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( t} ,l tf1ifu I: (;;J 11 ? .)

-X,cJE/Rt.9�t� ---------f'VXt-Vc/� X+d 1 L1X+L:\�

Vx - .6 � � X - -:, � �X - V cf

E u 1:, p:-, d ,;1 _=-r.i [ vx, v L 4:i:.4;,J

l X - � lo g Piaj r V'X -Lid " 6 x- V 1 ]

7 a ')' E rrai r:J I� 13 ( ti iJv44'} 3 f tlJ()) )J:. � Af t,\··t i.h { 11 �.

J", 1. E. Pej 1·· 5,-JA. � �j t ta·�-i °'Aft. t*f Et ch 1)t.J IJ '" '

t 45 7·· e �. :11. �, §}r3JJ�1* c "j. .- -

t:i1 � !r-tjll)J'"VJ� f Faj t[ X] = c x., x J:: { x E IR. I x � x � x 5

l i<- l , 0 E f +, -., X., 7 J t LJ� l: � ,,

[ X] 0 [ ':f ] = { X <> d [ X E= [ �] -> d � {d]]

J: U 1t /f,JJ,·j\'}., J [x] -1- [�] = [x+1, X+}]

[ X:] - [ �] � [ X - f, X -} ]

[xJ x [a]= [�;�{x], !t.t Xd� :r.)], maxfl�,?5,Jj, xiu- -1 1

[xJ-:: [d]:; L x] )( l-:::- -] o t ['d]) ) 1. )

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� s {:' t t t� .t 1 ·· ('j j =h tr) 1J , .• � � 0 1 ··

ix] = [ x,, x J c fR1 � j:j l ·c 0 f � J = [ v x , .ti x J c IF t L 7 " �. < � 1n\ � PaJ J ->z � 7"· Id j o E f + _, -> X., 7 3 ( � j :I L ,,

[!tJ�[�J = O([xJ o[�JJ t�1�.

*14'�:]f:ta.tMAf!w "{"'Id

[x] + [�]= { Q (x-+ � )., 4(x .. 8°J]

[x]-[d]= [v7(�-f)., 46 (X-1)]

[xJ x [';1] = [ �;>1 {vz�) vz�., vx.a.., v'iaJ., mAX{L1�1,6�<J.,Llx_a_,4xjj] [X] :- [)] = [ )'Hin f V �l'J.., fJ X If., Vx/1_, v'X/�J,

max { LlZ/i ., 6 'X/£ 4 x/J., 6 Vf J]

1� �'J 5� -l i1�1h\f.PJj 5t� ,� I2 irt 7 Gtl 1-htc r,J..°'�' 1tt E�J/.J'··� �ff�.

L t r � 1l-0,·· S, R� 0) tJJ'JJftJr,·�t,�R?/ ttlit.Jt '>A� °F<l"' & ?. t: �/ � Ffl� ())IA�,;,· r_·· t � t 1 �. ( 1� 1:: tr>-i >

1: ·c WL t ,� tt 11 �f';J 1:: 1-a, � t "'·7 HL,,e1> ��, ti1! at· 1 ..

�� /l {J fu1 � 15 L � L 1-., fJ'·· 1l· '2117" �. (°F 8t., F"�2, 3 f .�� J

PojZ �t§� A= (; �) E'5L�. (;) u, A2-3A�4E=(A+E)(A-4E)=O t�,�- (tr.t(··L. 1::1c1f.1i1iJ�)

(2) {(AtEJ(�)=(;) c1ittt: A(!)=4(:)

(A-ltEJG)= {�) t 1�rt: A ( j) = -(�) � ;j, e. (�lt gt t t:J �jf:J 4 .{)) /!1� "·1H�E ;}d) 3;�� f�(f) 'Jj t!U� )

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0 0 0 ()

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�/ i�f ��< tt�1J{ T-J�-'4- l- � lrt+ n.fI (Wt,ntcJff{)t. �� c t: r� \ , .

< I ) (: : r (: :) I: i;i·a ,'flf)h\'�'0 h 1' I>b\aXJfJ A =(: 1) t ,· 1 ? q<'k'>J.

