2"K 15 "#"$%&'( )"*+'(S /"4*+'( 8.)& 3&'(: )()(,)()( 00 nn xfxSxfxS ! !! !

)(;)(0

,,1,0;)(

xfFDNxfFD

1:)(: 1,,1,0;1 " " % nixxh iii

03/)(3

nnnn haaFDN

1,,2,1

111111

hhhaxxaha

iiiiiii

) "*4.'( 3MNM ?,"4A ?,7,OD 9)& "',&'( P(54A'((

L@: 000000 1/,5.0,21 &# zh

Q".J' 3J" ,7 1:)(: 1,,1,0 " ni

3R #5J47'( &G#SMI7'( "#"$%&'( )"*+'(:

],[,)()( 1) %xxxxSxS jjj

!"#: - 3

$%&'( )'% *+ !"#:

0,,2,1 ! nnj

)3/()(

3/)2(/)(

jjjjjjj

cchhaab

1,,1,0

;)()()( 32

" " "!

xxdxxcxxba jjjjjjj

!"#)10-4(: ,%(*-(# .*/*%0(# 12345- )-(# ,-60785+(# ,%0%9:-(# ,"%;<(# =24:: x 1 2 3

f(x) 0 1 4

1)3(,0)1( ! ! ff

!"#$: ) !"#!" $%& '(#)f *+,-!" ',./!" '(#) 0-12 (3.(

!"#: *+,-!" ',./!,+ ,/4:

1,1,4,1,0

FD0 = 0 , FDN = 1

6]0)13(4[3

61/)14(313

3031/)01(3

10021021

hhhaxxaha

7517/221

7/95.3/5.45.3/)2/36(

7/2)5.3/(1

5.35.0)13(21

2/3,5.0,21

]2,1[)1(43)1(

03/)21

]3,2[)2(49)2(

4111)( 32

1 &"""!"! xxxxxS

%+%546!" 7%89!" ,:;;76 <= >)% ?6!" @(89!" A,4 </ *;76!" ':-%( 6B+C.D/!".

!"# $%&'#()4(

(1-4):

) ( !"#$%&' (")& *+,-. /0,1xxy )( (")& 23#4)5&'x 2"&4%&': x = 1 (0.05) 1.15

("6 *4# 07)5y 67 2898&2"7:; (4 .2"<%=5&' !,7>&' ?,-. /0,1 @&4%&4#,.

)A( 25"6 ?51%B'03.1 C&D,:

E- @$F&' ?451%B+' ('-F%B4#.

G- H"$%B% 45 ?IJK#.

2&4L ?1 @J K$F&' -., ,.

!"#$%&' ("6 MN# @$N" @%O' ?,-.&'y = x cos x - .52 /""7:; /5"67&.

L' (")&' PDQ ('-F%B4#2&-4N53& A.,5 7D. 7RS /""7:; /"567& AB x cos x = 0.52

x 0.9 1 1.1

y 0.04 0.02 -0.02

) ( !"#$%&' A"7)%& 2N#'7&' 2.7-&' /5 2"&451%B' -,-L T7"81 / 0,U1:

xxxf 24 sin)( ! "

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) A( -VV=; !VV"#$%&' 2VV5"6 A4VVBL& PDVVQ -,-VVL&' T7VV"81 (-F%VVB'6

" x 70-VV6, Z

C%."%= @J K$F&'.

(1-4) s¹d9

(2-4) s¹d9

(3-4) s¹d9

lÐ«d�« qBH�« �UM¹d9

lÐ«d�« qBH�« �UM¹d9 ∫lÐ«d�« qBH�«4

(4-4):

!"#$%& '!( )*" +#*! +%,$%& -./0%&y = x sin x – 1 1!!234 1!5(2%.

+678*%& -,58$79& :,7; '&/<$7,"(inverse interpolation) 1!5(26% :67;& =%/,*5>% :0.5 2?0 2@AB =5!( 1!!234x sin x = 1.

x 1 1.1 1.2

y -0.16 -0.02 0.12

C52%& 1,8 &?D 1B E"F,G =!5,5H& .2I%& 2FJ5 K%D C52!: )(.)()(.)())(.)(( hxgxfxgxfxgxf ! ! "

ELM,I$>% =>",N5%& =0!$O%&. =0!$O%& P?Q 1!" R",3$%& S;9..

