[aumericol methods-~ 1 | Sduton of Algebraic and Trameend Equations. ae Given o equation flaj=o it ix posrible to find root, ln) becomes Ze exactly numerical methods + root of the : genecally not % auch that we dacus too find an approximate veal given equation. Equations jnvolvia algebraic quantity fees x,2,x3 ete. are Called a | polynomial @ algebrate equations. fa! (i) -4a- 9 =0 di) at +272 Bp ai ra 43athe=d. no- algebraic Equato ns that — involves Sinz , tan ete. quantities like ee bog, ore called aK eee equactions i) 1&-2=0 Gi) zlog, 1-120 (li) Fanx =24 numerical method, of Finding an) _appronimate root of the given equation o. repetitive type of procexd jenoe” iterodion Proven’, Ex 4AX ep. there 1 exits, hoo. value, Bey es #0) hay Op poaite #@=0, HO>0 then — ther alway, Sone Boxes one real ee Feet in the interval (a,4) Regul elk Pectt ial @ ioe ood ae falae poxth'on, the formula © approumoation 1 Ags , AOL tO) Fadi ese first i OM theese, » tee e yng 1N COLE) iA - cay Follows. Bea 26545) eb Rho) ORE the = precess 1k continued HII comuistency IETS. Cannenutive APPCOxt mation). foot ee eee Shere f9x-S26 > Cowect by) +thred Werinal a fe folai method. bet Mig) = 2234-5 °°" BHo)= 5 94-6, ead=+1 <0 HD =16>0 root “Wies in 023) Pt be observed that the of ¢@ of az=2 being -) i asm Wwe Gee a realcomparecl 0 £(3) = Ip Nearer +o Ze the foot iO the & we @ expect neighbourhood one, ee aball have the ateral (Bb) for applying the Method wach thot aCe) lisa armal enough Ov £(2-D= (2) —2(2.4) <5 = 0061 >0 Shan iteotectiengta 3(2: 2.1) a= sb pe) = epee be i £(b) eiCay sto: 66) Firat approximation , ee ad(b)- b Fla) Bas $(»)-F(2) 4, «CREST aa ee D £(e2.0942) = (2 094 8) i. a % = -0.003892 20 the ot fiex in (2.0942, 2. 1) ~G= 2.0942, f(a) = - 0.00392 be #1 , #(b) = 0-06/ zsLL A eustituting Ip viothe. Rue it @ we obtain the . second approximacion, = (20942) (0.061) — (2.1) 0.0039 2: ee ¢ 2.09455 9.061 - (-0.0049 2) % ond % Are Close. enough ; Xe in oe bette © OP Promotion than %,. arus the required approximate foot correct +> 3 decimal places is 2.096 2 compute the teal’ ¢vost © ef ace log x - 1.250 corcect 40 four decimals by applying Reduta - Fal snefhod . 4 ete eet) = x log 2 = 12 FI) = 2) 02) = 01640, $(a)= 0:23 >0 the feat root ren in G27 3) F(2-4) = 24 log | 2.¢- h2"= —-Ol0as3. £(2.8) = 2.8 aks 19, BS the ‘recta lex in (2.4. 2-8) Heration: 0029. > #(0.)= -0.0353 b=3.8 f(b) = 0-052 = oO F(b)- bfa) #(b)-$(a).. aefl) bf la) er f(b) - F(a) | mt, 0.60F a £(x = § (0.604) = 0.000386 ro Peg root | exw in =6(( 0-607, 0.4) fe Ber o> O-c0t += 0.00026 = Ot He) = “043352 ier aE eton - Raphson methad | Tn this method we lecate an approximate real foot es of the given | eguecion and improve its Accuracy | by etn iteratt ve process, to the roof The Fiat approximation / Xp is given by the opera formula. G= x, — TU) Plt) he = Second. approu mation } prpaes Peleg % tin the ens of ths ik obtained Ks continued = fill, we too consec ufive EEEeval a teal ooh of | cthe Equation 7 a3 ete cig. carmaeueee™ Ahee daecimat = Raphion, by. oreiynd newton, ar places , «interval (a, b)jn Again Ste FO = ep - fA) £'0n,) f'Q\) ee?) [Cony zn) SIO a sake [3@.1% 2) gimilarly % = 20146 ~ [@.0%46)" pee SO aeales 5]. 