Math : Numerical Methods
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1 7 EP-Progrm - Strisuks School - Roi-et Mth : Numericl Methods Dr.Wtt Toutip - Deprtmet of Mthemtics Kho Ke Uiversit :Wtt Toutip wttou@kku.c.th 7 Numericl methods If vlue cot be foud ectl, the pproimte vlue c ofte be obtied b umericl methods. 7. Solutio of equtios Suppose we wt to solve equtio f. A sequece of umbers,,,,, which gets closer to the solutio coverges to the solutio. If the sequece does ot get closer to fied vlue, the it diverges. Three methods to fid coverget sequece re s follow.. LINEAR INTERPOLATION METHOD Lier iterpoltio estimtes the rot of the equtio f b drwig stright lie betwee two kow vlues of the fuctio, d clcultig where tht stright lie crosses the -is If d b re er the root, the drw the stright lie betwee b, f b. If this lie crosses the -is t A, the b similr trigles: A b A, where f f b f b A f b bf f, f d f b f Fig 7. A b Fig7.
2 . NEWTON-RAPHSON METHOD The Newto-Rphso method obtis pproimtios to the root of f b drwig tget to the curve of f, d clcultig where tht tget crosses the -is. Pick first pproimtio, d successive pproimtios re give b:. PICARD S METHOD f ' (fig 7.) f Picrd s method rerrges the equtio f i the form g sequece of pproimtios to the root of pproimtio, Successive pproimtios re give b:. It the obtis g b the followig: Pick first g There re ofte severl ws to rrge equtio i the form diverges, tr other rrgemet. 7.. Emples g. If the itertio - Fig 7.. Show tht there is root of betwee d. Obti pproimtio for this root b lier iterpoltio. Solutio Let f The f d f. Hece the grph of must cross the -is t some poit betwee d. To fid pproimtio for ppl the formul bove, usig d b.. Use the Newto-Rphso formul to fid solutio to to 6 deciml plces. Solutio ccurte
3 Tke s the strtig poit. the itertio is givig: The lst two vlues gree to 6 deciml plces The solutio is.85. Show tht the equtio c be rrged i the followig ws: () (b) Appl Picrd s method to both these rrgemets, d hece fid the solutio to deciml plces. Solutio 7.. Eercises () Rewrite the equtio s. Tke the cube root of both sides. (b) Rewrite the equtio s. Divide both sides b I both cses let For rrgemet () The et terms re:.696,.9686,.5 The lst two gree to deciml plces The solutios is.. Show tht there is root of 6 betwee - d, d use lier iterpoltio to fid pproimtio to tht root. Use Newto-Rphso s itertio to fid the solutio to the equtio i Questio, ccurte to 6 deciml plces.. Fid pir of itegers betwee which there is root of. Use lier iterpoltio to fid pproimtio to tht root. Fid the solutio to the equtio of Questio, ccurte to 5 deciml plces, b mes of Newto-Rphso s itertio 5. Fid the solutio to the equtio e, ccurte to 6 deciml plces. 6. Solve to 6 deciml plces the equtio si ( is mesured i rdis)
4 7. Solve the equtio t, ccurte to 6 deciml plces. 8. Use Picrd s itertio to solve the equtio cos to deciml plces 9. Show tht the equtio () c be writte either s or s (b) Appl Picrd s itertio to the equtio, usig both of the forms () d (b). Which oe works? Fid the solutio of the equtio, give to deciml plces.. Show tht there is root of the equtio betwee d. Show tht the equtio c be writte i the followig forms: () (b) (c) Appl Picrd s itertio to ech of these. Hece solve the equtio to deciml plces.. Show tht there is root of the equtio e betwee d. (Here is mesured i rdis.) Rerrge the equtio i two forms, d hece use Picrd s method to solve the equtio to deciml plces. 7. Numericl itegrtio I m cses fuctio cot be ectl itegrted. There is method for fidig the defiite itegrl of f from to b. Divide the itervl b, ito equl itervls of legth h. b. Fig 7. b h. TRAPEZIUM RULE The trpezium rule pproimtes the itegrl b dividig the re ito trpezi. The formul is: b f d h f f b f f f If the grph of f b is cocve, s show i the figure, the the Trpezium rule will overestimte the true vlue. If the grph is cove the the rule will uderestimte the vlue