(6-4):

=!%,$%& E,O,!">% T/./;5%& .2I%& -./0 1U.V8:

x 0.0 0.1 0.2 0.3 0.4 0.5

y 1 1.32 1.68 2.08 2.52 3.00

/./6;%& T26!F8 =@!6A /,60!W =6!5,5H& .26I>% 1$.6!O 1.O,( U"# 'F) =6%9/"x (

" 25$ +$%& =5!( :,7;% =@!A%& P?Q '/<$7&. X #,NO%& P?Yy /O4x = 0.02.

666!"#$%& 1,6668 &?D R666OB '.666>*5%& 16665f(x) =66602/ 16665 /./666; T2666!F8 -666F5!n 1Z666G =665!( +66#*! 66!"#$%& &?66Q #,66NO )66*" '&/<$667," :.667;5%& T/./66;5%& .266I%& -./660

.2I%& '!( :7;! [?%& /.5*%& 2A,O4 -8 /O4 =$",Fn [?6%& /.5*%& 1ZG +%,$%,". X

.2I%& '!( :7;!1! n \&2,IAB +#*!.

) !!"#$%: =6602/ 1665 /./66;%& T266!F8 -66M,I$ 1B =66N!N; -66",N! &?66Qn //664n 1665 //4 ,Y>M,I$ ,5O!" X \,$",F \&2&/N5 +#*! E&25%&n+1 \&2IA +#*! E&25%& 15.(

(4-4) s¹d9

(5-4) s¹d9

(6-4) s¹d9

(7-4) s¹d9

!"# $%"&'( )"*+,p(x) -%"./*0 1"/2 34,5x = -1, 0, 1 -".6 7,1"8 9!,":; "< =-%"""./*0 >+"""?* 7"""(%.@*0 A1,1"""'@*0 B,C"""D*0: """< = 1,"""@2 C"""E%/2 1"""/2 F""";(%<*0 !""".*0 F"""(%;

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% 5Q1 2I \:; 9%.!.E!r 25#$. !Y %./ J S:R .1( T$U KQE%G1 9(::/,. 9%G5$*,)1%,1( -/U1%$ . 2I <I QQE % 5Q11+r ]%3?( *%G 2.rxxx ,,, 10 .

"1%R1( ZCY )8 "5OP& K:,8 -/ -!*:

rabh 0""

, - )(12

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rfhrxfrxf

fhxfxfh

//"+"+

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2 3, -

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... //++//+//"

+++++ "

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2 3, - trrt Exfxfxfxfxfh ++++++ " )()()(2)()(2

1 1210

)(2)(121

),(,)(12

bafrhtE

//"" //"

)()(12

.fabhtE //""

`##. V##3% ,5 =##*>1( 2I <I2

r . ZC##Y D##5$*, B(:##. QQ##E SE%##A, (C7##8 W1C##1!

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-(+3(7 : !"#$%& 8/9!$9 6+, Composite Simpson's Formula

QQE 2!3$.3 KQE%H D5$*, ::/ )/r "1(Q1( L5H "8:+.1 c%,R B(:.1( 2.f

Q E QQE 2r + 1 ]%3?( *%G 2.rxxxx 2210 ,,,, .

2I "1!_3$ B$d 2I 2/.5!:

2 3,2 3-4 5.)()()(

)()()()(2

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1253120

xfxfxfxf

xfxfxfxfxfxfh

dxxfdxxfI

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++++++

iii xxrabh 222,

2!!" " .

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xxfrhE rS

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bafhab

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'(7$:(3-6)

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)I ( S:R .1( T$U KQE%G1 25G5$*,) <Ir = 2 "$/:.1( S:R .1( T$U "456 )8.(

)V ( 4 B%G5$*,(r = 4) KQE%G01 %_3; .

)-/U1( :@ (.(

2 3, -

rabhfhab

xfxfxfxfxfhI rrr

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+++++ "

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)I (2%G5$*, )V (4 B%G5$*,

S:R .1( T$U KQE%G1 ::/,.1( D5$*,1(

)I ( 2%G5$*,(r = 2):

2 3, -

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//""++

fhabxfxfxfhI .

