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(H9/_ O0< . K&! N8()# O,? Q&]) C),- .M/97,- `,GR )$S! ([/A T7I,- J R/ Q! (
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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 ).
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O+E/ QG,- D/&%,-"3(+ 0*+,- $/V:) "(information processing) 3(#(/6,- D/% ",- '(7*+(given data) .0I&+,- 3(+ 0*+,- $B+)(input information) N8()#,- "
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WKš«b�« �U�uKF*«
unput information
WO�“—«u)«
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Włd<« �U�uKF*«
output information
WKšb*« ¡UDš_«
unput errors
WO�“—«u)« ¡UDš√
Algorithm errors
Włd<« ¡UDš_«
output errors
1ÍœbF�« qOKײ�«Ë W¹œbF�« ‚dD�«
23
W¹œbF�« �UÐU�(« ∫‰Ë_« qBH�«
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(ii) 854211 /%&'()* +, #- %010.
(iii) 3.1415926 / +, #- %2*3%-.
(iv) 4.265 /6&- 6&* #.
Types of Errors ¡UDš_« Ÿ«u½√
Round-off Errors V¹dI²�« ¡UDš√ ∫ÎôË√
!"#$:
7(6- 8# 0x: 95:'.;* 9 (exact number)
x: %&.&#<= %*& (approximation) 99)>/ 0x
? 6@%&'()*% +(,* (absolute error) A(B. CD#E)F&:
!xx "#$% 0
-*% +(,*%./01 (relative error) % 1)/,. CD#E)F&:
00
1 xx
xr"$%$%
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4#556 7*8 9) 556 2:$;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=.
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(iv) 4.26 !4.27
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<#;D 4C9 EFG+:
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37:*;:
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*<' ="#> '% ?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
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<#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 $'"
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(0.76 × 0.06) ÷ 0.06 = 0.05 ÷ 0.06 = 0.83
* 0.76
/ e K/%0134 #' d1;"34 7(834 # D: |Total round-off error| = 0.83 – 0.76 = 0.07
'%7"B26*'% C2"<3$'% =" # 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
1ÍœbF�« qOKײ�«Ë W¹œbF�« ‚dD�«
27
E4C%#"F 3 G&. HI 7'%& HI HB69 J@A #B
Max. relative error in a function of several variables
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W¹œbF�« �UÐU�(« ∫‰Ë_« qBH�«1
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(infinite process 2 finite process)V .3,;W+ B4,81N;) ./6*;&1 .3,;W+ .6+;D+D
Truncation Errors ŸUD²�ô« ¡UDš√ ∫ÎUO½UŁ
1ÍœbF�« qOKײ�«Ë W¹œbF�« ‚dD�«
29
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" 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
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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.
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Propagation of Errors ¡UDš_« b�«uð Ë√ —UA²½«
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1ÍœbF�« qOKײ�«Ë W¹œbF�« ‚dD�«
31
: 7& *& 7%5/ R-M( *5A: -::O LAPf(x) T5(:
:x 6Q5Q( **G.
:f Q5Q( 7& *57.
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! !xfxf 2."11
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power series) 7& *& _"4P% 3% (expansion of f) f [.
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W¹œbF�« �UÐU�(« ∫‰Ë_« qBH�«1
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) 7( *8";&' 50 !2#:-%&' 3#4516%&'(Sparse Matrices).
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J6#6!!,$-8 K#=!!I-8 K=!!BL 9:!!H(I3-8(finite-difference methods). 3= <#!!?3 6!!]# +!!>$ !!L C86!!!% J=!!!9L? @(Q#I!!!S$-8 EF!!!G P!!!3-8 !!!9=9=?3-8 K=!!!B-8 '8643!!!/(L +!!!,:- C86!!!% L!!!/(*$ P!!!G# M
Q#IS$:- f U-8 #& J=>(*3$-8 79LB-8 <$ 69I3/3.
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)0( <5#=& >?@&' *A+!)Gauss (G) Elimination Method
<93#!!!!B4-8 P!!!!Q !!!!9B4-8 @A6(!!!!7$-8 <!!!!$ '(!!!!)* +!!!!,- !!!!N9=B-8 EF!!!!G m4:3!!!!3<939-(3-8:
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9>:>$.
