Summary with Examples for Root finding Methods -Bisection -Newton Raphson -Secant

Transcription

1 Summary wth Eamples or Root ndng Methods -Bsecton -Newton Raphson -Secant

2 Nonlnear Equaton Solvers Bracketng Graphcal Open Methods Bsecton False Poston (Regula-Fals) Newton Raphson Secant All Iteratve

3 Bracketng Methods (Or, two pont methods or ndng roots) Two ntal guesses or the root are requred. These guesses must bracket or be on ether sde o the root. == > Fgure I one root o a real and contnuous uncton, ()=0, s bounded by values = l, = u then ( l ). ( u ) <0. (The uncton changes sgn on opposte sdes o the root)

4 Bass o Bsecton Method Theorem An equaton ()=0, where () s a real contnuous uncton, has at least one root between l and u ( l ) ( u ) < 0. () u At least one root ests between the two ponts the uncton s real, contnuous, and changes sgn. 4

5 Algorthm or Bsecton Method Step Choose and u as two guesses or the root such that ( ) ( u ) < 0, or n other words, () changes sgn between and u. Ths was demonstrated n Fgure. () u 5

6 Step Estmate the root, m o the equaton () = 0 as the md pont between and u as () m = u m u Estmate o m 6

7 Step Now check the ollowng 0 a) I l m, then the root les between and m ; then = ; u = m. 0 b) I l m, then the root les between m and u ; then = m ; u = u. 0 c) I l m ; then the root s m. Stop the algorthm ths s true. 7

8 Step 4 Fnd the new estmate o the root m = u Fnd the absolute relatve appromate error where a old m m new new m old m 00 prevousestmateo root new m current estmateo root 8

9 Step 5 Compare the absolute relatve appromate error error tolerance. s a wth the pre-speced Is a s? Yes No Go to Step usng new upper and lower guesses. Stop the algorthm Note one should also check whether the number o teratons s more than the mamum number o teratons allowed. I so, one needs to termnate the algorthm and noty the user about t. 9

10 Eample In the dagram shown the loatng ball has a specc gravty o 0.6 and has a radus o 5.5 cm. You are asked to nd the depth to whch the ball s submerged when loatng n water. Dagram o the loatng ball 0

11 Eample Cont. The equaton that gves the depth to whch the ball s submerged under water s gven by a) Use the bsecton method o ndng roots o equatons to nd the depth to whch the ball s submerged under water. Conduct three teratons to estmate the root o the above equaton. b) Fnd the absolute relatve appromate error at the end o each teraton, and the number o sgncant dgts at least correct at the end o each teraton.

12 Eample Cont. From the physcs o the problem, the ball would be submerged between = 0 and = R, that s where R = radus o the ball, R Dagram o the loatng ball

13 Eample Cont. Soluton To ad n the understandng o how ths method works to nd the root o an equaton, the graph o () s shown to the rght, where Graph o the uncton ()

14 Eample Cont. Let us assume u Check the uncton changes sgn between and u. Hence 4 4 l u l u So there s at least on root between and u, that s between 0 and 0. 4

15 Eample Cont. Graph demonstratng sgn change between ntal lmts 5

16 Eample Cont. Iteraton The estmate o the root s m m u l m Hence the root s bracketed between m and u, that s, between and 0.. So, the lower and upper lmts o the new bracket are l 0.055, 0. u At ths pont, the absolute relatve appromate error calculated as we do not have a prevous appromaton. a cannot be 6

17 Eample Cont. Estmate o the root or Iteraton 7

18 Eample Cont. m m u (0.085) l Iteraton The estmate o the root s m Hence the root s bracketed between and m, that s, between and So, the lower and upper lmts o the new bracket are l 0.055, u 8

19 Eample Cont. Estmate o the root or Iteraton 9

20 Eample Cont. The absolute relatve appromate error a at the end o Iteraton s a new m new m old m % 00 None o the sgncant dgts are at least correct n the estmate root o m = because the absolute relatve appromate error s greater than 5%. 0

21 Eample Cont. m m u l Iteraton The estmate o the root s m Hence the root s bracketed between and m, that s, between and So, the lower and upper lmts o the new bracket are l 0.055, u

22 Eample Cont. Estmate o the root or Iteraton

23 Eample Cont. The absolute relatve appromate error a at the end o Iteraton s a new m new m old m % 00 Stll none o the sgncant dgts are at least correct n the estmated root o the equaton as the absolute relatve appromate error s greater than 5%. Seven more teratons were conducted and these teratons are shown n Table.