)V ( 4 B%G5$*,(r = 4):

2 3, -

2 3, -)5.2()2()5.1(2)3()1(22/1

)()()(2)()(2 321404

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12598352251

:;<= ++++

8/9:#(5)+# ;+#<! =(>+)9.&) +(?3%& 8$ '@AB$%& C&#)D.& /0(

Richardson extrapolation (or Deferred approach to the limit)

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: 2rE "5$5:G,1( ".5G01 VR%6.1( =*>1(

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SCR 42 , EE:

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623)1()4(

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exactII

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1!2345'( 67- ?/@;AB; 0 ="3) $!A/4+C;34 1!2345 ( 3;3<34!2; 3!;"3D 80 E!"

21, II #"+=;'I $!=3 0 F+34$4!> G3!' H,0 80 I21, II !" 3J;+!, %&'($ )*%+,- .

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/0(12$34 5#$6'734

Singular Integrals

33'+;'( 354C'( /&0 K2&; (7* /4N," H,* #"+=;' #+23:

)0 ( 1S9 S"+="'( 1'(/'()+M;+2;C" $0 ( ?3D " 4T=0 $0 1"3D /,: 13U+M, V W<%;x QX'( Y/"#"+=;.

)Z ( 13U+M, V #"+=;'( [4;X.

/4!N," 4!3\ #!"+=;'( 0 ]4;N; #%N'( (7- QX F+2<+A [4$=7"'( 13//P'( K45'($nonsingular 0 80 I:

^- 1!!!!!!!!3"+_, 1!!!!!!!!9 S"+="'( 1!!!!!!!!'(/'((regular) ) 1!!!!!!!!3M;," 80finite #!!!!!!!!>+N;9' 1!!!!!!!!9<+D$differentiable 19!!!!%;"$continuous #!!!!"+=;'( Y/!!!!" Q!!!!X +!!!!M93T"; !!!!="3 80 I

"< 4$93+; 19A9A;Taylor series Y/"'( (7- QX(.

`- 13M;," #"+=;'( [4;X$(finite).

H!!93$&; /!!P< /4!!N,"'( #!!"+=;'( 1!!"3D Z+!!A&' 1!!3//P'( K4!!5'( 67!!- K!!3<5; !!="3$/4N," 43\ #"+=; )'* .#3$&;'( (7M' ?/@;A; 0 ="3 Q;'( K45'( ]P< Q93 +"3X$.

89,- : :,234 ;$ 5#$6'734)-:( ) ?3D ]P< /,: 13U+M, V 19"+="'( 1'(/'(x(

/a4N;'( #b0(singularity) 13'+;'( K45'( Y/&c<:

1- <*,=734 (Substitution)

#"+=;'( !"1

cos dxxxI /,: /4N,"x = 0

]3$P;'( ?/@;A( 2tx "

!! ""1

2cos22.cos dttdttt

/4N," 43\ #"+=; (7-$.

2- >?>?7$34 @,'1$(Series expansion)

K<+A'( #"+=;'( HAN,

cos dxxxI

()*)*"

( )*)*"

2/112/72/3

!6!421

!6!4211

0 *)**"

2/112/72/3

!6!422

limlim1

dxxxxI

xdxxdxx

/4N," 43\ #"+=; (7-$.

!"#$% &'#()*$% +,-.* / 0 12345 !"# ,6 &'#()* 7$8 45:

3- ABCD34 E$6'734(Partial integration)

#+T": K<+A'( #"+=;'( HAN,

dxxxxx

)sin(.22/1

sin21cos2 dxxx

/4N," 43\ #"+=; (7-$.

86*26F : :,234 ;$ 5#$6'734)G(

<*,=734(Substitution)

]3$P;'( ?/@;A(:

xettx *"*" ,ln

!! *"*

1.1).ln( tt

dtdtttt

+=; (7-$/4N," 43\ #") . 0 _&V0lnlim0

H00=34 E$6'734 I>J $6J !"#$:

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(upper bound) 84!!!!!!_,'( R!!!!!!5@9') #!!!!!!T" 1!!!!!!34_,'( g3!!!!!!%'( !!!!!!" Z!!!!!!A&B3 87!!!!!!'($

)()(12

4fabh 55*.

