Root Finding
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1 Root Fndng
2 What s Computer Scence? Computer scence s a dscplne that spans theory and practce. It requres thnkng both n abstract terms and n concrete terms. The practcal sde o computng can be seen everywhere. Computer scence also has strong connectons to other dscplnes. Source:
3 What s Computer Scence? Computer Scence s practced by mathematcans, scentsts and engneers. Mathematcs, the orgns o Computer Scence, provdes reason and logc. Scence provdes the methodology or learnng and renement. Engneerng provdes the technques or buldng hardware and sotware. Source:
4 Numercal Computng/Analyss s the study o algorthms that use numercal appromaton (as opposed to general symbolc manpulatons) or the problems o mathematcal analyss.
5 Root Fndng Gven a real valued uncton o one varable (say ), the dea s to nd an such that: () =
6 Root Fndng Eamples ) Fnd real such that: ) Agan: 3) Agan: tanh 3 0 cos 2 0 Observatons on, 2 and 3?? (two are trck questons)
7 Root Fndng Eamples สมการต วแปรเด ยว = () () = + => root (*) = 0 => * = 0- = - สมการต วแปรไม เช งเส น () = => root (+2)(+) = 0 => * = -,
8 Requrements For An Algorthmc Approach Idea: nd a sequence o, 2, 3, 4. so that or some N, N s close to a root..e. ( N ) <tolerance What do we need?
9 Requrements For Such a Root-Fndng Scheme Intal guess: Relatonshp between n+ and n and possbly n-, n-2, n-3, When to stop the successve guesses?
10 Some Alternatve Methods Bracketng Method (ประมาณแบบช วงคาตอบ) Graphcal Method Incremental Search Method Bsecton False-Poston Method Open Method Newton s Method (ประมาณแบบช วงจ ด: ว งบนความช น) Avodng dervatves n Newton method
11 Intermedate value theorem When you have two ponts connected by a contnuous curve: one pont below the lne the other pont above the lne... then there wll be at least one place where the curve crosses the lne! The ntermedate value theorem tells us that a contnuous uncton s postve at one end o an nterval and s negatve at the other end o the nterval then there s a root somewhere n the nterval
12 Theorem Theorem: An equaton ()=0, where () s a real contnuous uncton, has at least one root between l and u ( l ) ( u ) < 0. () u
13 Theorem I uncton () n ()=0 does not change sgn between two ponts, roots may stll est between the two ponts. () u
14 Theorem I the uncton () n ()=0 does not change sgn between two ponts, there may not be any roots between the two ponts. () () u u
15 Theorem I the uncton () n ()=0 changes sgn between two ponts, more than one root may est between the two ponts. () u
16
17
18 Incremental Search Method I ( ) and ( 2 ) have opposte sgns, there s at least one root n the nterval (, 2 ). I the nterval s small enough, t s lkely contan a sngle root. The zeroes o () or root can be detected by evaluatng the uncton at the nterval d and lookng or changng n sgn. Wrte a program
19 Eample Use ncremental search wth d = 0.2 to bracket the smallest postve zero o () =
20 clear all; close all; clc; =.^3 2*.^2 + 5* - 0'; = nlne(); a = 0; % lower boundary b = 3; % upper boundary d = 0. = [a:d:b]; = (); [; ]' %plot(, ); dsp('press Enter to contnue') pause; Numercal Computng: 2/
21 lag = true; (a)*(b) < 0 = ; c = a + *d; whle (lag) [sgn((a)) sgn((c))] (sgn((a)) ~= sgn((c))) [ c (c)] dsp 'ound root'; lag = alse; end = +; c = a + *d; end else (a)*(b) == 0 dsp 'root s at the boundary'; else dsp 'could not detect root'; end Numercal Computng: 2/2558 2
22 Assgnment-0 Mody Incremental Search code or detectng all roots ound on the gven lst and specyng locaton o each root
23 Algorthm or Bsecton Method
24
25
26 Step Choose and u as two guesses or the root such that ( ) ( u ) < 0, or n other words, () changes sgn between and u. () u
27 Step 2 Estmate the root, m o the equaton () = 0 as the md-pont between and u as () m = 2 u u
28 Step 3 Now check the ollowng () I ( ) ( m ) < 0, then the root les between and m ; then = ; u = m. m u I ( ) ( m ) > 0, then the root les between m and u ; then = m ; u = u. I ( ) ( m ) = 0; then the root s m. Stop the algorthm ths s true.