2.0946 (3(2.0946)— 4] Thus the required. approximate foot CorrecE = “ts B decimal places i, 2.095 2) Find a al ‘root Ber jhe equator ai+a?+ 3x44 20 by perforin +is0 iteratioms, uslu newton®- Raphston method sf Let fs 24a sry flo)=4 | (=! , fC9=-6 <0 “a feal root lies i G,-) and let x=-1 fi) = 3x 7p2x 43, — $e) gre) = Sete 4'Ge) FC) 2 ed pee ss e 1 rm 2S iteration: § %, = % ¢ ation ale) 125 fGise ce i 3224 ~ ga. 25) rl.223#'(6. 8528) My 20-8526. Ly = 0-826. hse): Fea tre real foot of the equation a ye -2 =O corre ct +o three decimal ees, a. Fpiging: Seeton: 5 eal method. ¢ fee #0) = 2e™-2 £0) =r 20, #0) = 0.4183 70 Bie. tootiliearins (0,1) % let %oms pi(a) = re‘te* = et (241) ie tee £(%) = iipaate |) St 61783 ee pe =0.867 £'(%) £10) e'(2) a | | = 0.3679- £(0-864 : oes LOSERS iro 39-[0-86796 2] fi(o-8679) Seer aT e 6 4699 41) L, = 0.8528 ee 0.38522 — £(0.8523)is . of the equat, FN Pisa tneverceaiy toate 0 on OSINA+t CoRR HO of HCL) = HK SINX + COAX.’ #1) = Neoaxn t+ sinz -Sinx = XCOAX tee aem in radians, X= x, = Pl%) » gam Fin) ¥(%) '(n) = W-(NSINA +conn) TcoAn = aos) = 95a x a 2.3933 28922 {2 8233 S10 (2.8223) + 04 (2.8233)] enue ice 8°“ 2: 823 3c0s(2. 8243) Ar = 2.7786. we get mn = 2.7994 ; ae TOUGH “foot of 22 Uatag Newton ~ Raphson method X= Yoo => vo oo Coy 220 a5 so/”-, 2,465 4.zz. Affe cory eA, : | Conider a function ye fix) Beet Xe, 2, = XK th 1 y= Xth ar. In=Xi.th “be ahet of oo REE poin&A ak q common interval h. Let the Correspondin | voluew of a Ce) ie, Yo= £(x.), Y= #(x) 40D. --. y= £(%) we define forward and backward. differences Forward differences the eee ae difference of f(x) peel bert la dehmed ay follows. ATG) = £ (th)- #0) AR 1% colled the forward — difference oO arm a on hove for the values 1,4, gee RFs) = $ (44h) 40) @ , Ae uo io. BS (0) = Barth) =F) ot Ag= ys-y Af (a2) = #4 Qath) - £(%) Q BPP Aetna AiFerence gf thet Fiat! forarel ore called.y= Ay,- Ay, ary Stay, - ay, ete Por wor di Sference | table pasfiy [oy J La Xs | do a dh ay | Dy n Xs = Js ; | Ay, orhet MEIKY arenttien samy. | table MOI ARAN ARNG ace called the acon forward ol ferences Backward, — ditferences the firat backword difference of $0) denoted by vf) in defined | on Follows !Fie £(2)- $(4-h) called : ke backia rol Oi ference operator. st Za%n VFA) = $x) = $Cancth) a da- day BF he%p, ond Fy \0 ye mete C= FG Fie a) @ eon dn- SAS? mh Xn and FOGS) Pda BR to AT F(R.) = Fin) — 8%, ,) @ Bei, ye) ae etc eetst). 