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)( dxxy ?(/@;!!!!A+<

$A<"A [/:+D. x 1 1.5 2 2.5 3 3.5 4 4.5 5

y(x) 0 0.41 0.69 0.92 1.10 1.25 1.39 1.50 1.61

)ii( 1!J3;, Q!X h+!5;DV( R!5@ 1!"3D ZA&((i) 12<+!A'( .+!,+3<'( 0 ?!9: (7* 1'(/'( .+,+3< Q-y(x) = ln x.

)i( #!"+=;'( 1!"3D Z+!A&' $!A<"A [/:+D$ O4&,"'( H<C [/:+D " Fd= ?/@;A(:

13'+;'( .+,+3<'( ?(/@;A+<:

x 0 126

f(x) 0 .25882 .5 .70711 .86603 .96593 1

)ii( +3< Q- 12<+A'( .+,+3<'( 0 ?9: (7* 1'(/'( .+,f(x)=sin x R!5@'( 1"3D ZA&+X QX Q9PN'((i).

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134C: ?+D40 1A"@' . $A<"A [/:+2' 1<A,'+< #(iA'( #& ?T.

4N, 0 ]: !*

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(2-6) s¹d9

(3-6) s¹d9

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"#$%&' 2":; <9=' !

11 dxx

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() *+,-:

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"#$%&' <9='I 2:+15 7#;+) 2I0+R S#=:=E@#A%;B' CA8& 2F0#6"&' 2":6&' 4U.). N.

)i( 2G#9"&' 2":; MV #"h <#9=& 2"WX&'

2lnxdx

9#0 2:+15 7#;+) 2I0+R S#=:=ET (.90"9 345#; 7'48%

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(5-6) s¹d9

(6-6) s¹d9

(4-6) s¹d9

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2&B40J .K.

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7:; 4U.) IKJ ,, .+:46%&' MG MFI,&' CA8&' <9='.I `" (+#;. N(i).

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2lnxdx

T >+=-"&' /01 345#; 7'48%9#0 2:+15 7#;+) 2KXK& S#=:=E

)ii( "#$%&' 2":; <9=' "2

>+=-"&' /01 345#; 7'48%9#0 S'L8%" N:

)) (h = 0.5 )< (h = 0.25

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"#$%&' 2":; 2:+15 7#;+) 2KXK& <9=' 2/

"#$%&' 2":; 2:+15 7#;+) 2KXK& <9=' *

(8-6) s¹d9

(9-6) s¹d9

(7-6) s¹d9

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)i( %&' 34.4="&' d#-#:0&' (" "#$%&' 2":; +4; 2:&# "2

)( dxxyI

>+=-"&' /01 345#; 7'48%9#0 x 1.0 1.2 1.4 1.6 1.8 2.0

y(x) 1.0 0.8333 0.7143 0.6250 0.5556 0.5

)ii( 2&'4&' e8% d#-#:0&' fLV d-#$ 'LQx

xy 1)( " MG MFI,&' CA8&' <9=#G(i).

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x 1.1 1.3 1.5 1.7 1.9

y(x) 0.9091 0.7692 0.6667 0.5882 0.5263

)iv( M!!!%U:%- 7'48%!!!9#0(i) .(iii) .!!!E=F& (.!!!94+#1%:+& [#,:%!!!9B' 2!!!6:+A \!!!0A6% JF5 "#$%F& 4:4U <:+I .T2:+15 7#;+) 2I0+R h:=E .V V

)v( !"#$%&' 2!":; <#!9=& (.90"9 345#; \0AI 3+G.!%"&' d#!-#:0&' !$ S#"48%!9" N>+=-"&' /01 345#60 #a:F5 dFE= M%&' 2U:%-&' `" 2U:%-&' (+#;..

"#$%&' 2":; <9=' +

1sin1 dxtt

"#$%&' 2":; <9=' ME;#-&' (elliptic integral) *2/

2sin411

(.90"9 345#; 748%9' , -16#"h

)i( 2&'4&'f 2:&#%&' A#6-&' 4-5 #a":; 7.FI": x 1 1.2 1.4 1.6 1.8 2.0

f(x) 2.718 3.320 4.055 4.953 6.050 7.389

(10-6) s¹d9

(11-6) s¹d9

(12-6) s¹d9

(13-6) s¹d9

&' 2":; <#9=& (.90"9 345#; \:0A% #--$": V "#$%: "2

)( dxxfI

T 3#AI"&' d#-#:0&' 7'48%9#0 _%0#UQ d-#$ (Q : 2":; <9=#G 7I-I 34!5#6&' fLa0.