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J=94Y8 +L] -6(7$-8 '>) 9*(>-8 5& ( $9] 0:" +S,* (U*$#2x:
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PQ L#=H$ 0-#Y8 -6(7$-8 EFG ZH&:
13121 ,,, naaa ,,,
']= -6(7$-8 0-1: 2 , 3 , ... , n
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(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, :
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24232 ,,, naaa ! ! !
FG* &28$9%2# 123: 3, 4, ..., n
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11
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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$.
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22
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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# =>:
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5E:
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xxx /1!!!. MV!!!)5/502 6!!!*+(+#C57<2.
) ( ! "#$ %&'()*+,, (Elimination without Normalization):
34 (=,/Z" 60,X2 (01#$"02 YZD
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132
%&'()* +,-.)* :/0-,)* 1.2.$)* 3-!,)* 4") 152()* 6+&70)*
(Back Substitution)
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!"#$ %&'$( )*+,- ./ )+0"1'$( 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 >$&, Q9VH 6"*U&>R /:
[.\'( /),4&5 BB5 /A&KN'(: 3
3 23 nnn #
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A&######KN'(" [.######\'( /)######,4&5 BB######5 >R JR)].######M'(" S######&U'( ^'X######$" ( 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.)####;
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)###*+,-.- .###,,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.
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) ( !"#$ %&'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$/ VLPP$', 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',
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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):
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3,.0:
%)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 $#372 J)&' 1-3.(
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uy ! 41030000. !" 41034714. !" 510471. !" 0.136
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= .994 – 1 = – .006
E = 3 C – 1./2. = .501 – .500 = .001
Q = E/d = –.001/.66 = –.166
3–34
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Clamped cubic spline algorithm
Clamped spline
Closed formula
Complex number
Complex root
Compatible
Composite integration formulas
ÍeOK$ù« `KDB*« wÐdF�« `KDB*«
(4) r�— o×K*«Index W¹eOK$ù«Ë WOÐdF�« �U×KDB*« qO�œ