�!,.l�. tP> � A E M1i(2) (f.ll.h\"�'0 hJJ'� A·� 1t><1tu51i11) tJ,·JtJi l 1

f (A)= afl,A.,.t CJ..,_, A ',t-� ... -t C L A +a6f = o l G·� t:r? ( t 7 t irt? >

0 �J}i]x)0 l EE E 754 tl) ;Jt1J1J,it�, 0 N. J. Hjh.a>x : Acc,a·a.cr cu1d Sfa/,; /; 1 df Nu,,,erica I A�ordlt.,,,s.,0 SIAM (1916)

0 �faEJjfi,ffr 0 R.Ha1'1mer_, HJlocks., U./t:ulisk anol D.Ro.fi:

0 N<Americal Tool box for Verified Co1JtfUfit13 I., Srri1tjev-Vevfa3 ( !P93)

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� 't"' r.,1�1::it L, cr 'J �11t [{ < / g�;tf {r

< gt l 9 �:: t-<J,·· ¥�it�. 1::.·r: e� 1�f-t5l. t.1�f5ifc�l 1'�.

© �iHj1t/z ,� Pi]l lt�-t;·t. llA! 8tJn---c-·1a,�1\-�itii..1ti ( �11t1At. etc.> ,Q ... 4� t,� ll <[iJ I l ,; h �. 1"' tcd? ,�t1131t# I� f(1Jt. L k. f }l. 1r J J< JJlfe.h111�. c�··<.-fP�'E�1;'��J

(I) Lu �S�;"t,A �> lh <1-i �.if I�· J

l � .tJ. L u l,j Af 5t:r· la PA= Lu t1) J, {� 1�fij1A E i: =- J1J1i§'J L l: l: =-fiJ4j fJ U ,�� '�, J.

L1 = Prb ·· · ca.)

f. �f 11 11/ f ;J<·V),? t"I�ux = � ... <-C)

t �t II l X � �<&") � .

13-� t� �--�, � J? I�, (a) t (-6.) fj =. ]1J1jj'J f1MJ. lcL <\, � P) 7:·/ Psi !f t��1;:ttt < ftj GtJ! 7.·i1¥,1 �-

(I)

t�t:. c 2 J ( � ; ) 7# = (: ) t lj· � } E 4; ¥> d.

(31 (: :)� = '!I E�f� Z ;:ii>(;) E�<JSJ"

6 )

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L D}J'�li ;t I: fA l, 1 , :l l, A 1- �"flit ., A J, 0- ,Z ?j§'J-1-j� �' t .. ld1 �-

( J �:z i- �'�f,J . .iiffJ?jy� A t:i1G A= L L t t �'PJf9�·) '/..1,

0

-J... �J'J }iZ.. t ld,fjj:J;f! I:: 0/J'·-JJ,, � t�lJR) 3 � _ 1t t t t J t·'Jt c t:tPJ A�"�,-· ��.:i�fi]h\'�' J·; t 2 \, �g J 1d'c" t�J.

c 2) 1111iJ 1g p,�,a ( &Jri � ;t )

JTI � 1.! icL de t CA -;i. E ) = o t1> fi]f._

- ;t,"' tJf t,,··�� ,:'cJ�. (s�iq� �,�� ) __,__ ___ _

'Jll >< 1t Jlt,fiffe'JA rJ A=.SVS* c 3,�fH'.J�tc_�'�; C.ft �- r�r�··L, s Id 2 =- 1'.),1'jJ�T· Ute! .t =-r-J1jj'J.

L /J' i, ff 9 J O Q) i1 PJ h\1]" /J''. A 0 rAf]1A

& fi P-ri 5! (c1 ,.i& 11,{B1 {: Schwl' �' PJf E ;j<J; j 0-5i 7"·J7>?3.

( 1t 1:: 'cf�,( la, 181}�1-l�\ A.�, t;, f\'11., ?·· ( Hesse.,he1) 1H� 'cr l: M1·· )

1£ � i 1 Jt 1 r c,,

cz)' m� 1& 0) 13 fi. ttl!\ti··? rJ 0 1�, 11 }'J A 0 ®�1A � �t> 'J � ,, � l: t� � .. ·· 1nt1t;

Z.J!f) ( Gersh.3ovi .-. )

fj §'J A� I!] �1! /j �t ;t f@J e7) 'f �;;t0).[1$ d) t r::.!fft. 9 �

k :: .0 {, E C / 1 f -O.i.i } � z I �.(� I ?,1. :. J {::J,il:#..t j

1v i!1. "'ti4.t d) t Q) �'6 (c) !f !c· -& 0.;.i J: 1 � tri r:: r, , ai�,

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Jil{11! P>-0-li it fi\ E 6''1iii t J. 0.2. 0.3 - 2

;lJj! ( Collaii Q) titJi' )

A E 1tx n 't1 i=J:tUj J � t 1 �-

( V;

� . V:) i/\''<.{1)8(�"t;'·· 01·'1d' 1 > ,{t.,,'�-'1> �1t.:,)�ft.KJ�l'v 1.r= �

1:: i� L l , A yr = (!: ) H � .