'L#"& i+1#G j+8R' d-#$ ( k'. 2":; <9='. NI #aG+I% j+8) 26:+A OC0.

)ii( 2&'4&' 7:; S#:FIG MV 260#9&' d#-#:0&'xexf ")( .FI,&' CA8&' 2":; 4U.) MG M 2":6& _+:46%I MG(i).

)iii( #al%:A5Z) 4; 2:&#%&' 34:4U&' d#-#:0&' () *+G'

x 1.1 1.3 1.5 1.7 1.9 2.1

f(x) 3.004 3.669 4.482 5.474 6.686 8.166

!!!!"#$%&' 2!!!!":; <!!!!9=' 2

)( dxxf d#!!!!-#:0&' !!!!$0 S#-:I%!!!!9" (.!!!!90"9 34!!!!5#60

(m' _:4& 3+G.%"&' .;+R' 445 4U.). 2=:=E&' 2:+1I&' 7#(N) _%0#UQ MG.

)iv( 2G#!!!9"F& #!!!a0 h"!!!9: 2!!!":; +!!!0$) 4!!!U.)h !!!"#$%&' 2!!!":; <#!!!9= (!!!$": n!!!:=0I

44!!!IF& S#=:=!!!E >+!!!=-"&' /0!!!1 34!!!5#60N /!!!9,- M!!!G #!!!"$ 2:+!!!1I&' 7#!!!;+R' (!!!"(iii) . 2":; `" (+#;h MG (.90"9 345#; 2&#= MG 2"48%9"&'(iii).

)i( =' 2:&#%&' d#-#:0&' ("2":; <9 "2/

(.90"9 345#; \:0A%0.

x 0 12#

f(x) 0 .25882 .5 .70711 .86603 .96593 1 .96593

)ii( M!!!G 2!!!0#Ug' M!!!G 2=:=!!!E&' 2:+!!!1I&' 7#!!!;+R' 44!!!5 7!!!$(i) .4!!!U&' () 7!!!F5 'LQ 2&'4&' d#-#:0 MAI: \0#9&'f(x) = sin x T

)iii( !!"#$%&' 2!!":; <!!9=- () #!!-4+) 'LQI .!!V #!!"G N >+!!=-"&' /0!!1 34!!5#; \!!:0A%0 3.!!A8&' 7!!U=h 2!!":; (.!!$% M!!$ <.!!FA"&'I 44!!IF& 2=:=!!E /!!9,- (!!" 7#!!;+R'

MG #"$ 2:+1I&'(ii) T 2":; `" (+#;h MG 3#AI"&'(i).

(14-6) s¹d9

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/%FU5 dF ?UZ9. *+R' (" o.+#E \FAZ)a ("W!&' 4-5t – 2!H=& (!" S#!9#6" \X!!Ag'– 2!!:-#K (:I0!!9 .) X!!8 7:!!; M!!AI: M&#!!%&' .4!!U&'. Na 2FU!!9"&') 2!!9#6"

+%"&#0/2:-#Kp ( 7:; 4-5t 2,F%8"&') n:=t 9#6"2:-#K&#0 2.( t(sec) 0 10 20 30 40 50 60 70

a(m/sec2) 30.0 31.7 33.6 35.7 38.0 40.7 43.7 47.1

)) ( o.+#E&' 25+9v (" $ 4-5 t = 10 .t = 50.

)< ( (:-"W&' (:0 o.+#E&' #aIA; M%&' 2G#9"&'t = 10 .t = 50.

() *+,-: dttexexy

)9% 2&'4&' fLV (Z9.Z4 "#$% J"Dawson's integral.(

(i) ))( 3.!!!A8F& 2!!!":; +!!!0$) M!!!V #!!"h 2!!!&'4&' 2!!!":; <#!!!9=& #!!!V+#:%8' _!!-$":y

#"4-55.0"x ]2!":; <#9=& O) )5.0(y [ (::+!15 (:";+!& 2=:=!E>+=-"&' /01 345#60.

)<( 2!!":; <!!9=')5.0(y !!E /0!!1 34!!5#; 7'48%!!9#0 (::+!!15 (:";+!!& 2=:=>+=-"&'.

