Lab II Numerical Methods

Transcription

1 Lab II Numerical Mehods Peer Brinkmann March, Numerical mehods By now, you have seen many eamples of iniial value problems of he form d d = f(,), ( ) =. () You have also learned some echniques ha yield eplici soluions of iniial value problems. Bu in fac for mos problems here is lile hope of finding an eplici soluion; his is where numerical mehods come in.. An eample Le s eamine he iniial value problem d d =, () =, in oher words f(,) = and =, =. Le () be he soluion. We already know how o find he eac soluion () = e (how?!), bu for he sake of argumen, le s preend we don know much abou i and see wha we can consruc numerically. Copyrigh c, The Triode (iode@mah.uiuc.edu). Permission is graned o copy, disribue and/or modify his documen under he erms of he GNU Free Documenaion License, Version. or any laer version published by he Free Sofware Foundaion; wih no Invarian Secions, no Fron-Cover Tes, and no Back-Cover Tes. A copy of he license is available a hp:// This documen is your guide hrough he second lab session wih Iode. I is no a homework assignmen, and you do no have o urn in any work.

2 The firs plo of Figure shows he direcion field of our iniial value problem, wih a mark a he poin (, ). The direcion field line segmen a his poin has slope f(, ) =, and so in he ne plo of Figure we show he whole line hrough (, ) wih slope. If we sep along his line by going across uni, hen we have o go up uni also o say on he line. This brings us o he poin (, ), as shown in he hird plo. This poin ough o be reasonably near he eac soluion curve, since he eac soluion curve also has slope f(, ) = a he poin (, ) where we sared. Now le s repea he process. The line segmen a (, ) has slope f(, ) =. The fourh plo in Figure shows he whole line hrough (, ) wih slope. If we sep along his line by going across uni, hen we have o go up unis o say on he line. This brings us o he poin (, ), as shown in he fifh plo. Again, his poin ough o be reasonably near he eac soluion curve. Repea he process again o arrive a he poin (, ) in he sevenh plo. We now have a lis of four poins (, ), (, ), (, ), and (, ) ha we epec o lie fairly close o he poins (,()), (,()), (,()), and (,()) on he eac soluion curve. The eighh plo shows hose four poins, conneced by line segmens o give an approimae soluion curve. Then he ninh plo shows our approimae soluion ogeher wih he eac soluion, () = e. While our approimae soluion isn perfec, i s no bad for a firs aemp! This mehod of obaining approimae soluions is Euler s mehod. We describe i more generally below, in Secion... Numerical mehods wih Iode Sar he direcion fields module of Iode and inpu he differenial equaion d d =, ha is, Relabel variables so ha is he independen variable and is he dependen variable, hen Ener differenial equaion and ype in for f(,). Then Change display parameers so ha he -values range from o and he -values range from o. We are going o consider he iniial condiion =, =. The eac soluion of d = saisfying his iniial d

3 ..... Eample..... Eample..... Eample..... Eample..... Eample..... Eample..... Eample..... Eample..... Eample Figure : The plos for Secion

4 condiion is = e, so plo = ep() using Plo arbirary funcion (in he Equaion menu). [Alernaive. Use he Soluion mehod buon o choose he Eac soluion mehod, hen ener = and = and click he Plo soluion buon. This alernaive will work provided your Malab insallaion includes he Symbolic Toolbo.] Le s see how Euler s mehod performs, compared o he eac soluion. Change he Soluion mehod o Euler, and ener sep size. Now plo a numerical soluion by clicking on he poin (, ) (or else ener = and = and click he Plo soluion buon). Check ha your resuling Euler plo looks similar o he ninh plo in Figure. How does he Euler numerical soluion compare o he eac soluion? Jo down your observaions on he las page of his lab! e.g. A which -values does he graph of he numerical soluion have corners? Does he numerical soluion graph ge closer o he eac soluion as increases, or does i ge furher away? Reduce he sep size o. and plo anoher Euler numerical soluion wih he same iniial condiions. How does he soluion change, now ha you have reduced he sep size? Repea his wih sep sizes. and.. Jo down your observaions on he las page. Finally, change o he Runge-Kua soluion mehod wih sep size.. Runge Kua is a more sophisicaed numerical mehod han Euler. Plo a Runge Kua numerical soluion wih he same iniial condiions. (Firs choose a differen color for his plo, o make i easier o follow.) Wha do you see? How does he accuracy of Runge Kua wih sep size. compare o Euler s mehod wih sep size.? o Euler wih sep size.? Again jo down your observaions on he las page.. Euler s mehod Take a uniformly spaced sequence,,,..., which means here is a sep size h > such ha +h =, and +h =, and so on. (In Secion, he sep size was.) Le () be a soluion of Equaion saisfying he iniial condiion ( ) =. We wan o compue a sequence of -values,..., n such ha k is reasonably close o ( k ), for each k =,,...,n. Here s how we do i, based on our eample in Secion...

