Unit I. 5 (rANSCEnds 8 84 ty \ the fy 4 Witter, Nhe i} gotttion +? Erroyy ‘oo major ¢ 1 ihre major CatexOring “meri, ' 1) Inherent erroy . ay for Vos jata, for Which the po 8® err the inherent error person TH ape, ihe statement of the ca be oblem Def (2) Round off error procedure. While all C8 erry, inconvenient to wor TE with approximations to an all de approximations lead tg litate. off errors. The round off tier ¢ f added to‘ the finite , ese é epresentati SWantity a ion of a ¢ make it the true Tepresentation of the ; Sonoma 6 DUM ber Example (1) 28. 63243 becomes 28, 63 correctly to four signi 00243. ignificant figures. The mund of when toun (2) 18, 9673 becomes 18. 97 when rounded vif correctly ty fou significant figures. The round off error is - 0027 : (3) Truncation error : These are errors caused approximate formula in computation. For example. we i 3 ty infinite series, ¢= 1+ 7 +7 qt $ Fthe above infinite sens Suppose we find the sum of iit an the first 10 terms- The terms = on the calculation of & This type of | i merical error. rors encountered in nu a, Bxplain any tH tos Ap 98 methods with exam? Solution + See question 1.2 516 NUMERICAL METHODS IN SCIENCE AND ENGG, “s fit) =f lng) +(e —X9) F (ys Xy) + Ot Xp) (2 = AyIf Cy , Xp, 49) + CD) Here xp = 141 = 2, 22 = 43 f (tg) = 43 f lq, £1) = 1, Fa, 1,42) = All the remaining helper order differences are zero. Substituting these values in (1), fix)=4+(8-1)4 &-1)@-2)1 9. State Lagrange’s interpolation formula (Ap 95 Bharathidasan Uty ; Oct 97 B. Tech M.LU. ; Oct 95 WLU.) Solution : Let »=f(x) be a function which takes the values Yo sR se 3p, Corresponding to x=2y , 27, .--%p "s interpolation formula is (x —24) (2-29) «.. l2— 2) .) = ——<—— rade ‘ (xq — Xq) (tq —Xg) .... (39 — Xp) 78 (ty —%q) (4 - r (2 —Xy) IX — x4) 2. (2 —Xy 1) pe Oy &,—%5) &,—*) - @,—*, 9) 2" 10. What is the Lagrange’s formula to find y, if three sets of values (xp, yo) ,(xy 471) and (x2 ,y2) are given? (Oct 97 MU.) Solution : Pe (may) ap) ea) = H2) Gp—%y) 32) 2°* Gy xq) =) 7 x= a5) =x) * Gey a0) ==)7 pi > IN SCIENC. ICAL METHODS IN SCIENCE AND i 50 NUMER Eng, quation by elimings; =a: 2 +b: 8%, (Noy from the relation yx 48. Form the difference © aandb M.K.U, ; Nov 97 M.S.Uty) Same as worked example 2 on page $12 of the main tox ‘am (1) gtly py. get? Sey EOS =2'a +363"... (2) x Solution : Given y,=a@-2'+6°3° and ¥ =4a2°+9b3"....(3) We can eliminate the quantities a 2°, 63° from (1), (2) andj, | by using determinates. The result is " |sre 49 Jx+1 2 3) =0 % 1 ie. 4 2(2-3)—yy4y (4-9) + yy (12-18) =0 10. ~Jx42+5 yy 41-69, =0 OF Yx+ 25941 #6 y= 0 49. Form difference equation eliminating A and B in Jn=A 2"+B-5", (Oct 95 MU) Solution : Given My = As 2" 4B. 50 (1) Na 1 AQ™ ly Bare =2A2"4525" (9) and y, ,y=A-2"*2) peneo F4AQ"425R5" 1g) We can liminate the ; and (3) by using detersinenee na and B “8? from (1), ©) ° 3 : result is Yneg 4.25 a Yn+1 2 5leg w. Faaf sh ree Se wet yale { prianesntlt At ‘ pgroninniatien af the I pwr ills peut We use the a : - ya (dy fig at , h ide mn , , Wy ‘ Then Ved v, hf vy) {fe renee appron : ve the PP di , We can alse UP yo Le fay WD 1 eh fey? wad Then ¥) 2) ise the contral diffe approx We can also 1 ence app