)q( M!!!!G _!!!!%0#UQ M!!!!G C!!!!A8&' 2!!!!":; <!!!!9=')< ( 2!!!!":; () 7!!!!F5 'LQ)5.0(y

.'4!U&' M!G 34.!U." M!V #!"$ 2:+15 7#;+) 2I09& 2=:=E 2A.0Y"&' O.#9% 2:9#:6&'0.4244364.

(ii)4+,-"&' "#$%&' 2":; <9='

(15-6) s¹d9

(16-6) s¹d9

!"#$$%&'() *$$+),-"'( !"#$$%&'( .$$/0 #$$12345& 6$$%"3 7$$"#%&/' *$$3,,8'( 9:$$5'( *"3; ,#+3< 7"#%&/' *3,,8'( 9:5'( =,>? 9345&' @!A"B C *"#0 D,,8&"'(

dxdyyxI

)'( 9:$$54 *>3>$$E'( F$$&"3; G#$$H> 6$$%"3 IJ$$'()*$$"32'( K$$583 L$$3> *3$$H#32'( 7$$"#%&

55.790"exI (D:)E'( ./0 7"#%&'( G&%M:

" dxxgI

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) !( AB)*+ %*(C D6 EBF*+ "C: 110 !k

G&1)8*+ %*(C D6 EBF*+ "C: k!"105

) G( f1(x) = .33333 , f1(y) = .71428

!"#$%& '!()%& !"$*%& #!+%& ,"-#%& .-/%& 012)%& .-/%&

Operation Result Actual

value Absolute error Relative error

yx# 11010476. " 22/21 410190. !" 410182. !" xy ! 38095. 8/21 510238. !" 510625. !"

yx$ 23809. 5/21 510524. !" 410220. !" xy % 11021428. " 15/7 410571. !" 410267. !"

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) S( f1(x) = .71425 , f1(&) = .98765 " 105

!"#$%& '!()%& !"$*%& #!+%& ,"-#%& .-/%& 012)%& .-/%&

Operation Result Actual value Absolute

error Relative error

uy ! 41030000. !" 41034714. !" 510471. !" 0.136

wuy %! )( 11027000. " 11031243. " 424. 0.136

vuy $! )( 11029629. " 41034285. !" 465. 0.136

vu# 51098765. " 51098766. " 110161. " 410163. !"

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(i) 41051000

1000* !((!p

999.5 < p* < 1000.5

(ii) 4997.5 < p* < 5002.5

(iii) 9985.005 < p* < 9994.995

(iv) 9995 < p* < 10005

)G( p .1 .5 100 1000 5000 9990 10000

*max pp! .00005 .00025 .05 .5 2.5 4.995 5.

)!( x 2x

3x 26x 3x

Exact 4.71 22.1841 104.487111 133.1046 14.13

3-digit

(chopping) 4.71 22.1 104. 132. 14.1

3-digit

(rounding) 4.71 22.2 105. 133. 14.1

%B-2=>*+ %>&)*+: Exact: 149.13.141046.133487111.104)71.4( ! !)f

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(11-1) s¹d9

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(chopping) ;0.14!)

R(91! %^U^* G&1)8*+: 3-digit: 149.1.14.133.105)71.4( ! !)f

(rounding) ;0.14!)

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0.14636489.14 *!

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(chopping) ,4.15149.71.4)371.4)671.4(()71.4( !)! !)f

DB5& R(91! %^U^* G&1)8*+-: f(4.71) = -14.6

D4& (>M &8*(C*+ > ,M D6 "&"#*+ D2'$*+ EBF*+ b2W& D*(8*(2-: 3-digit

(chopping) 0093.

636489.145.14636489.14 *

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3-digit

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636489.146.14636489.14 *

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(++) ?6++C#$ &++:-@ (++) (6++C,#$ D++!E#$ 7&++B",)? ( (++) *++,N)- ( 7- H++#G ++O28) 7++: P++1 4++1 Q&6&++C'#$ 44++N4 @ ?2++L Q&++8J:N3 ?2++L (++98J:N H++#G ++:O Q&++8J:N

@3 +:O Q&8J:N . 44+N P+8J"9 (+R D+!E#$ $S+R P+8J"9# T2+!#$ U4+'G 7- A+L$@#$ 7+:@D!E#$ $SR 4#@9 (9#$ Q&6&C'#$.