ÎU¹b???−Ð√ ÎU???³???Oðdð W???³ðd???� W¹e???OK$ù« W???GK�U?Ð �U??×?KDB?*UÐ W??L?zU??� w?K¹ U???L??O???�ÆWOÐdF�« WGK�« w� UN� WKÐUI*« �U×KDB*«Ë
434
ÍeOK$ù« `KDB*« wÐdF�« `KDB*«W³�d*« Êu�³LÝ WGO�
W³�d*« ·d×M*« t³ý WGO�…dFI� W�«œ
◊dA�« œbŽ—«dL²ÝUÐ q{UH²K� WKÐU� W�«œ
»—UIð»—UI²�« qO−Fð
W¹d¹dJ²�« ‚dD�« »—UIðWÐb×� W�«œ
©d�«d�® …bŽU�©�Ëd�® WI¹dÞWO³OFJð W�œUF�
WO�ULJ²Ý« WO³OFJð W×¹dýWO³OFJ²�« W×¹dA�UÐ ‰ULJ²Ýô«
©sÝËœ® q�UJðWHO¦� W�uHB�
WI²A�…œb×�
W¹dDI�« …dDO��«‚ËdH�« jD��
‚ËdH�« ‰Ëbł‚ËdH�«
WOK{UH²�«WOK{UH²�« W‡DÝu²*« WLOI�« W¹dE½
q{UH²�«q{UH²�« q�UF�
�U�uHBLK� dýU³*« qOKײ�«…dýU³� WI¹dÞ
WKB²� dOž W�«œbŽU³ð
©q²O�Ëœ® WI¹dÞnÓŽUC� —cł
WOð«– WLO�wð«– t−²�
dOOF²�« l� ·c(«dOOFð ÊËœ ·c(«
QD)«
Composite Simpson formula
Composite trapezoidal formula
Concave function
Condition number
Continuously differentiable function
Convergence
Convergence acceleration
Convergence of iterative methods
Convex function
Cramer's rule
Crout's method
Cubic equation
Cubic spline interpolant
Cubic spline interpolation
Dawson integral
Dense matrix
Derivative
Determinant
Diagonal dominance
Difference scheme
Difference table
Differences
Differentiability
Differential mean value theorem
Differentiation
Differentiation operator
Direct factorization of matrices
Direct method
Discontinuous function
Divergence
Doolittle's method
Double root
Eigenvalue
Eigenvector
Elimination with normalization
Elimination without normalization
Error
6ÍœbF�« qOKײ�«Ë W¹œbF�« ‚dD�«
435
ÍeOK$ù« `KDB*« wÐdF�« `KDB*«QD)« bŠ
QD)« d¹bIðQD)« u/
QD)« —UA²½«ÍbOK�ù« —UOF*«
rOOIð©�dH¹≈® WGO�◊u³C*« œbF�«◊u³C*« q(«
œułËQD�K� wÝ√ u/
¡UHO²Ýô«W¹UNM�« v�≈ ¡UHO²Ýô«
W�uHB� ©‰ULŽ≈ Ë√® qOK%W¹dDI�« WOŁö¦�« �U�uHB*« qOK%
nz«e�« l{u�« WI¹dÞ…œËb;« ‚ËdH�« WI¹dÞ©WON²M�® …œËb×� …d²�
WON²M� WOKLŽW²ÐU¦�« WDIM�« WI¹dÞ
©ÁU&ô«® XÐU¦�« lÞUI�«©ÁU&ô«® XÐU¦�« ”UL*«
WLzUF�« WDIM�« WGO�»cÐcð
WO�U�_« ‚ËdH�« q�UF�w�U�_« i¹uF²�«
d(« b(«”ËU' ‚ËdH�« WGO�
”ËU' ·c(« WO�“—«uš”ËU' ·c(« WI¹dÞ
wze'« “UJð—ôUÐ ·c×K� ”ËUł WI¹dÞq�UJ²K� ”ËUł WGO�
Ê«œ—uł ≠ ”ËUł WI¹dÞ‰b¹“ ≠ ”ËUł WI¹dÞ
wÝbM¼ »—UIðf½U−²� ÂUE½
d½—u¼ WI¹dÞ
Error bound
Error estimate
Error growth
Error propagation
Euclidean norm
Evaluation
Everett formula
Exact number
Exact solution
Existence
Exponential growth of error
Extrapolation
Extrapolation to the limit
Factorization of a matrix
Factorization of tridiagonal matrices
False position method
Finite difference method
Finite interval
Finite process
Fixed point method
Fixed secant
Fixed tangent
Floating point form
Fluctuation
Forward difference operator
Forward substitution
Free boundary
Gauss difference formula
Gauss elimination algorithm
Gauss elimination method
Gauss elimination with partial pivoting