2 vy, = Wey, difference of the Firat = backward Neer are. Known aS Lecond backward ences . they are ay Follows 9 Von- Vigie , Wyn rs nena Me4a | entice, in i fable nornely the last TWYn, Vda 0 yy WY, Ole called — the aera backward = bebo grins Construct. OL finite. difgerence | table for the function fG@)= x%+24! where %~ tokes the value& 0,1.2.5, S. 6. _ and Volertify the leading £0 nage and backwatd alifferences,S|... Se eee a Fo Beye CERT by data prom thin we obtain Blo 1, PCS Se Kee ses) fl > 69 Ps) 213) (6) #223 the fintte difference toble iA a follows Me Oar O8O 2 70.8 h 0.05 from the table Vy, = 0.0540, VY, = 0.001% %, Vy, =0-000n vy, = 0.0002 on substitution in the — formula. | we get B(0-26) = ©. 26602.” fr aya? a ¥ “ha inle26 = Basa) S| Given Sin 4S* = 0.7071, sin SO= 0.7660 | Sin $5°=0-8192, Sin 60+ 0.8660, find Sin St using an appoumabioate interpolation formula cape have do find the value of * ae ‘ ° : : Aine Bae sinn at %=52° which is ea po ee 60° and hente Nexston4 interpolation fomula i é table “ (orear. Oppropriate. The diterence as Follorat.eS a et . | ns | 0.4041 | i | | 0.0539 | | on eres | -0-0057) | | 00532 poeue | O.oh6S | | ~ & 0. 8660) | Inz Co 9, © | | ) } | | Here De eae oe = Sees — One | i s | Substituting in the formula a in (S49) 2 0.8387 2 sing? 6, Find ‘the interpola tig polynomial F(a) Aotigghed Flo)=0 , f2)=4 (hk) = 56, f(6)= 204 £(8)2446, #(10)2980 and hence find f@), #(5) and £G) a the inte rpolati ng polynomiad can be found from either of the two interpolation Pain we shall use Newtons ; forooth interpolation — formula. ' the differnce toble is aw follows. || 3| From the follavoing table efind the é i number of students who have obtained (@ lew than bS marks (b) between ho Ausmarcy r 2 ae Marks | | 30=Ko | 40-90 | 50-60 | eo-F0 mae N' o.0F Students) 31 | go | 5; 35 BI | Ee we = ahall recontitute the given table with #2) representing the number of Students ley than xa marks. That 4, les than 4o marks 31 Students, less = than 5 markKh 31t425 43° Students los than 60 Marks F3+51= 124 Students les than 46 mons Mbrse.169 Students les than 80 marks, 1644317 Ito Student we have the new table ei with a forward differences. ej oy AY BY as| ee eek POR EG OPP ying ¥ Neroton'a forioata interpol ation — formu, HS-HO - 0.8 rere ee oe ; 10 on substitution ia +he 4 (45) = 44.86 2 4B shus the los than 4S Marks \ forrnu! & data 1 Cig & ; Buk £40) ==! by Hence: 74S) FCS) = es Blais sthus the number of Students Scoring between 4o and ks th IA, Exercise problems, UAE on appro priate interpol ation formula 40 est mote g Berta) for ahe given value of 2. : aT — a a 1.3 | a) Bef eet 2, 2 | 6,050|6. 686 | 4.389 | 8.166 | 4.029 (185) =) number of students obtaint,, iq AB ‘ need toh toe (45 )-F (4c) 5 24Interpol ation formulae -for uneégual intervals Divided — differences Let # Oe), FOU), #02) ,.., # (xn) be the values of an unknow ry funchion ye fx) correspon oo te THE valused Grice Tee ee. My GE UNnegual tatervalk the fimé divided differences ace defined as follows # (4%) = £O)- #0) # G,,2)= $GD-#H) X= %e BGeX, eke the second order divided difference are defined ar follow’. F (oj othy Ma = ees ale #n, 25, %) = OAI= £00.88) X32; similarly the ef@ other higher order divided differences are defined . the tolular arrangement of these — values ip called the divided difference | table and Ww an follovo.s. i2 [Se60[ Foo Te sd LU] po] Xo | £(%.)