5 The iniial poin (, ) lies on he graph of (), and he graph has slope f(, ) a ha poin. So if we jus follow he angen line and sep across by h and up by hf(, ), hen he new poin should sill be fairly close o he graph. Tha is, if we sep across and up o he poin = + h = + h f(, ) hen he poin (, ) should be fairly close o he graph of he soluion. [Check: The slope of he line from (, ) o (, ) equals = h f(, ) h = f(, ), which is eacly he slope of he direcion field line segmen a (, ).] Repeaing his reasoning, we obain (, ) by leing = + h = + h f(, ) The general rule of his ype is he Euler updae formula: i+ = i + h f( i, i ), () which ells us how o compue once we know, and hen how o compue once we know, and so on. Graphically, he Euler poins ( i, i ) are he corners in he Euler approimae soluion.. Approimaion error in Euler s mehod To deepen your undersanding of he Euler mehod, ener he following differenial equaion ino Iode: d d = sin( / ) wih display parameer < < and < <. Plo a numerical soluion using he Euler mehod in Iode, for sep size and iniial condiion ( ) =. Also plo he Runge Kua soluion wih sep size., which we will regard as being he rue soluion curve (because he error is known o be very small when using he Runge Kua mehod). Now answer he quesions a he boom of he las page.

6 >> c=:.: c = >> c=sin(pi c) c = >> plo(c,c) >> lengh(c) ans = >> c() ans =. >> c() ans =. Figure : Represening and ploing funcions wih Malab Programming Euler s mehod In Projec II you will implemen your own numerical mehod. So you need o undersand how Malab and Ocave represen numerical soluions inernally. The code in Figure illusraes some of he main poins, and you can ype he code yourself a he promp in he command window of Malab or Ocave. (Bu you will need o qui ou of Iode before doing so, or else he plos migh no show up correcly.) The firs command, c=:.:, creaes a vecor of -values, ranging from o wih sep size., i.e., i creaes he vecor (,.,.,.,., ). The second command, c=sin(pi c), creaes a vecor of -values by evaluaing he funcion sin(π) a all he enries of he vecor c. The hird command, plo(c,c), inerpres c as a lis of coordinaes on he horizonal ais and c as a lis of corresponding coordinaes on he verical ais. I plos all si poins (, ) and connecs adjacen poins wih sraigh line segmens. The resuling graph looks like a rough approimaion of a par of a sine curve. We can obain a beer picure by decreasing he sep size: if you replace he firs line by c=:.:; and repea he remaining wo seps, hen you ll see

7 For each i, compue: h = i+ i k = f( i, i ) i+ = i + h k, and hen append i+ o he vecor of -values funcion c=euler(fs,,c); =; c=[]; for i=:(lengh(c) ) h=c(i+) c(i); k=feval(fs,c( i ), ); =+h k; c=[c,]; end; Figure : Euler s mehod and Iode s implemenaion of i a plo ha looks very much like a piece of a sine curve. The las few commands in Figure show how o access cerain informaion abou he vecor c, such as is lengh (i.e., he number of enries), and he values of is individual enries, numbered from o. Remark. The above way of represening a funcion numerically as a vecor of -values and a vecor of -values fis beauifully ino our discussion of numerical mehods: he Euler mehod akes an iniial value and a vecor,,..., n of -values, and compues a vecor,,..., n of -values. Now we are ready o inspec Iode s implemenaion of Euler s mehod. Use he Open menu iem in he Malab main window (no he Iode window!) o open he file euler.m. Figure shows he conens of euler.m, wihou he commen lines (which begin wih a percenage sign). We ll go hrough i line by line. Line defines a new funcion calledeuler. When his funcion is called, i epecs o receive hree parameers; he parameer fs represens he funcion f(,) from our differenial equaion, is he iniial -value, and c is a vecor of -coordinaes, like we have seen before (in paricular he firs enry of c is ). Line also indicaes ha his funcion will reurn a value in he variable c, which will urn ou o be he desired vecor of -coordinaes compued by Euler s mehod. Line iniializes he value of he variable o be. The variable always conains our curren numerical approimaion i. Line creaes he If you end a line wih a semicolon, Malab will no prin he resul of he operaion. In paricular i is wise o use a semicolon when he resul is a vecor wih many enries, like in his case, and you don wan o cluer up he screen wih los of numbers.