)!( 14.2 D2'$*+ EBF*+ 4103.1 !"

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?-?e 86.625, 39.375, 119.50, 71.50

%2-'C>*+ %>&)*+ %C&CW*+ %>&)*+ n Computed pn Correct value pn

0 1.000 1.000

1 .33333 .33333

2 .11110 .11111

3 11037000. !" 11037037. !"

4 11012230. !" 11012346. !"

5 21037660. !" 21041152. !"

6 31032300. !" 21013717. !"

7 21026893. !"! 31045725. !"

8 21092872. !"! 31015242. !"

?-?f n np

0 1.000

1 .33333

2 .11111

3 .037037

4 .012346

5 .0041153

6 .0013718

7 .00045727

8 .00015242

A = 1./3. = .333

B = 3./4. – A = .750 – .333 = .417

(12-1) s¹d9

(13-1) s¹d9

(14-1) s¹d9

(15-1) s¹d9

(16-1) s¹d9

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C = 1./2. – A = .500 – .333 = .167

D = (4 × B – 2 × A) – 1 = (4 × .417 – 2 × .333) – 1

= (1.668 – .666) – 1 + (1.66 – .666) – 1

= .994 – 1 = – .006

E = 3 C – 1./2. = .501 – .500 = .001

Q = E/d = –.001/.66 = –.166

3–34

p(x) = (( x - 9 ) x + 27) x – 27

p(2.95) = (- 17.8 + 27) (2.95) – 27 = 27.1 – 27 = 0.100

p(2.99) = (- 17.9 + 27) (2.99) – 27 = 27.2 – 27 = 0.200

p(3.01) = (- 18.0 + 27) (3.01) – 27 = 27.0 – 27 = 0.000

p(3.05) = (- 18.1 + 27) (3.05) – 27 = 27.1 – 27 = 0.100

%B-2=>*+ J(2(#g+: p(x) = (x – 3 )3

p(2.95) = (- 0.05)3 = - 0.000 125

p(2.99) = (- 0.01)3 = - 0.000001

p(3.01) = 0.000001

p(3.05) = 0.000 125

3 - 35

p(x) = ((x – 3) x + 3) x – 1

PQ(8$*+ .4< ,WC$ R(91! %^U^* G&1)8*(2-: p(.9) = - 0.001

p(.99) = 0.00

p(1.01) = 0.00

p(1.1) = 0.001

%B-2=>*+ R&)*+ : p(x) = (x – 1)3

p(.9) = - 0.001

p(.99) = - 1x10-6

p(1.01) = 1x10-6

p(1.1) = 0.001

3 – 36

(17-1) s¹d9

(18-1) s¹d9

(19-1) s¹d9

3 – 78

i) , - 01.100495.

995.00.100495. )

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8999.4950

24902500

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24902500

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W¹dDI�« WOŁöŁ W�uHB�W¹dDI�« wŁöŁ ÂUE½

WFD²I� Èu� WK�K�²�5ðd� q{UH²K� WKÐU�

bOŠË qŠdI²�� dOž

ÍuKF�« b(«UOKF�« W¹UNM�«

WO�u� WO¦K¦� W�uHB�w�u� w¦K¦� ÂUE½

t−²*« —UOF�„uK��« ÈuÓÝ

◊dA�« WM�Š W�uHB�◊dA�« s�Š ÂUE½W�œUF� Ë√ W�«œ dH�

Spectral radius of a matrix

Spline interpolant

Splitting of a matrix

Stability

Stable

Steffensen algorithm

Step-size

Strictly diagonally dominant matrix

Substitution

Successive approximations method

Symmetric matrix

Synthetic division

System of (linear) equations

System of nonlinear equations

Taylor series

Tolerance

Transcendental equation

Trapezoidal rule

Triangular decomposition

Triangular system

Tridiagonal matrix

Tridiagonal system

Truncated power series

Twice differentiable

Unique solution

Unstable

Upper bound

Upper limit

Upper triangular matrix

Upper triangular system

Variable secant

Variable tangent

Vector norm

Well-behaved

Well-conditioned matrix

Well-conditioned system

Zero of a function (or of an equation)