yay Mad =f (4 I) i A Then yy.1=Ij-1 +h FG) wen) If we replace a by the difference the method of obtaining ¥j is this method, the number of i of y is only one. If we use the difference intervals separating the two This method of solut method.ND ENGG. ticular ease order! ular case of predictor al value they do nat penis tes 1 the step size ations of the information r points and ood estimate it methods j., Nov 97 ie conditio® we have J gictor formula, jrathiar Uty) @ & A - NUMERICAL So, i "AL SOLUTION OF ¢ Z 1 = a ITAL DIFFERENTIAL Bountions a ™ ng the formula (1), we ean calculate ¥4 ,¥2 5% late v4.2 0¥a+ snowing th va : jnowing the value of yo, This is called an explicit me e have the thod. To solye the same di ed i , proved Euler method aco int aan ver 1=VWAG [Fl y+ FO Feed] veel 2) In (2), there is a difficultly to caleulate y+ 1 S* the 3 of the equation. We therefore n implicit value of y;41 Occurs on both sixe: cannot solve for ¥;41 directly. This is known 35 al method. Note : Implicit and explicit methods will also arise in numerical solution of partial differential equations. 42. State the difference between single step and multistep methods used to solve ordinary differential ically. (Ap 96 M.U.). ly the equations numeri to solve numerical in the interv al : Suppose we want nber of Solution + dy _ % Y fe, y)giveny Xo) erval into a finite nur! differential equation We divide this ints [xp - 8) of x. subintervals by the points ei eearee (tO< such that ay aja =1.2.0™) Th x=suti? FAO sPre Nd : etermine @ number 3, which is an ical methods, We ge , : ae to the value of the solution y@) at the point x, yy is the numerical approximation : Pap numbers Oy al equatios-Solution : Runge-Kutta methods » siete jiowing formulas is 4 Kutta formulas of the second , b) Picard’. cs modi ormula dd) Milne's » (Nov 95 formula is @ si Runge-Kutta methods and p thods for solution of 5, Nov 97 Bharathidasan Upy, Compar Cerrector me problem. ‘Ap 9QA NUMERICAL SOL ingen ty Nu sol WF shift fh ” PARTIAL MRK IAA, Iwan = cy Wa te the bhai | ‘To uae Aduan's ietHead, we wearl nt beat waates Of y peter tothe required uatie ot y (hp 9% Uhunaihidannn iy Solution + Muay Wh. Bay Trae on Malia!» Mile’ vmethond ii ee well ate bing ivettont (Nov 04 Hhuruthidowwn Clx Solution : The statement in aloo U6, State Milne’s predictor and currector formulae We solve y= f(a 95 y (ty) Yye atky 1) = Yer WR eBhin ve y ligt BAy= yy ab are ag 14h, (Ap a MUO Solution 1 Milie's predictor formulit b anh news (ay v0 84 |! ue yi where elie holween x aig: Milne's eurroetor formula ti h fay wsvorg (vor aries) yg! ti where € Hien between ag and ny 1. What will you do, If there bea connidenuble difference i ined value and corredled value, 1 pretlictor corrector methods? iton ¢ If there is a coraideralle difference betwnen -eornooted value, We Lake the corrected wale and find! out the new corrected: value Jill there i no growl difference betweonWenn AL METHODS TS SCIENCE Arty nt ‘ 2 n® 4 ody & +e i ono J Gao" tee Nt i lnteron oy ody yy y whee ie ae 2 Stete Milnes canreetor formula, (eg v7 May, i iM Selution : Ser quest 2 20) How mony prior values re required le 7 nae vaies in Milne's method? Prodi iy. Solution © Four prive valies W) Piek out the correct answer ; The error teem in Milne's Pretictor Formula ts W Sie! ih 19h a) " A Wh iq Ay a ag Of 4) b Solution = The error term in tat, SU. What is the error term in Mil Solution : The error tern ie 32. Write AN (Oet 96 MU), Bharathia; Solution t; WAIT AMD RENTIAL EQUATION What is a predictor-corrector method of sealving ® differential equation? (Oct 97 MUL. lution = Predictor-correetor require the values of y at x, ,% hy methods are methe 14 Rage ne Ot value of y at x, , 4. We firet use a formula te find y at x, ) and this is known aa a predictor formula of ¥ so vot is improved or corrected by another fortress © as corrector formula. 26. Say “True or False’ : dy Euler's formulas for the solution of a =f %.3 provide a pair of predictor corrector formulas (Nev 95 Bharathidasan Uty) Solution : The statement is true. 27. Exhibit the Euler formulas for the solution of a =fix,)) a3 a-pair of predictor corrector farmulas- Solution : Euler's formula is yin eset fea yt=t a eee) Improved Euler's formula i= ' hb aa yer evir g | fe ths 1 HV | () is the predictor and (2) is the corrector. 28. Write Milne’s Pp’ ictor corrector formula. (Nov 94 Part time B-E. ; ™ ’s.Uty ; Ap 96, Oct 96 M.U.; Ap 97 Bharathidasan Uty). Solution : Milne’s predictor formula is where € lies between Xq-gandxy sr Milne’s corrector formula is4 \) METHODS IN SCIENCE AND ENGG ( G ‘ ANOMER! # Ay vain the second interval is computed in ane the same tour formulas, tiling tha Values + of ag Mp respectively ne place of u Fil up che blank : x svolve the computation of Fe. Fe : a _. (Ap 96 Bharathidasa, Uty Selution : Caleulation of higher order derivatives of fix,yy, » State the special advantage of Runge-Kutta Method over Taylor series method. (Nov 95, Ap 97 Bharathidasan Uty ; Ap 96, Oct 96 M.UL). Solution : Runge-Kutta methods do not require prioz ot higher derivatives of y (x), as the Taylor method the differential equations a using in Applications are sted. the calculation of derivatives may be difficult the Runge-Kutta formulas involve the computation of various positions, instead of derivatives anc this ction occurs in the given equation, 21. Is Euler’s modified formula, a particular case of second order Runge-Kutta method? (Oct 97 M.U.) Solution : Yes, Euler's modified fo second order Runge-Kutta method, 22. Say ‘True or False’ : rmula is a particular case of Modified Euler's method ig the Runge-Kutta method of second order, (Ap 97 Bharathidasan Uty) Solution : The statement is true. 23. Compare and cont Runge-Ku differen rast Taylor series method am! {ta method in solving a first order ordina”Y tial equation numerically, (Oct 95 M.U.): Solution + Refer question 20, 2 24. Which is better-Taylor’s method or RK. method why? (Oct 95 M.U,) ; Ds Solutian « ‘Rafise ascectl. _——— TT wit pill Alt thi vy wnothiodd tie Chel hee ayy y a ght Mutorertlal Hq UAL LA, ‘Ay a situate anette iy Mt wi sanadiann A a EH peiy) two HE 4 tlie js cttmivontial eqniallon WP Oy, yp, ‘od iy Win the Walia niet tial te offense ht Vy og OP ” uy iti he written y Mire can alii viv thy norte Joe foo Hh State modified Kuler algoriiiim to salve VFO volvo tht ig th Oot 0 MAU) Solution: When vs iy tA, toby ee yy h Hee Thom yy yy t a fing wady F099 Me Fill up the blank The improved Buloy method in based on the avercigen (Nov 97 MAK,U) ti ith Solution + Slopes 15. Till up the blank ; The mottified Ruler method ig bared on the average of Solution ; Points, 16. Write the Runge. order , “Kutta al cond solving y « piy Ys yay) a "— : (Oot 4 Ny as Tech, M.U) onote the intery Vithioy are ( "T betwoon I) foranutgy 2% tte frat increment in ¥a ani Then 2 Then 2) = age hwy eye Ay wom a The merement is + eo valae™ T Manner weg the seoond interval expat 2 th : e samie tires formulas. asthe uy y we > . : Re PACS of xy wy Tespectiwedy “oo 17. State the third erder RK, method algortihm fie Linterentiol the numerical solution of the fist order + equation. (Nov 94 Part time KE. MS.UG?: Solution : To solve the ferential equate * fe third erder RK. methad, we use the tollowmg aigormt het bh =kfie.y) = &, } tes fjxt8iy+2 , 2 z kg th fixe hye Dhy— hy) and avedty +4828) 1S. Write,down the Runge-Kutta formula of fourth order to solve St a fr.s) with wine) = Sa (Ap 96 M.U. ; New 35 MSU: Ap 95 Part time BE, MS UTy. ; Nev 97 M.K.UO Solution : Let k denote the interval between equidistant vahies of x. If the initial values aTe (%).Ng) the first increment mt is computed from the formulas. ky HAS %o.20) : h & ) Ry= hf] 2s +98" 3 | ‘ ky) aynhe| sot 30707 : kg=h Fixgth yet a) and ay= Jer t2has 2b reea MU Net MTA, tlh ltr Chee nk iy. atity “Wily ‘he ay is aa Win i THe Bute) "Wty 4, rele cbifferentiad equa ‘Ap iy HOD tas Wty) Solution i yi od Yn VK ft, Ymiin oh) 3.4 Where the VON ity Hential eeopunsdesa ja af the sa Given ¥ «xy, 944) “Vind yO 1) hy Butera metho (Ap 96 Mig) Solution ; Por the Alporuthiny in Yen fia ! Z Euler Aifferertial + epaation fi . | - Mn ¥ Aft, Ym!) 4 Here y et+y. Bo fa, yy rey Initial condition is x ~O,yelie Take he 1 * Ing m= 0,9) = ¥94 Wf leg. vo) From (1), taking SP ia el+ 1+) a1-} § = State aiiae on Belew sg vmall, the method ix too sh 10. dd ph 1s nit ; In Euler's method e ate inaccurate = ty nd. if h is large, 8 (Ap ty a y= 9, 5law VN De solution af dhediasen amet armenia! EETev Ad) Dagan td darter — ABE SBR ABN ot Ate anniedra tone Het! peastbal 1 Whata mt Tee wnentiied aetad tenn shite | went annie Consider the first onder differential equation oY Pov gde iba the exact solution vue) of the differential equation ia Che eee age Sx, be represented by the fill Hine ahower ay the eltaarane We divide the range «,, — 2 inte a number of stopmizes of weqntaal length. Ler Sea a. Kya At eWanowdn, Bre Che varia step Joeations. Fer each xj, approximate valuen of the dependent variable y() are calenlated using a suitably reeursive foervtatoa These values. are Yoo ¥1yd2 een Noy and these are shown by F ‘Let the boundary condition be y (vo) = vo Computation rox value yy ce My be known as numerical ae be for the first orde ; "Part time ILE. MESUty) solution af the equationpods IN ancap Mere NI ERI \ 3 h he ye AP oy : io ‘aie Ht ¥ 1 # =“ of ¥ wit x at the poing Jonotes th th qorivative denote s tin) sy gg » Taylor method ats 4 jemerits of the - 3, Write the ae oe dine 3.6. M.S.Uty $ Oct 96 MU) ion, (NOY , ae main text book. f the Taylor series method?, Bharathidasan Uty) 1 text book, solu fer to sy oon page 341.0 sadvant M.U.) (AP 96 341 of the mait solution + Re : i F n ga. What is the di age I (Oct 96, Oot v7 Solution + Refer to $7 on page 4. Fill in the blank = s method will’ BE Very useful to give some ful ngnerical methods such as Runge Kutta = method etc. (Ap 95 Bharathidasan Uty). Solution ¢ initial starting values. 