29 Step 4 New estmate m = 2 Absolute Relatve Appromate Error a m new new m u old m 00 old m prevousestmateo root new m current estmateo root
30 Step 5 Check absolute relatve appromate error s less than prespeced tolerance or mamum number o teratons s reached. Yes No Stop Usng the new upper and lower guesses rom Step 3, go to Step
31 Eample Use Bsecton method to nd the root o () = [wthn -:0]
32 clear all; close all; clc; = '.^3-0*.^2+5'; = nlne(); a = 0; % lower boundary b = ; % upper boundary d = 0. = [a:d:b]; = (); [; ]' %plot(, ); dsp('press Enter to contnue') pause; Numercal Computng: 2/
33 lag = true; delta = ; (a)*(b) < 0 = 0; whle (lag) = +; m = (a+b)/2 (m) == 0 dsp 'ound root'; lag = alse; else (a)*(m) > 0 a = m; else b = m; end end [ a b m (m)] % (b-a)/(2.^) < delta end else (a)*(b)== 0 dsp 'root s at the boundary'; else dsp 'could not detect root o non-lnear equaton ; end Numercal Computng: 2/
34 Assgnment-02 Mody Bsecton code or detectng all roots ound on the gven lst and specyng locaton o each root
35 Advantages Always convergent The root bracket gets halved wth each teraton - guaranteed
36 Drawbacks Slow convergence
37 Drawbacks (contnued) I one o the ntal guesses s close to the root, the convergence s slower
38 Drawbacks (contnued) I a uncton () s such that t just touches the -as t wll be unable to nd the lower and upper guesses. ()
39 Bsecton Convergence Rate Every tme we splt the nterval we reduce the search nterval by a actor o two..e. a b a b 0 0 k k 2 k
40 False-Poston Method o Solvng a Nonlnear Equaton Major: All Engneerng Majors Authors: Duc Nguyen Numercal Methods or STEM undergraduates 40
41 False-Poston Method or Regula Fals Method Because o the convergence o Bsecton method s slow. We assume that (a) and (b) have opposte sgns. The Bsecton method used the mdpont o the nterval [a, b] as net terate. A better appromaton s obtaned we nd the pont (c, 0) where the secant lne L jonng the ponts (a, (a)) and (b, (b)) crosses the -as.
42 Introducton ( ) 0 () In the Bsecton method U ( )* ( ) L U 0 (2) O 42 L r U Eact root L Fgure False-Poston Method r L 2 U (3)
43 (c)
44 False-Poston Method Based on two smlar trangles, shown n Fgure, one gets: r ( L ) L r ( U ) The sgns or both sdes o Eq. (4) s consstent, snce: U (4) U ( ( L U ) 0; ) 0; r r L U 0 0 Eact root O L r U 44 L
45 45 L U r U L r U L r U L L U From Eq. (4), one obtans The above equaton can be solved to obtan the net predcted root U L U L L U r r, as (5)
46 Step-By-Step False-Poston Algorthms 46. Choose L that and U 2. Estmate the root, as two guesses or the root such 0 L U m 3. Now check the ollowng (a) I and (b) I and m U 0 L m ; then ; then U L L U L, then the root les between L L and U m 0 L m, then the root les between L m and U U U L m
47 47 (c) I 0 L m, then the root s Stop the algorthm ths s true. 4. Fnd the new estmate o the root m U L L U L U m. Fnd the absolute relatve appromate error as a new m new m old m 00
48 48 where new m old m = estmated root rom present teraton = estmated root rom prevous teraton 5. say 0 3 s I a s, then go to step 3, else stop the algorthm. Notes: The False-Poston and Bsecton algorthms are qute smlar. The only derence s the ormula used to calculate the new estmate o the root, m shown n steps #2 and 4!