::">t�,

11'un ';J� -E ;;t � max �*-1 ���" 14 - - ,��!� �

t hr� 9 A� t�tfu1A t1\··/3Ji � j

(2) 11

,u� 1! 1�8'.[; 1, ��t1i 'I1::··tJ11t c� �-

( 11fli�r1,L l �1Jfl c IJ tW �{J·\)qj§'J A t::i=) l 1, AtA )

U) ffl.f-J�� (/) /t,-1-- d) '- I:

(3) lY<fl) �t iL5t\C E /R., � A t. m >< h. .fj j � ( 1t > � ) � , 3. b E /R � c -; J

t�fl� ttii., P111@_t'J B 8�JlJlt P= c tx= t C-J°'Xit=J

ltb�t:J Ax==b , x�o (xi?o,J°:f.,2, ···.,n. J{/) Y.. ?. "f1t.. :t:A t 1 � ft��"> .: t.if lv};f t 11 � 1�··1' �151; 1J t ·· ,r 1\1"' �It 11, �- (tftFfJ �)

8

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® Jl1t;( U) J� r.n 1 EE E 7Slf. ll# 1"1J. tt \1! JJ--·· #i /£ 1�tl c..h � (;) tJ 1'ul �'J 5'12.lt

t.-¥ � ift 0> � ,�:td', 1 "i ,%--�i (TJ /lJ /k (l) 1ll tc( :f� Jl 11?fJ;E 1:J 2 T· g tl 9 � fr: JI) l � /ff:' h\..

crd' < 1 t<l 'd· ;rd'� 1• (-%·4}-517··-3 'J - 0 1��.,,-1)

(I) J) i� i\ 0 �'5' (_JJ-.-{i- ;l)�

J)iffil � P: /R--+ IR � P(t)\� t t ,( ,�i1L 1 ra

f(t)::(--- (�t+R,-, t)t+ .. ·+ f,}-,f(J

t7> fl� 1: l- 1 �t Jt '�.

17'Jilf· P(t) � t3 � 2tz

+ 3 t t 9 = (t1+2t

l

-{3t)+4'

=- (t 2+2t+3)t +Y.

= (C t + 2 ) t + 3 ) t + <;-

cz> m � IJ' 1a- r,.111 P) �t

/J,-j - 5l-z··/j -=*-t t <JD� 8�,ff. ,f ··l( n § 1 ·· t}t��;P>1! '8 .;}"&,J � t�\:·tfl �.( ttt Wiii.i t, Er� A¥-i1r � Jrju l ;}"<"&, � � 111 J )

llJ fl f: JR � JR .o,·)� � /J' �r �151� 1a. 9J ii! it. f· Jili flJ U:_ix_) L, 11Jl f_ 1<"k')j: t.lJ�']J "'.

( 7 j'' J / :;·2 If ffl. fqj ., :::_. ;z - t° / J,l f,JJ 1 4 z t ., :_, I 7 ii/J" r,�1

�' t ") PlJ/1 f (t> ,�iJ � 1, 11.1 1 J C1) t �- a 11 P) f. t", t,., ... ., t" ,� 71111d

-=r 11 -i Pl) IJ' � f c td u� o, , ., ... ., vt J , er t 1 � ��=* 1, , a 1:: 9 � c. � t-t3· p.i ( t ) = • II . .

t t ( A. = 0, I, .. ·, h ) l: d>' �

3=t., a,1. 4 - i n

Pct> = Z f ( tJ 1,(· ( t J . ,i: I

I:: LJ:t!, P(tr� fct;Q)··1t7;,;,,.�kt·�iti11!;:t· f:tJef-J .. ·· ?

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• -=. :i - � � j,1j· fJ] (j 5 ,.-7;, ::-".1 #J fa1 P> g,Jk}t

• .-fx. t·� z. 7J,rn·r11j IJ, -t:z:t··;,:i. 7 ;Jt�i\ T1t (t),. �(1t�t> o,�o

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I � 2. k (2. P(t)=-zCo -t LCtTl(t) C�= �f(t)� J.t�TL) �=-1 J 1-\-t I ,<=o 2Jt+

t L fr. J: �, PC t ) E f Ct ) 0> fr. t ·· > z. 7 ifii'Pa J J Jr-� � t. of .J ,··.