5. Fill up the blank : The use solve numerically differential _ts restricted by the labour involved in the ie tion of _____» (Ap 96 Bharathidasan Uty). Solution : higher order derivatives. 6. Write down the fourth order Tayl i (Noy 97 M.S.Uty). i al of Taylor sertes t Solution a ee elution $y) 4.4 Ym th Ym + Fy nt ™ tere y", denotes the r'” derivative of y warct. x at the poiltof 4 ANT VIM Witit wneegntad, 1EBEGAE mn ; ‘ 4 PUES, Bary Mitte nappa megirA tiered 4 Melee ay “Hera Oo, pi Wn. Ce oe Vel eObberee Mie peroy : 7 CO) OF inn ee cine ben weeniitet OH yan ey Ihe Guy ' NY son de Fo ey ed Te Bi me tie Hikh= mags fas,5 oy hve MUO ia iW ee WL HP up the blank af qualiar Uhre woh Ntion of the difference ‘ Psa ‘eae ity). ME Hy 0 4 tty, melt tAp. Khurathidasan Uty Solution © Whe piven ena con hae verter a UE - 2B 4 tu, bd The ataib e ¥ tquation in We - 2H +10 he. = 17 © O5(R Wie aye Ba 1,1 So the solution is ie fey eegny lV" = eye eqn 59. Find the solution of the difference equation 4yn,2-4u4,,,+4u, 2". (Nov 97 Bharathidasan Uty). Solution : The piven equation can be written as (EB? -4 E+ 4), =2" The complementary funetion U,, is (E*-4£+4)U,=0 found such that The auxiliary equation is B48 + 4-0 ie. (E=2)7 30. E=2,2 2 Uy = (ey enn) 2" The particular integral v, is (B°-4 EB +4) 0, =2" 1 Pu aesa found such thatMERICAL merHops IN SCENT ae ENGg NUMEBICA” 1 534 : 1 ; 2M (B-2)° — gt =a gn(n-1) os Ph Q latent 6-1] 2 (E-a) -9" Hencg the general solution is Uy Un +p 2" ?a@a)) =(cy+egn) 2r+ ay 2 60. Firid the particular integral of (E"-7E+12), yn (Nov 97 M.K.U,). Solution : The particular integral Uy is found such that (E?-7 B+ 12) u, =2" Lia -7TE+12 ea gn) gr gnt 2-1x2412 2 61. Fill up the blank : Sth = The particular intergral of the solution of the difference equation yn49-6y,4,+9y=3" ig (Nov 95 Bharathiar Uty) Solution : The given equation can be written as (E°-6£+9)y, =3" The particular integral u, is found such that (E* -6E+9)u, =3"te vO MIiNAtiy sh *. (Noy a7 ny tent book 2) and (3) ing Aand B in from (1), (2) Wey INT Ry. DVD a Nat ‘ LN Mitte uxegeiay INTERVALS, st Me Yeap i NTE prey FQUATIONS : ¥ on ‘ yao mened “4+ ¥, (20 ~ 80) « O ory va 1804, 66 wYnag=Bpy : _ bo, saat Hy, 0 Form the and § from Oct 96 atts Solution t tifferenoe equation by eliminating the relution Yn a2 he 2)": Oct 95 , SiN Some ate an cy Vabite vel ' Book. « O36 (oop et) and Yeeg=a Eb imeee “AaB sdb (- 2 iy i (uhtities a 2" and 6 (—2y" from (1), (2 Y using determinants, The result is [esq 4! 