Gaussian quadrature formula
Gauss-Jordan method
Gauss-Seidal method
Geometric convergence
Homogeneous system
Horner scheme
W¹eOK$ù«Ë WOÐdF�« �U×KDB*« qO�œ ∫(4)r�— o×K*«
436
ÍeOK$ù« `KDB*« wÐdF�« `KDB*«◊dA�« WKOKŽ W�uHB�
◊dA�« WKOKŽ W�Q��◊dA�« qOKŽ ÂUE½
Àb×� —UOF�¡«dI²Ýô«
WOzUN½ ô WOKLŽ·UDF½« WDI½wz«b²Ð« QDš
—«dI²Ýô« ÂbŽWOK�UJ²�« WDÝu²*« WLOI�« W¹dE½
‰ULJ²Ýô«…d²�
w�JF�« ‰ULJ²Ýô«d¹dJ²�«
d¹dJ²�« W�uHB�Íd¹dJ²�« 5�ײ�«
©W¹—«dJð Ë√® W¹d¹dJð WI¹dÞW¹d¹dJ²�« WOHB²�«
©wÐu�Uł® WI¹dÞ!«dłô W¹œËbŠÍuKŽ bŠ dG�√
W¹UN½QD�K� wDš u/
wD)« ‰ULJ²Ýô«wDš q�UF�
wD)« ©¡UCH�« Ë√® ⁄«dH�«wDš ÂUE½
—Ëc'« l�«u�wKH��« b(«
WOKH��« W¹UNM�«WOKHÝ WO¦K¦� W�uHB�W�uHB� ”uJF� œU−¹≈
W�uHB*« —UOF�w³�½ QDš d³�√
WDÝu²*« WLOI�« q�UF�WDÝu²*« WLOI�« W¹dE½WDÝu²*« WDIM�« …bŽU�
Ill-conditioned matrix
Ill-conditioned problem
Ill-conditioned system
Induced norm
Induction
Infinite process
Inflection point
Initial error
Instability
Integral Mean Value Theorem
Interpolation
Interval
Inverse interpolation
Iteration
Iteration matrix
Iterative improvement
Iterative method
Iterative refinement
Jacobi's method
Lagrange polynomial
Least upper bound
Limit
Linear growth of error
Linear interpolation
Linear operator
Linear space
Linear system
Localization of roots
Lower bound
Lower limit
Lower triangular matrix
Matrix inversion
Matrix norm
Maximum relative error
Mean value operator
Mean value theorem
Midpoint rule
6ÍœbF�« qOKײ�«Ë W¹œbF�« ‚dD�«
437
ÍeOK$ù« `KDB*« wÐdF�« `KDB*«©—dJ�® nŽUC²�
©—dJ�® nŽUC²� —cł©Á—«dJð �«d� œbŽ® —cł nŽUCð
»—UC�«WOFO³D�« WO³OFJ²�« W×¹dA�« WO�“—«uš
wFO³Þ —UOF�WOFO³D�« W×¹dA�«
qš«b²*« »dC�«WOHK)« ‚ËdHK� sðuO½ WGO�
fðu� ≠ sðuO½ WGO�WO�U�_« ‚ËdHK� sðuO½ WGO�
sðuO½ WI¹dÞ�ôœUF� rEM� sðuO½ WI¹dÞ
Êu��«— ≠ sðuO½ WI¹dÞWODš dOž �ôœUF� ÂUE½
…œdHM� dOž �ôœUF�œdHM� dOž q�UJð
…œdHM� dOž W�uHB�©ÍdH� dOž® t�Uð dOž qŠ
W�uHB*« —UOF�t−²*« —UOF�
dOOF²�«d]OF�
ÍœbF�« ¡UG�ù«W¹œbF�« �UÐU�(«ÍœbF�« q{UH²�«ÍœbF�« q�UJ²�«ÍœbF�« q�UJ²�«WŠu²H� WGO�
WOÐU�(« �UOKLF�« VOðdðwÐcÐcð
wze'« q�UJ²�«wze'« “UJð—ô«
∆e&w¾¹e& wDš ‰ULJ²Ý«
�U¹œËb(UÐ w¾¹e'« V¹dI²�«w¾¹e& wFOÐdð ‰ULJ²Ý«
Multiple
Multiple root
Multiplicity of a root
Multiplier
Natural cubic spline algorithm
Natural norm
Natural spline
Nested multiplication
Newton backward difference formula
Newton-Cotes formulas
Newton forward difference formula
Newton method
Newton method for systems of equations
Newton - Raphson method
Nonlinear system of equations
Nonsingular equations
Nonsingular integral
Nonsingular matrix
Nontrivial solution
Norm of a matrix
Norm of a vector
Normalization
Normalized
Numerical cancellation
Numerical computations
Numerical differentiation
Numerical integration