Fo1 7 -tz. t .. 51- 7 11 tll i\T.,.(t)= ccq_ ('ll�t) (11.=0,/,2, .. ,)

I::: 7 " 1 , Ti ( t ) � T� { t J, 1 ( t J � t {l) j J �ffil � {/) fi; T·i<: t! .

Q) # �-- "Jl�{j�JtJ4 (.:. ;i -�;, 'J- ':} "> � i\.>;J15- '°Z JD - 1} ;,� ��

etc. )

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- �i (Tr!1J,€i, t � ) )

+� f Ii 1> j 5 :; -

ii" (!J � f�"' t .o,� 1Q;-f:�r t?�i �I��� L, p,,��1:A> t·· 1 A:ll Jff.;h t! r� 'J,, 1Ja1. Lr� 'J l 1,,

� (jJ ·:JI I? / z_'J. ll' t, ..1,.,;@ -:::t � a_ t. :!t; / ,fl "'t ,t/-..:·iA...'JE. t 1r--11�c;,i.( iu ':I tJl�J �i<0i&v1"71.�}k.

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,.t. {t) = . IT . . t - ta: ( J. = o, I, ... , n) t ch· 21=1,3; ... t..i-ta

P<t)= Lf<t;),t(t> ;1..: I

I:: L f: t �. Pct) E .f Ct> 0) 7 t·5: �;�;t;m·P,fJ t 0:t lfr·:

: : 7. .. I d, 5 : [ (l , e J � /f(, 1/,· }� � �' (... � (/) 7 7 .. 7 / :,;'�inn 'f ,?I p1-r· + 0' ec. · , m 11,i{ F -J L .t" � t 1 :a . · , . � ( t�: , I f <tJ - />(t) I� cn.;o ! 1/F"'

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b'" h �' �. ( it,b lit-Ji, �fq1l1*-�J�i e. ti1AiJt f f. 122)

f I /. 7. I .:.. :z - /<,, . J - ·:; 0) �� 4,'1 �c. 7 ?"5:. S,"�imr?ej � 1t � 1,

. f' fctJ dt = ffrtJolt +£ � J? I� &f-1,fu 1 � 6 :i E.� l �. : .ft E,, ;: 2 - /- :,, · J - 'J 0 t rt· � 11j,J ._:·

Q (D fj' ii� i: ,t," (t,-,,peloiJo..P rule)

0ipctJ

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E. =Se ( .f(t) - rct)) dt � s fJ.(t-Q )(t-C) f (t)- P(t) �a. o. (t-a>(t-C>

: : 1 ··. :ti �-i' Cl> 11; .2 Jft� 1A ��.I£ --------

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--�S:S(t)Lct>d% = h,{S) fsrudl F.1f?t., g(t):(t-a.)(t-6) (Ct)= f(tJ-f(tJ 1::L 1 " (t-a )(t-8)

,,.

l: = (t-a.)(t-e.)ctt = ;-(S)-r(S) "{- (C-o.JJ t - f'csJ-P<SJ s.-e r b

(5-a>(s-CJ a. cs-axs-e.J 6 J

' .f<s J - f (s > I r'' ? ( s ( = -2 J ( ! J E 'd' a ; E [ a , e.] 1; v·.fJ,li 9 ?J °' 7. ..

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-a l:M 7, E = -1. ftt J xf- re -a.iJ = - f<JJ{.i--aJ3

2, 6 /2. ,t:., ff-� I ' � . f 4-1:J ./ L,� o. -------------

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5f �l)<'l/lJjf} f z [o.,CJ-IR ,�t}l. �-P(t) :i;x�t Ptt)= (t- t)(t-cJ fra.)

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+ ( t- (). )( t -,e_)f ( C )

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:; / 7 0 '/ / P) �t\ ------------�

(-C.

f<t) dt =�( J'(to)�4f(t,)-f f(t2))- _9/.; r'nJ l )a, 3 0

t a -t e , fr-a. ) 1.----______ (frJc""L, to =a, L=C- ) , =

2. .,h.=

2

()ttr (/) � :- 7 ° '1 / 0 ti\; t, Bil��� t\� t 2 c /�J<1c��)1� )(�)

@ .:.:i:-�/·:l-') O) i:�

© ,@ !. ; � {l) 1t ;i;, 7 r 7 :,,- :/1imJ" PIJ� 1··.�t � t, ::=. J - f. :- ·"J -'1 If)� i\; E�a.

r f<t>olt =Aw. fct,)t E

t�r�.-t,

1-A4. = h, (" £ (S-t )··· (S-�tl){.S-�-1) ... (S-11-) JSJo �(i-1 )··· (�-it I){ �-°R-1) ... (� -11.,)