49 Assgnment-03 Wrte False Poston code or detectng all roots ound on the gven lst and specyng locaton o each root
50 Newton-Raphson Method Slope Method or Fndng Roots
51 - + Basc Ideas Slope= () A y=() ()=dy/d C B ()-0 X +2 X +
52 Newton-Raphson Method () ( ), = - ( ) ( ) ( - ) +2 + X 52 Geometrcal llustraton o the Newton-Raphson method.
53 Dervaton () ( ) B tan( AB AC '( ) ( ) C + A X ( ) ( ) Dervaton o the Newton-Raphson method. 53
54 Algorthm or Newton-Raphson Method 54
55 Step Evaluate () symbolcally. 55
56 Step 2 Use an ntal guess o the root,, to estmate the new value o the root,, as =
57 Step 3 Fnd the absolute relatve appromate error as a - a =
58 Step 4 Compare the absolute relatve appromate error wth the pre-speced relatve error tolerance. s Is a s? Yes No Go to Step 2 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. 58
59 Eample You are workng or DOWN THE TOILET COMPANY that makes loats or ABC commodes. 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. 59 Floatng ball problem.
60 Eample Cont. The equaton that gves the depth n meters to whch the ball s submerged under water s gven by Floatng ball problem. Use the Newton s method o ndng roots o equatons to nd a) the depth to whch the ball s submerged under water. Conduct three teratons to estmate the root o the above equaton. b) The absolute relatve appromate error at the end o each teraton, and c) The number o sgncant dgts at least correct at the end o each teraton. 60
61 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 () 6
62 Eample Cont. Solve or ' ' Let us assume the ntal guess o the root o s 0.05m 0. Ths s a reasonable guess (dscuss why 0 and 0.m are not good choces) as the etreme values o the depth would be 0 and the dameter (0. m) o the ball
63 Eample Cont. Iteraton The estmate o the root s '
64 Eample Cont. Fgure 5 Estmate o the root or the rst teraton. 64
65 Eample Cont. 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. 65
66 Eample Cont. Iteraton 2 The estmate o the root s 2 '
67 Eample Cont. Fgure 6 Estmate o the root or the Iteraton 2. 67
68 Eample Cont. The absolute relatve appromate error a at the end o Iteraton 2 s a % 00 2m The mamum value o m or whch a 0.50 s Hence, the number o sgncant dgts at least correct n the answer s 2. 68
69 Eample Cont. Iteraton 3 The estmate o the root s 3 2 '
70 Eample Cont. Estmate o the root or the Iteraton 3. 70
71 Eample Cont. The absolute relatve appromate error a at the end o Iteraton 3 s a % The number o sgncant dgts at least correct s 4, as only 4 sgncant dgts are carred through all the calculatons. 7
72 ต วอย าง หาค ารากของสมการ 2e + 4 = 0 ว ธ ท า ให () = 2e + 4 () = 2e + จากส ตรการท าซ า ก าหนด 0 = 0 r r (r ) ( ) r ( 0 ) = 2e = -2 ( 0 ) = 2e 0 + = 3 = 0 (-2/3) = 0.67
73 ( ) = (0.67) = ( ) = = 0.67 (0.578/4.908) = ( 2 ) = ( 2 ) = = ( /4.473) = ( 3 ) = 4.82 * 0-5 ( 3 ) = = (4.82*0-5/4.454) =
74 Algorthm ของการหาค ารากของสมการ ()=0 ด วยระเบ ยบว ธ น วต น-ราฟส น Input : 0 เป นค าเร มต น = ขอบเขตของความคลาดเคล อน N = จ านวนคร งส งส ดของการทาซ า = ค าขอบเขตของฟ งก ช น Output : ค ารากของ ()=0
75 Algorthm r = 0 Do r < N then r+ = r ( r ) / ( r ) r = r+ else error( Out o Range ) Untl r r- < and ( r ) < Return r
76 Advantages and Drawbacks o Newton Raphson Method 76
77 Advantages Converges ast (quadratc convergence), t converges. We wll show that the rate o convergence s much aster than the bsecton method. However as always, there s a catch. The method uses a local lnear appromaton, whch clearly breaks down near a turnng pont. Small ( n ) makes the lnear model very lat and wll send the search ar away Requres only one guess 77