5 5 7.. Emples. Evlute si d, usig the trpezium rule with itervls Solutio Here, b, h. Appl the formul: 7 si d si si si si 8 si d.9. is fuctio of : the tble below gives si vlues for d : Fig Use the trpezium rule to fid pproimtio for uderestimte Solutio Appl the trpezium rule with, b 5,, h. 5 5 d d 5 d Is our swer over or A sketch of the grph is show. Note tht it is cocve The swer is overestimte 7.. Eercise. Evlute. Evlute d, usig the trpezium rule with four itervls. si d, usig the trpezium rule with 6 itervls
6 . Evlute. Evlute cos d, d,, usig the trpezium rule with 8 itervls () b ect itegrtio, (b) b the trpezium rule with itervls. Wht coclusio do ou mke bout the ccurc of the rule? 5. is fuctio of, with vlues give b the followig tble: Use the trpezium rule to pproimte d 6. is give i terms of, b the followig tble: Approimte d,usig the trpezium rule. 7. is give i terms of, b the tble: Fid pproimtio for 5 7 d, usig the trpezium rule. Write out etr row for the vlues of, d hece fid pproimtio for d, usig the trpezium rule 8. B cosiderig the shpe of the grph, show tht our swer to Questio is overestimte of the true vlue. 9. B cosiderig the shpe of the grph, show tht evlutio of cosd b the trpezium rule will be uderestimte. Is this cofirmed i our swer to Questio? 7. Emitio questio. The equtio of curve is 5. A regio is bouded b the curve, the es d the ordite t. (i) B usig the trpezium rule with four strips of equl width, fid pproimte vlue of the re of the regio. (ii) Use sketch to show wh the pproimte vlue of the re of the regio is greter th the ctul vlue. () Rerrg the cubic equtio b Stte the vlues of the costts d b. 6 ito the form 5
7 (b) Use the itertive formul b With d our vlues of d b to fid the pproimte positive solutio of the equtio, to pproprite degree of ccurc. Show ll our itermedite swers.. Show tht the cubic equtio hs ol oe rel root d further tht the root lies betwee d. Two possible itertive schemes for fidig the root re (i) d (ii) / Show tht ol oe of these schemes coverges from iitil estimte of d hece fid the root correct to d.p., justifig the ccurc of our swer.. Drw digrm to show wh, if is pproimte solutio of the equtio f, the f ' f is, i geerl, better oe. Prove tht the cubic equtio 6 hs root lig betwee d, d fid it correct to two deciml plces.. Show tht e d e Show tht use of the trpezium rule with 5 strips(6 ordites) gives estimte tht is bout.8% to high. Epli wh pproimte evlutio of this itegrl usig the trpezium rule will lws result i overestimte, however m strips re used.. () B sketchig the curves with equtio equtio d e hs oe egtive root d oe positive root (b) Use the itertio formul / e, show tht the e with to fid i tur,, d d hece write dow pproimte to the egtive root of the equtio, givig our swer to deciml plces. (c) Describe the result of such ttempt. Commo errors. Ide ottio Do ot be cofused b the ottio, For emple, do ot cofuse with, is the secod pproimtio i the sequece which strted with. Solutio of equtios I ll the methods be ver creful with rithmetic. It is ver es to mke mistke, especill with the egtive sigs i the lier iterpoltio d Newto-Rphso formule.
8 Whe doig severl successive pproimtios, It is good ide to keep i the memor of our clcultor, d the replce it with fter ech pproimtio is completed.. Numericl itegrtio () Mke sure tht our itervls re correct. The must be equl i legth. (b) Whe pproimtig somethig like d, mke sure tht ou squre the vlues before ou use the formul. Be creful of the followigs: d d Solutio (to eercise) ,, () (b) ,7.65 =========================================================== Refereces: Solomo, R.C. (997), A Level: Mthemtics ( th Editio), Gret Briti, Hillm Priters(Frome) Ltd.
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