24 Table Cont. Table Root o ()=0 as uncton o number o teratons or bsecton method. Iteraton u m a % ( m )

25 5 Most wdely used method. Based on Taylor seres epanson: ) ( ) ( ) ( 0 Rearrangn g, 0 ) when ( the value o The root s! ) ( ) ( ) ( ) ( ) ( ) ( O Newton-Raphson ormula Solve or Newton-Raphson Method

26 A convenent method or unctons whose dervatves can be evaluated analytcally. It may not be convenent or unctons whose dervatves cannot be evaluated analytcally. 6

27 Newton-Raphson Method () ( ), = - ( ) ( ) ( - ) + + X 7 Geometrcal llustraton o the Newton-Raphson method.

28 Dervaton () ( ) B tan( AB AC '( ) ( ) C + A X ( ) ( ) Dervaton o the Newton-Raphson method. 8

29 Algorthm or Newton - Raphson Method Step Evaluate () symbolcally. Step Use an ntal guess o the root,, as = -, to estmate the new value o the root, Step Fnd the absolute relatve appromate error a as - a =

30 Step 4 Compare the absolute relatve appromate error wth the pre-speced relatve error tolerance. s Is a s? Yes No Go to Step usng new estmate o the root. Stop the algorthm Also, check the number o teratons has eceeded the mamum number o teratons allowed. I so, one needs to termnate the algorthm and noty the user. 0

31 Step Eample ' Solve Eample usng Newton-Raphson Methods. Solve or ' Let us assume the ntal guess o the root o s. Ths s a reasonable guess! and 0 0.m are not good choces) as the etreme values o the depth would be 0 and the dameter (0. m) o the ball m

32 Eample Cont. Step Iteraton The estmate o the root s '

33 Eample Cont. Estmate o the root or the rst teraton.

34 Eample Cont. Step The absolute relatve appromate error a at the end o Iteraton s a % 00 The number o sgncant dgts at least correct s 0, as you need an absolute relatve appromate error o 5% or less or at least one sgncant dgts to be correct n your result. 4

35 Eample Cont. Iteraton The estmate o the root s '

36 Eample Cont. Estmate o the root or the Iteraton. 6

37 Eample Cont. The absolute relatve appromate error a at the end o Iteraton s a % 00 m The mamum value o m or whch a 0.50 s.844. Hence, the number o sgncant dgts at least correct n the answer s. 7

38 Eample Cont. Iteraton The estmate o the root s '

39 Eample Cont. Estmate o the root or the Iteraton. 9

40 Eample Cont. The absolute relatve appromate error a at the end o Iteraton s a % The number o sgncant dgts at least correct s 4, as only 4 sgncant dgts are carred through all the calculatons. 40

41 Secant Method-Dervaton () ( ) ( - ) + + Geometrcal llustraton o Newton-Raphson method., the X Newton s Method ( ) = - ( ) ( ) ( ) ( ( ( )( ) ) ( () Appromate the dervatve () Substtutng Equaton () nto Equaton () gves the Secant method ) ) 4

42 Secant Method Dervaton The secant method can also be derved rom geometry: () ( ) ( - ) C E D A + - B X The Geometrc Smlar Trangles AB DC AE DE can be wrtten as ( ) ( ) On rearrangng, the secant method s gven as Geometrcal representaton o Secant method. the ( ( )( ) ( ) ) 4

43 4 Step 00 - = a Calculate the net estmate o the root rom two ntal guesses Fnd the absolute relatve appromate error ) ( ) ( ) )( ( Algorthm or Secant Method

44 Step Fnd the absolute relatve appromate error s greater than the prespeced relatve error tolerance. I so, go back to step, else stop the algorthm. Also check the number o teratons has eceeded the mamum number o teratons. 44

45 Eample In Eample, the equaton that gves the depth to whch the ball s submerged under water s gven by Use the Secant method o ndng roots o equatons to nd the depth to whch the ball s submerged under water. Conduct three teratons to estmate the root o the above equaton. Fnd the absolute relatve appromate error and the number o sgncant dgts at least correct at the end o each teraton. 45

46 Eample Cont. Step Let us assume the ntal guesses o the root o as 0.0and Iteraton The estmate o the root s

47 Eample Cont. Step The absolute relatve appromate error Iteraton s a % 00 a at the end o The number o sgncant dgts at least correct s 0, as you need an absolute relatve appromate error o 5% or less or one sgncant dgts to be correct n your result. 47

48 Eample Cont. Graph o results o Iteraton. 48

49 Eample Cont. Step Iteraton The estmate o the root s

50 Eample Cont. The absolute relatve appromate error Iteraton s a % 00 a at the end o The number o sgncant dgts at least correct s, as you need an absolute relatve appromate error o 5% or less. 50

51 Eample Cont. Graph o results o Iteraton. 5

52 Eample Cont. Iteraton The estmate o the root s

53 Eample Cont. The absolute relatve appromate error Iteraton s a % 00 a at the end o The number o sgncant dgts at least correct s 5, as you need an absolute relatve appromate error o 0.5% or less. 5