78 Drawbacks: Avodng dervatves. Dvergence at nlecton ponts Selecton o the ntal guess or an teraton value o the root that s close to the nlecton pont o the uncton may start dvergng away rom the root n ther Newton-Raphson method. 0 For eample, to nd the root o the equaton The Newton-Raphson method reduces to Table shows the terated values o the root o the equaton. The root starts to dverge at Iteraton 6 because the prevous estmate o s close to the nlecton pont o. Eventually ater 2 more teratons the root converges to the eact value o
79 Drawbacks Inlecton Ponts Table Dvergence near nlecton pont. Iteraton Number Dvergence at nlecton pont or
80 Drawbacks Dvson by Zero 2. Dvson by zero For the equaton the Newton-Raphson method reduces to or For, the denomnator wll equal zero. Ptall o dvson by zero or near a zero number 80
81 Drawbacks Oscllatons near local mamum and mnmum 3. Oscllatons near local mamum and mnmum Results obtaned rom the Newton-Raphson method may oscllate about the local mamum or mnmum wthout convergng on a root but convergng on the local mamum or mnmum. Eventually, t may lead to dvson by a number close to zero and may dverge. For eample or roots the equaton has no real 8
82 Drawbacks Oscllatons near local mamum and mnmum Table 3 Oscllatons near local mama and mmma n Newton-Raphson method. Iteraton Number % a () Oscllatons around local mnma or
83 Drawbacks Root Jumpng 4. Root Jumpng In some cases where the uncton s oscllatng and has a number o roots, one may choose an ntal guess close to a root. However, the guesses may jump and converge to some other root. For eample sn 0 Choose It wll converge to nstead o () Root jumpng rom ntended locaton o root or sn. 0 83
84 Secant Method Roots o a Nonlnear Equaton /5/
85 Secant Method The Newton-Raphson algorthm s based on the evaluaton o dervaton. The Newton-Raphson algorthm requres the evaluaton o two unctons per teraton, () and (). It s desrable to have method that converge as ast as Newton s method yet nvolves only evaluatons o () and not o ().
86 Secant Method ) ( ) ( - = () ( ) ( - ) +2 + X, ) ( ) ( ) ( ) ( ) ( ) )( ( Newton s Method Appromate the dervatve
87 Secant Method ) ( ) ( ) )( ( Geometrc Smlar Trangles () ( ) ( - ) + - X B C E D A ) ( ) ( DE DC AE AB
88 Algorthm or Secant Method
89 Step 00 - = a Calculate the net estmate o the root rom two ntal guesses Fnd the absolute relatve appromate error ) ( ) ( ) )( (
90 Step 2 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.
91 Eample To nd the nverse o a number a, one can use the equaton ( ) a 0 where s the nverse o a.
92 Soluton Use the Secant method o ndng roots o equatons to Fnd the nverse o a = 2.5. Conduct three teratons to estmate the root o the above equaton. 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.
93 0 a Graph o uncton () () ( ) 0. ) ( a a a a ) ( ) ( ) ( a a ) ( a
94 Iteraton # ( ) ( ) ( ) 7.5 secant( ) ( ) , ( a ) (2.5(0.6) ) ' () ', (rst guess) 0, (prevous guess) Secant lne, (new guess) a 9.09%
95 Iteraton #2 ( ) ( ) ( ) secant( ) ( ) , ( a ) (2.5(0.55) ) () (guess) 0 (prevous guess) Secant lne 2 (new guess) a 69.23%
96 Iteraton # Entered uncton along gv en nterv al w th current and net root and the tangent lne o the curv e at the current root ( ) ( ) ( ) secant( ) ( ) , (2.5(0.325 )) a 2 2 ( a % ) () 2 (guess) (prevous guess) Secant lne 3 (new guess)
97 Advantages Converges ast, t converges Requres two guesses that do not need to bracket the root
98 Summary We have looked at our ways to nd the root o a sngle valued, sngle parameter uncton We consdered a robust, but slow bsecton method and then a aster but less robust Newton s and Secant method
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