4 Yui 2 |=0 Me tt We can eliminate the and (3), b UP Sag 2(2+3) Le. dy, po- 16 y, = ory, vocdy, 51. Solve y, , 9- 8yy.1415y,=0 (Nov 94 Part time B.E. ; MS.Uty ) Solution : Refor worked Ex. 3 on page 312 of the main text book, 52. Solve y, ,o-4y,=0. (Ap 95 Part time BEL; MS.Uty) Solution : The given equation can be written as (B°- dy, =0. - The auxiliary equation is E° fai)». es ie(B-2)(E+2)=0 8s -2,2. Ba . So the solution.is 4 =C)-2"+Cy (-2)" where Ce arbitrary constants, :peo IN ATIC a ‘ i ANn aa pcrtlev kWe, ieee My, Ne Rey VOOM capeabiede Can dhe Soltis Whe in Wr tte, 4 wi \ u tho anxitine. equation a” GW Bie hin’ Nh =e Real , (BO a Where Goa tha Ohat RON 1a By 1 rey ra HEU CU EY comtetin be Sa. Dick up the correct answer t The solution of the difference equation UE" — 5 BG) Hy 0 iy Mey enya! WC eeyms" ey Cy, an + (9! Od) Ce eas " a) C) (<2) 4gt= ay" (Ap 95 Bharathidasan Uty) Solution : The auxiliary equation is R®-5 846 0 ie (R-2) (8 -3)20).0.8 Bde Sethe solution ivy, Cy 2" 409 3" or False’ ; The solution of ¥¢ 44 Mea pty =O is 2nx TX sR sin y= Acos ?® (April 95 Bharathidasan Uty ; Ap 96 MU) Solution : Refer worked Ex on page 313 ain text book. Tho statement is correct, ae . eo 56. Fill in the blank ; If ati Pare the roo ts af the auxiliary equation of difference equation, then th Bree ¢ complementary function is —— (Ap 95 Bharathidasan Ut! Solution : : CF. i¢ =r (Cos. x 0+ Cysin x @) where Te eaneea tae 5: oQ & A~ INTERPOLATION w 527 DIVIDED, NUM, a TH UNEQUAL INTERVALS, ' INTE. DIFFE, EQUATIONS This i * known a8 the general Gauss - Lagendre integration formula. For derivation refer to page 304 of the main text book Putting n=1 and omitting g i es in the shite WH g Second and hyper difference’ a , . J+fyda=} ifaysfoshi)=h +9 which is the trapezoidal rule, (For details, refer to page 394)- 37(b). How is Simpson's one third rule a special c@8¢ of Newton Cotes quadrature formula? Solution + See deduction 2 on page 304 of the main text book- 37(c). Establish Simpson's three eighths rule. Solution : Putting n=3 and omitting fourth and hyper differences in the above Newton-cotes formula, we get a+3h J ydx=3h 1+ GA, 3x3 2, 3 3 27 go tao jo 3 3 2,147.1) wan) 14pe-ns fe +g@-) jr 28h (1s38+3E +E fo) =8Ai p+3fe+n+afas2h)+f(ar Shy) =} (nt8n+301% ) Similarly a+6h J ydseSb (urdyst 83u*9) ardh a+bh J ydan 5 (I-24 8% -14 8964 I0 <1) atin- 3A Adding all auch expressions from a toa+nh=, when n is a multiple of 3, we get‘w BIS NUMERICAL METHODS IN SCIENC vGG. nal A [(xy=0 Bp My f&) hth tina. en a a) (xg — 0p) + Ot — Fn) (x =x) 8 ay) oe 4) OO x) (Xo ~*1 07 *n fx) ey ee (py —X) (y — X0) os On Fn v (Refer result of question (2) on page 513) fy ted + * ax) &—Xy) Hq) 70) (xg- £1) »-- @o-%n) ‘ fn) ee * @—a,) &p— 20) 2 Hn Fn - vd Multiplying throughout by (*-*0) (x- x1) -@— Xn) We get (x — 3) (&— 49) (x=*n) __@- 4) e- fa)= ==) GSE ao A yscoovassasnsinotesane SP esunarnensonsqseseenee 2 (x — 29) (% — 41) ++ @~%n- » (Gq —%0) Gn 41) + On Fn - v fp) This is clearly Lagrange’s interpolation formula. 16. Explain the difference between Newton’s divided difference formula and Lagrange’s interpolation formula. (Ap 97 Bharathidasan Uty). Solution : Lagrange’s interpolation formula is merely a variant of Newton's divided difference formula. This is fully proved in question 15. : = 17, What is ‘inverse interpolation’? Solution : Suppose we are given a table of values of x andy. Direct interpolation is the process of finding the values of y corresponding to a value of x, not present in the table. Inverse interpolation