n�RLK� V²�»uÝU(« rKŽË �UO{U¹d�« w�

Æ 1992 X¹uJ�« ÆÆ rKI�« —«œ ¨4 ◊ ¨Ê«dð—uH�« WGKÐ VÝU(« W−�dÐ ≠ 1

Æ 1993 X¹uJ�« ÆÆ rKI�« —«œ ¨2 ◊ ¨�«dHA�« WžUO�Ë �U�uKF*« W¹dE½ w� W�bI� ≠ 2

Æ 1986 X¹uJ�« ÆÆ rKI�« —«œ ¨WOL�d�« �UJ³A�« ≠ 3

Æ 1988 X¹uJ�« ÆÆ rKI�« —«œ ¨ÍœbF�« qOK×²�« ≠ 4

Æ 1995 X¹uJ�« ÆÆ rKI�« —«œ ¨2 ◊ ¨ wD)« d³'« ≠ 5

Æ 1999 X¹uJ�« ÆÆ rKI�« —«œ ¨ 2◊ ¨‰UJÝU³�« WGKÐ VÝU(« W−�dÐ ≠ 6

Æ1994 X¹uJ�« ÆÆ rKI�« —«œ ¨Ê«uý— …eLŠ Æœ l� ¨‰UJÝU³�« WGKÐ W�bI²*« W−�d³�« ≠ 7

1997 X¹uJ�« ÆÆ rKI�« —«œ ¨ ‰UJÝU³�« WGKÐ W−�d³�«Ë �UO�“—«u)« ≠ 9

Æ1998 X¹uJ�« ÆÆ rKI�« —«œ ¨ Ê«uý— …eLŠ Æœ l� ¨ �UODF*« vMÐ ≠10

Æ 2000 X¹uJ�« ÆÆ rKI�« —«œ ¨ W¹œUF�« WOK{UH²�« �ôœUFLK� W¹œbF�« ‰uK(« ≠11

Æ 2001 X¹uJ�« ÆÆ rKI�« —«œ ¨ WOze'« WOK{UH²�« �ôœUFLK� W¹œbF�« ‰uK(« ≠12

Æ 2002 X¹uJ�« ÆÆ rKI�« —«œ ¨C++ WGKÐ VÝU(« W−�dÐ ≠13

Æ 2003 X¹uJ�« ÆÆ rKI�« —«œ ¨C++ WGKÐ W�bI²*« W−�d³�« ≠ 14

Æ 2005 X¹uJ�« ÆÆ ÕöH�« W³²J� ¨ »uÝU(« rKŽ w� WFDI²*« �UO{U¹d�« ≠ 15

2006 X¹uJ�« ÆÆ ÕöH�« W³²J� ¨ Ê«uý— …eLŠ Æœ l� ¨C++ WGKÐ �U½UO³�« q�UO¼ ≠ 16

Æ 2006 X¹uJ�« ÆÆ √d�« —«œ ¨C WGKÐ W�bI²*« W−�d³�« ≠ 17

Æ 2007 X¹uJ�« ÆÆ √d�« —«œ ¨C WGKÐ »uÝU(« W−�dÐ ≠ 18

Æ 2010 X¹uJ�« ÆÆ ÕöH�« W³²J� ¨ »uÝU(« rE½ ≠ 19

Æ 2010 X¹uJ�« ÆÆ ÕöH�« W³²J� ¨ WOKJA�« �UGK�«Ë WOðU�uðË_«Ë W³Ýu(« W¹dE½ ≠ 20

Æ 2012 ÆÆ X¹uJ�« ÆÆ ÕöH�« W³²J� ¨ WOKJA�« �UGK�«Ë WOðU�uðË_«Ë W³Ýu(« w� W�bI²*« W¹dEM�« ≠ 21

Æ 2012 X¹uJ�« ÆÆ ÕöH�« W³²J� ¨ ÍœbF�« qOK×²�«Ë W¹œbF�« ‚dD�« ≠ 22

NUMERICAL METHODSand

NUMERICAL ANALYSIS

Dr. Abu-Bakr Ahmad El-SayedDepartment of Computer Science

University of Kuwait

ÍœbF « qOK×² «Ë W¹œbF « ‚dDmathcsbooks.com/assets/files/الطرق العددية...

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Transcript of ÍœbF « qOK×² «Ë W¹œbF « ‚dDmathcsbooks.com/assets/files/الطرق العددية...

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