| i | | | #(%e,%) ce | ect) | # (2,42) Xr) FO) +#(%r, 4%) ee =] $06, %i-2} #0n-2) 2 Cys ,Xa-s) Fh =3, Ans 4ng) 1, Xpy-ota) tea? £n-) | #0ta 3.400, %n) | | | 00) ae =: | | SPMD A= aoe ee! | pacton's 29 difference Bhi ; | newton’, general interpolation formula, oe Gp FC), FA), £03). FOr) be a er er vers ot a unknown function 40) Corresponding fo the values of % tt, ioe a OE unequal intervals, then Xs, Xa, -- fi $0) = # Go) + (1-%.) $ (%,%)) + (X-%o) (x-21) Fo, %, Xa) . bet) Ct) Rag) F (ort, 22%)ahd D.D | 3d D.D [a x | 4 | ip.p | Loe 2 | F(%e) #0) £22) | Sein | pe $ (oi, 22 ioo-43 ,14| ye N= eres eal | ee ee ed (0, %, 23,25) $(%y, %2) pea ob | 198-96. 109, } fer? SEER Og (06; 2, X3) ‘ | isy-too 224 po |For, \7 6h 2 (My 12,%3,%4) | [Os a gaa se 2? aie | as0-i96. eu B-4 | | | 7 4es ae Xu) ; q -15! PaaS, “tan oa oy | $(%3,24) A412, 43 74,45) 1 B68-3Sp = 254 ' bs -35 25 ; | B-6 wee | £3, %4 Xs) es Wy 8 | Flay)2868 4349-254. us lo-é | £(Ay As) | | 136-8684 4 | | | lo-@ 110 #1Z)AFracy eer ee formed Firat Lx [osteo] st o.p is p.0o| tp. o ||%=0 |f{xe)=-5) (te, X1) || | | | -lk- Gs) =-9 | | 1-0 P(%, 21 ,X2) X21 | POR)=y| FE 2 (00,208.25) j # (2%), 22) Be-F)_ . | j-12S-CW) = eo eer | i $ (0% 4) |) = | Ga) 25) 26-C39,.g) ~ | (4, %D eA F337, 1,)| Bigtieie op a4 oq / | + en a2 | j | ~ || $a, Zs, %y) lo-] | %,=8) tlxs)--21 ' oe. | + (4s, 4) oe io- | 255 -(Daoh ? ’ | . x p10. = 3 ‘ | xe10 | Fax)-355| £ | : Hes TN f the — fourth order differences are zero since the third = order differences or AamMe. on AUbAT HHO in Newton's ee in-ter polation formulQ . a ie eye) (+)? S fp = oxiaattex—s #()= 2 (2-12 (o)* 460) —5 = ~hee- Had GA a polynomiar in x for the followin data wring Vests divided difference formula. 1 # | ey i = | Ca eerae s | ‘ (248 Poll 6 | 9.1335 | sf The divided Odfierence tate cts formed frat. bc y =£) i p.p ao = ad Ae ] “ugh a | F+te)=1248 tte, 24) | oe = B3A2uS | | rt tals | i £(u)233 Oa | elt tat) | (4%) | (02 Sete) | oe ork © £ Gea 10%) aT oe | 134% 23] 2428 =I peese #2) 25 al (4%, 3%) #04, %5) | BB+lo =13) | sa Ko SH Lo H%)24 ae 4 (Xs, %4) | 4G) | |Miertumeieak Sies eae Steg f a mn let ar fia the intervals Ta, iy divided into n equal parts of width = h_ where, h= bra. n VE OF TS th aS th a) = zh as belt ther polats lief -the division Alo , let Yo? Fle), Ye FO) =2 Yo = FG) be the Cores ponding values of y= #00. Now we have a xzet of values of Ye tO,.ot _ equidistant pointy of % Pitta. woleeie Oe) ae. obuleted. pia t= a.