is the process of finding the values of * corresponding to a value of y, not present he table.Qk a_ INty DIVIpy: ERPOL jr ED, yy, ATION Ss ) NUM gh Wet “a 18. Give the j M. & Ip DntBtIAL INTEMIAS a Nive, PE RQUATIONS (Nov 97 iM Veree of ty MIATION S.Utyy MRT anKe's interpolation formula. Solution : z=Y-y)iy~yy i Oo-y) Grd IDO -y,). (vo aa ay yi 20) Y= Hg) 1s Fn) YO) 1 = Yn) os 1 —Ynd On =o) Wn =)» On In -1) 19, Wh im has Lagrange’s formula over + Sn = : See question 11. In addition to the advantage men ioned there, Lagrange’s formula can be used for inverse interpolation also, while Newton’s formulae cannot be used. 20. State Newton’s formula to find f’(x) using the forward differences (Oct 95 M.U.). Solution : Let y=f(x) be a function Yor ¥1 In corresponding to %9,%1,---*n Let the values of x be at equidistant interval: taking the values of the independent variable x. s of size h. y 12:3 Then 2p-1 3p" - 6p +2) 3 panes} soe Esher HS x-X9 where P=}, (1) j di chi . (a) gi the value of at any x, Which is a non tabular value. : Pages =0, Then putting p =0 in (D, we have ‘icular, at 7-702 =° pm In particul ; : 2 wv) “peyet| an-Sanr tn | dz j20vox IN SCIENCE AND ENGGIE grt +i re oo OIE ayes * + ya] eahths rule $m, What te the order OF cere’ in Trapezoidal formata ihe Traperondal formula is of the This i= Sunpsan’: Thee Solution ¢ Error in h 49, What is the order of error in Simpson's er OT MAL; Ap 97 Bharathidasan tty) Selution ; Error « the Simpson formula is of the ord 40. State the local errer term in Simpson's rule, (AP 97 Bharathidasan Uty) Solution : Principal part of the error an the int where vy; is the value of yand yf | the denvative of vata sexy 41. What is the local error term in T, Solution : Principal part of the error | where 9, 48 the value of y and yy 1a derwative of yatasay iideperdent varvable x, dilterences af the a Examples)! a, <Qe 4 Ln NTE; Diving e POLATION yy TDED, NUy TION WITH UNEQUAL INTERVALS What adva © INTE, DIFFS. EQUATIONS New n = wtGn? tage has Lagrange’s formul ii, a over Soluti ‘olution ; 7 Newton mee forw, Variable x i x are ard and backward interpolation formulae of © Use ts eu only when the values of the independent E dependent ¥ HY spaced can also when the differences of -lgrange’s interncanna y become smaller ultimately. But of x, the indepe ee formula can be used whether the values whether the fe lent variable are equally spaced or not and erence of y become smaller or not. ay ‘True or False’ : To fing find the inte ‘ : Ordre : 3 he inte rpolating polynomi ~» Lagrange’s method when the e, ; en the x's are not equally spaced. 12, §, ‘al for the given data can be applied only (Nov 95 Bharathidasan Uty) Solution : The statement is false. ; 13, What is the disadvantage in practice in ap, _-~” Lagrange’s interpolation formula? plying Solution ; Though Lagrange’s formula is simple and easy to remember, its application is not speedy. It requires close attention to sign and there is always a chance of committing some error due to a number of positive and negative signs in the numerator and the denominator. 