E=

Jt+3 f (h+2)(3) (h h <n+2)! )

0 (S-z).S(S-1) ... (S-n)r/..S

_h, __ l_5J S(S-1) ··· ($-'h) d.sI h12 f (h+IJ1<, ih

(n.-tf)! 0

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I t,c1, -��',,,t2+1MA"' 5t.l! E ,J# &m 1·�1J-.. �,) µ ") �;\;Id BJJ � t,, (" {f". st. ffe i.ren /,)," ;j; L�,f.,j Jf_ E JG� L fr

11 1

J 1i� 1 tr) i; € ( T;j'fl3' t � ) ::;>(D

{fJ 1t= 3 ") ii.g ( ',,;,. 7' '};, ,,1;; =- � �) ( :,;, 7" ');.."' ; �4 H •.:j IJ:/f �) s: frt>Jt = f ( f{t.J+ 3Ht.>+ 3 f<t,>+ frt,i>-W, f"4H J

1fl 1t:: Lf ()) i'§' ( 7··-1v· ·II) ��) ( Boo le's rule J

S)t)d.t= :� (7frt.)t 32 s<t.)t ,2f1t. )+32f(t,)+7fft.J)- �1 f"h)

Poi 1 f ct> = Ai>,, n, t t f ff31 [ o, 1 J 1 � ls l $ .

SI

S' . - ]' 2 S<tJcft=- ,4,v,,iT[tdt= [-Lcosn.t =:::- z·iJJ.

0 O Tu o ll.i

� Pa� [ o, t J � 6 �}l L 7 )t'Sff Se? t�t) ti\:© }!if j t,

ff ftJ dt• Ip ff (o)+ 2J(f )+2f(:) +2 f(1) +2f {i:J+2fff)+f<�+ EI 3 (..1/ �

E = + [1 { !L'AM,. 7l, J, + /L �7[3, + ... + iC' ,t..,. \} d? I £1 � f 4 J[, < 'if

7 J'). i = f { f (0)+2f{l} t.Zf({) + 2f{1)+2f{t} + 2 f({ Jtfci!}tE.

:O)�tMiL./! - []/ < 2� 1"ih�.

D 1:�1 ra �at �t·� J ( +61t<�1tJ .. 1kvi a ... )

4 ·

..I

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§ II. 7. 2. 417 - · 17 O- � ./ (l) i-0 ti\; t ;ftjt t\ ">#j/1, (Jo�[)

f3 it��;\ Id, Jr,]!� Jl1fl 0 tti �' 'a� t"T� it �-�I)f�%:JJ··· t <'l � : 'c/J''°t" � lt 1 " � . � (1) � t f I �A, I: 1,{j - r 17 a - 9 / P>tat �� f �. ( it� 11 tit. P. /32)

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f ll. 1. 3

i� � /J' �r Pl]ft J: [ o, 1 J ---) JR,�?" c, 1 r;, 1,. t�i�' �/5.ta. '

gcxJ: _f'f ltJdt ( Ot X ! 1 )

;t �! �, t.& 1A. �t-JA. 1 a r,,�� t ,�, /J'',J; /fib Al,·:.'- -6.

rr f',..'1t"° < M ·Q" . .to. a', t 1: 1 Jt. ,,r: ---�

I j 3 (xJ - frrtJ.U I < (:;2)! x ••2

�����·�--7"-·----�----�---���_; 1:Jiii\'"> J.t1l.1l�1t7 1 ,1"21<1 �ttJ'··,-it·!t.'tt�.

\ 17iJ tin I Ji 1�-1v J j 1,� t( 1d' d) 1 ·•

) te\tt = xex -e+c �J:i 1:,r1.Jt"-'M0ail �i)J--�;J, ct-a 0ta, -�i,�1c1?; 1 .. 'd�".

(it{�' Galois �b� ? �� 1�11. i j,0, ? )

§11.8 (1fl>1C)

, �ttf' 1o't"())}Wi ,�? 1 1 7/'J .filttJt1�ii' ( tt.tqJJ[�'{ij/l X�(t)>

i.=L··�h

l J4J " 7 rI1 �'J;i? ,t 1:: /,s ,*v· E /ivJ 'J ii. i 1 #J , l � � 5l. ) 'J· c··, -1!% c-h l 1 • a.. '

. .4Ptl ff�� ti. iV. (�;II ( l<L i /l, 7,=f(J,?., tii / � 1tJt t: Tu·{� J

�·�� 111 (.\I�'

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