| %, x2 | ig EE | ae8 | Pp ge . eee he. eS + orking “procedure or obi rep 1) \ aG@iven re definite - iotegrad ee Si da for evaluation fiat divide interval [a,b] ioto appropri adeeS Trape 20id | as [(Y4u 4) + eS R evaluate e t( i rule. given 2.81 Here Pate Alko 4, = 2-105 Hence sae 3 [ 4) dx= 1 hs Evaluate yale. Here Step [ ength tere a ule 209) +454 Ie Le csi) yl 1) de ite the tapeasida ch eet, fee boul & £.859| F481) 4.464] =I h eel Ne Diy De 2 seer = 4,464. Yo trape soi deal cule dvs oe a ad ee [Rest 4, 46a) + 2%(2-808+ 3.614) RCO 8S 4.57) ] waiing the trape.svidal 6 ae I+a > oO c en f(x) =a kK) iEmploying 4cthe drapedoidal mie, | “Evaluate ( setmotde oy avi clin a | she interval (05, 2a) to fen eq a parts ¥ Et Here the interval » [0, 2 | = [0,40'] t best to divided info §=6 ten equal Cub -iatervals. Accordingly, we take he (Yo) CR cme eee ete Ev aluake P(x SINX, the values ef are xzhown in the table. I teller sy i —— | [o [4 [Is [2a] 36 | est S62 | 2} 81"! | 90") 0.9571 |0.284 9 |-000 ae | Y| © |o.se o.209]ousy 0-5876|0-70n1 | 0-864] 0-891 the trapegcidal ule now gives. Pet Oe | ‘ : b | f Sind Ave 2 [Wor 4.) + 2 (44.444 ey, | é 444 495+4,] | 2 [o+ ( + 20 1000) 4 20.1564, + 01304 + 0-454 4 0.58484 040741 + 0.869 4 0.841 Foss + 0.4994 |[V008 4 OC xs.863)] = (0-219 655) x | = Ones (takin Roe ie | d a | wotet the , Stack value of the given integ cod 1A CORO =. | | = as : poe wees a bo 5 uAerg sthe Arde dal rule Evaluate ¢ ae by uous | the ‘ene [o, | tato io equal parts . Compare with the exact yesult. s we devide ae interval Co, J ite ten equal = parts by aki “9 ho. 1 We compute the Toh as ot ye fas et for %.=0 = 1 © [o1 ]o2/[o3 [oy] Oe) og o-7 0.8 0.4 |bbe FoUd 0-3 0-24.08 | 0.6 F054 0-0 0. [cea Bite once) Se de- Fh [iy.4 40) TAG a4 ast y+ Ye + Y, , + 4x44, t4e) = 0. 632635 The exact Value ee. [ee eB (= 0.3649 = O63212, _d now, we have the air oO ase d Peta Si Mpaon'a yo e ‘a | We | Che ay | Ay Lo | A | | Uri ete z | rule for ney in gin Py i) fae by of dx = BL (Yetyy) +4 (94 9s) + 24a] + Stee = Ae [eingy rs (He) 2G] Ap deduce She ‘wolue of 7 =f ban a per BARD: Pe) aS we must have, eg @ 124 (0.7854) — Evaluate SR taking seven dvtdinates by ‘applying simpsons ug rule. Hence deduce an appropriate yolue ot § log, 2. :Sa era % cule 1 evaluate ( ax : 5 { To apply _simpson'y ee rule, 4 1 must be 'a multiple of 3 and we Ahall take nes itael f poe wenn bee . 2uNE ie | 22 hy dee Sh (ey) datos { o Paton! | tlt ail Sie ania! yee ™| 27183 |. 6484 | 3956 1-28 40 Be a a lave thn He er |e oe substituting these values in the with het, tule 3 De 3 [6.4133 +1.2840) + 3 6u87+ 12958)! 5 : =e 70023 + 4,132 ewe? i [E= 4.9259|| eee a =a obtain &\ use Rrepacns “gh pole AS ; | the Gpprouimake value of firs yds | by seat 3 equal parts Ierterva la, gat Here he 6-2 = 0-3-0 =0.) AI. Se = | WHS oe Slee ees Lae =(i- 3874 4 i 0.9645, 0.8854 | ; L 3 ge [ake | ‘ . . \ 3th rule substihing in simoeaons ¥% for n=3. G.4al> | “Vu-sy* 2 3 = D[ (I+. 3) ¢3(0-996+0.96%0) “= +2fer38s4)| oa [! 3.43 (1.4635)]+ rlo-80%4)] iW use — Simpaon’a Peele to find es z al fat Ab: by taking é eb: inte”! 0