14. Say ‘True or False’ : fange’s interpolation formula can be used for equal intervals. (Ap 95 Bharathidasan Uty) Solution : The statement is true. 15. Show that Lagrange’s interpolation formula is merely a variant of Newton’s divided difference formula. Jution : Let feo) sf ep Fn) be the values of fix) corresponding to the arguments Xp ,X) ,--4n- If f(x) is a its (n+ Dthe divided differences will be polynomial of Jre®{ @Q@ ™ " 5 INTERIOLATION Writ tiNKQUAL ITERN a“ VIDED NUM & INTE inbEE KQUATIONS Fe Anaya YO Pity ay he tag ae ito oe YAR = tal ty YD = wy hf org, tye tg to ' ' ‘ TUR Ri) 0 a9) cass 0 — a EN TA i) OR Dy TDP ta De fa od I the funet the in + 198 ( fon fo) in a polynomial of degree fy divided din “ronee will be zere R : 0 So the list term in CD) ie. fe stg tbs ma Hence (1 bees on PW) © flay) ata Xo) f(x 44) 4 = xy) OF a FO ha eat ry) Cr om 100 ay ay) ty fe The formula (2) is called Newton's dtorded difference titerpolation formula. 7. Say ‘True ¢ Falne' : Newton's divided difference formula is applicable only for equally spaced intervals, (Ap 97 Bharathidasan Uty) Solution : The statement is false, 8 Find the second degree polynomial fitting the following data: (Ap 96 M.U,) a 1 2 4 i 4 5 13 Solution : The divided difference table is as follows x Ve AY, A” ¥, l 4 hed 2-1"! ¢ 4&8 aha 4-1" hh Berea 4 4 13 N '4 divided difference interpolation formula isant NUMERICAL METHODS IN SCIENCE AND BNGG. Ry 5H. When will iteration method suce Soluti AUy, Ry i® the fang Yittteny, tn ondor that the iteyation’ may succeed, enpation of Ry i the tig ORR: As 7 i a ate eee i Heal sap H pA Ry inereaag§ ea Mossad teny the others in that equation) and the 7 ew attached to a different unknown in that ation. This fe Voy ok Cent will be got when the large coefficients are along Ro incresgeg nil nh ah Owe the leading diagonal of the matrix of the ene ts. t “Rady i 57, Explain relaxation method to solve a system of ate 0d ee » simultaneous ns. (Ap 96 M.U.) rade ina ME tig Salution : Refer §8 page 147 of the main text book, cach stayy MPa ge My et ES With sere 58 Write down the relaxation table for the solution of ‘The actual the acta ys in th ae Seed tae the system x+2y B,2x+ 25 y= 15. ual Telatation Ate nee at (Ap 97 Bharathidasan Uty), . i§ shown in the floning table x lerwing table Solution : The residuals ry, ry are given by y F 0 ee 8 ryex42y43 ‘ C % 2x4 By 15 i Tag at ‘The operation table | 0 ts Y Te ea Explanation : (1) Ling j f RO «(see calculating r),r», (1) is obtained by taking s-0=y and R, 0 D @ (2) The r : i wuidate thi ly largest residual is rp(=-9). We try to 59. Solve the equations 10x-2y=8, -*+10y=9 by the Viquidate this by applying a eae ye or relaxation method. (May 97 Anna Uty). we apply 1-Rp, From . bya ie ted Solution + The residuals ry, rp are given by , ~2, tory ry respestivaly ond 0, 1 to the values of x and y respectively. Line | ry=1x-2y-8 tent, (2 gives the current values of x,y and ry=-x+1y-9 } (3) The muimer anion ints ts We have the operation table : | this we apply 1-2, From the operation table, we fd that his adds 10, -1tory,7 respestively and 1, 0 to the values: wee AS seandy respective. Line (3) gives current values of 5,7 and Rk, 1 i at ' the residuals, R 0 4 =2 20 As the residuals ‘pre 2ere, we get the exact solution: on x=l=y. ezt | act | ,