8 vecor c ha will conain our numerical approimaions; iniially, i only conains he value. In paricular, he firs enry of c is, i.e., we have c()= and c()=. Line is he beginning of a loop ha les he variable i range over all numbers from o he lengh of he vecor c minus one. The body of his loop, lines, consiues he Euler updae sep and will be eecued for each value of i. Line compues he sep size h by compuing he difference beween he (i + )-s enry of c and he i-h enry of c. The epression c(i) sands for he i-h enry of c, which we regard as he curren -value. Now, he variable conains he numerical approimaion i of he soluion a he curren poin i, and Line compues he slope of he soluion a his poin by evaluaing he funcion given by fs a he poin wih coordinaes c(i) and. The variable k conains he resul of his slope compuaion. Then Line compues he Euler approimae soluion value a he ne poin c(i+). Line appends his value o he vecor of -values, and ha s i! Remark. In Iode Projec II, when you wrie your own numerical rouine for he Improved Euler Mehod, you can keep he framework of euler.m. You only need o change Line in Figure, and hen Lines which compue he updae formula. Remark (Opional: for advanced users). The module euler.m can be used wihou he Iode inerface. Figure shows an eample of his. The iniial value problem in his eample is d d = sin(), () =, and he Malab code in Figure compues and plos an Euler soluion wih sep size. on he inerval [, ] of -values. The firs line in he figure creaes a new funcion fn, wih he firs parameer being a sring conaining he epression for he funcion ( *sin() ), and he remaining wo parameers being he names of he independen and dependen variables. We have only discussed Euler s mehod for uniformly spaced -values, and so you may be wondering why he program compues he value of h a every sep. The answer is ha we wan our numerical rouine o work even if he vecor c is no uniformly spaced. Some versions of Ocave do no know inline funcions, bu he disribuion of Iode comes wih an approimaion of Malab s inline feaure ha works well enough for he purposes of Iode.

9 >> fn=inline( sin(),, ); >> c=:.:; >> c=euler(fn,,c); >> plo(c,c); Figure : An eample of euler.m in acion on is own. To ry his, firs make sure you sared Malab or Ocave from your Iode direcory, so ha i can find he file euler.m. Mahemaical epressions in Malab, Ocave For simple epressions, we use he usual keyboard characers: * means, (^-)/ means ( )/, pi means π. Buil-in funcions. ep() eponenial, e log() naural logarihm, ln log() base logarihm, log abs() absolue value, sqr() square roo, sign() signum funcion, which equals + if > if = if < sin() sinh() cos() rigonomeric cosh() hyperbolic an() funcions anh() rigonomeric co() ( in radians) coh() funcions Eample. sin(ep(y))^ means sin (e y ), acos(ep()^(-)) means arccos(e ).

10 A Observaions (no for handing in) Noes for Euler s mehod, sep size Noes for Euler s mehod, sep size. Noes for Euler s mehod, sep size. Noes for Euler s mehod, sep size. Noes for Runge Kua, sep size. Noes for d/d = sin( / ) (see Secion.): on your screen plo of he direcion field, locae he Euler poins (, ), (, ), (, ),... Graphically, hese are he corner poins in he Euler approimae soluion. which of hese Euler poins lie on he rue soluion curve? is he slope of he Euler segmen saring a = i he same as he slope of he rue soluion curve a = i? why or why no? is he Euler approimae soluion always below he rue soluion curve? does he error (heigh difference) beween he Euler approimae soluion and he rue soluion ge bigger wih every sep o he righ? Conras ha observaion wih wha happened in Figure for he differenial equaion d d =.