Numerical quadrature
Open formula
Order of arithmetic operations
Oscillatory
Partial integration
Partial pivoting
Partition
Piecewise linear interpolation
Piecewise polynomial approximation
Piecewise quadratic interpolation
W¹eOK$ù«Ë WOÐdF�« �U×KDB*« qO�œ ∫(4)r�— o×K*«
438
ÍeOK$ù« `KDB*« wÐdF�« `KDB*« ÍbðË Ë√ ÍeJðd� dBMŽ
“UJð—ô«©œËbŠ …dO¦�® W¹œËbŠ
b¹bײ�UÐ W³łu� W�uHB�¡UDš_« ©b�«uð Ë√® —UA²½«
wFOÐdð »—UIðwFOÐdð ‰ULJ²Ý«WOFOÐdð W¹œËbŠ
©Wł—b�«® WOŽUЗ W�œUF�WNO³A�« sðuO½ WI¹dÞ
»—UI²�« ‰bF�rOI�« WOIOIŠ W�«œ
©W¹œuŽ® W¹œ«bð—« WGO�nz«e�« l{u�« WI¹dÞ
WO�UE½ W�«œw³�M�« QD)«
w³�M�« QD)« bŠ—cł W�«“≈
©w³Ý«d�« Ë√® wI³²*« t−²*«ÊuÝœ—UA²¹— ¡UHO²Ý«
W�œUF� —cłV¹dI²�«
V¹dI²�« QDšWO½U¦�« ‚ËdH�«
WK�K�²*« „uJH�©WK�KÝ® WO�U²²�
WŠ«“ù« q�UF�W¹uMF� ÂU�—√
Êu�³LÝ …bŽU�Êu�³L�� 3/8 …bŽU�
WODš WO½¬ �ôœUF�WODš dOž WO½¬ �ôœUF�
œdHM� q�UJðœ^dHð
…œdHM� W�uHB��ôœUF*« qŠ
WA¼ Ë√ …dŁUM²� W�uHB�
Pivot element
Pivoting
Polynomial
Positive definite matrix
Propagation of errors
Quadratic convergence
Quadratic interpolation
Quadratic polynomial
Quartic equation
Quasi-Newton method
Rate of convergence
Real valued function
Recursive formula
Regula Falsi method
Regular function
Relative error
Relative error bound
Removal of a root
Residual vector
Richardson extrapolation
Root of an equation
Rounding
Rounding error (or Round-off error)
Second differences
Series expansion
Sequence
Shifting operator
Significant digits
Simpson rule
Simpson three-eights rule
Simultaneous linear equations
Simultaneous nonlinear equations
Singular integral
Singularity
Singular matrix
Solution of equations
Sparse matrix
6ÍœbF�« qOKײ�«Ë W¹œbF�« ‚dD�«
439
ÍeOK$ù« `KDB*« wÐdF�« `KDB*«W�uHB* wHOD�« dDI�« nB½
WO�ULJ²Ý« W×¹dýW�uHB*« —UDA½«
—«dI²Ýô«dI²��
¢s�MH²Ý ¢ WI¹dÞ…uD)« r−Š
W¹dDI�« …dDO��« WOFD� W�uHB�i¹uF²�«
WFÐU²²*« �U³¹dI²�« WI¹dÞWKŁUL²� W�uHB�WO³O�d²�« WL�I�«
©WODš® �ôœUF� ÂUE½WODš dOž �ôœUF� ÂUE½
—uK¹Uð WK�K�²�`�U�ð ≠ �ËUHð
WO�U�²� W�œUF�·d×M*« t³ý …bŽU�
w¦K¦*« ©o¹dH²�« Ë√® qOKײ�«w¦K¦� ÂUE½
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
©ÁU&ô«® dOG²*« lÞUI�«©ÁU&ô«® dOG²*« ”UL*«
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)
W¹eOK$ù«Ë WOÐdF�« �U×KDB*« qO�œ ∫(4)r�— o×K*«
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�« qOKײ�« ≠ 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
Æ1993 X¹uJ�« ÆÆ rKI�« —«œ ¨ ©WLłdð® WOL�d�« WK�UJ²*« dz«Ëb�« ≠ 8
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�« qOKײ�«Ë W¹œbF�« ‚dD�« ≠ 22
ÔrOJÓÚ(« ÔrOKÓFÚ�« ÓX½Ó√ Óp]½≈ UÓMÓ²ÚL]KÓŽ UÓ� ]ô≈ UÓMÓ� ÓrÚKŽ Óô ÓpÓ½UÓ×Ú³ÔÝ˚©33∫ …dI³�« …—uÝ®
NUMERICAL METHODSand
NUMERICAL ANALYSIS
Dr. Abu-Bakr Ahmad El-SayedDepartment of Computer Science
University of Kuwait