Numerical Differentiation

Transcription

1 College o Egeerg ad Computer Scece Mecacal Egeerg Departmet Numercal Aalyss Notes November 4, 7 Istructor: Larry Caretto Numercal Deretato Itroducto Tese otes provde a basc troducto to umercal deretato usg te- derece grds. Tey cosder te terplay betwee trucato error ad roudo error. Fte-derece grds I a te-derece grd, a rego s subdvded to a set o dscrete pots. Te spacg betwee te pots may be uorm or o-uorm. For eample, a grd te drecto, m ma may be wrtte as ollows. Frst, we place a seres o N+ odes umbered rom zero to N ts rego. Te coordate o te rst ode, equals m. Te al grd ode, N = ma. Te spacg betwee ay two grd odes, ad -, as te symbol Δ. Tese relatos are summarzed as equato []. = m N = ma - = Δ [] A o-uorm grd, wt deret spacg betwee deret odes, s llustrated below ~ ~ N- N- N For a uorm grd, all values o Δ are te same. I ts case, te uorm grd spacg, a oedmesoal problem s usually gve te symbol. I.e., = - or all values o. I tese otes, we wll lmt our cosderato to oe-dmesoal te-derece problems. However, advaced courses cosder multple space dmesos dscussed equatos [] ad [] below. I two space dmesos a grd s requred or bot te ad y, drectos, wc results te ollowg grd ad geometry detos, assumg tat tere are M+ grd odes te y drecto. = m N = ma - = Δ y = ymj ym = ymay yj yj- = Δyj [] For a tree-dmesoal traset problem tere would be our depedet varables: te tree space dmesos,, y ad z, ad tme. Eac o tese varables would be deed at dscrete pots,.e. = m N = ma - = Δ y = ymj ym = ymay yj yj- = Δyj [] z = zmk zk = zmaz zk zk- = Δzk t = tm tl = tmay t t- = Δt Jacarada Hall 4 Mal Code Poe: N/A Emal: lcaretto@csu.edu 848 Fa:

2 Numercal Deretato L. S. Caretto, November 4, 7 Page Ay depedet varable suc as u(,y,z,t) a cotuous represetato would be deed oly at dscrete grd pots a te-derece represetato. Te ollowg otato s used or a oedmesoal problem. ) [4] k Ts otato ca be eteded to problems more ta oe dmeso cludg traset problems. I te most comple case te otato u, y, z, t ) ( k u s used to deote te jk ( j k value o te depedet at a partcular pot te rego, (, yj, zk, t) were te varable s deed. Fte-derece Epressos Derved rom Taylor Seres Te Taylor seres provdes a smple tool or dervg te-derece appromatos. It also gves a dcato o te error caused by te te derece epresso. Recall tat te Taylor seres or a ucto o oe varable, (), epaded about some pot = a, s gve by te te seres, ( ) ( a) d a ( a)! d a ( - a)! d a ( - a)... [5] Te = a subscrpt o te dervatves reorces te act tat tese dervatves are evaluated at te epaso pot, = a. We ca wrte te te seres usg a summato otato as ollows: d ( ) ( - a) []! a I te equato above, we use te detos o! =! = ad te deto o te zerot dervatve as te ucto tsel. I.e., d / =a = (a). I te seres s trucated ater some te umber o terms, say m terms, te omtted terms are called te trucato error. Tese omtted terms are also a te seres. Ts s llustrated below. ( ) m d d ( - a) a ( - )! m a! a Terms used Trucato error [7] I ts equato te secod sum represets te trucato error, εm, rom trucatg te seres ater m terms. d m ( - a) [8]! m Te teorem o te mea ca be used to sow tat te te-seres trucato error ca be epressed terms o te rst term te trucato error, tat s a

3 Numercal metods ME 9, L. S. Caretto, November 4, 7 Page m d m ( - a) m )! m [9] ( m Here te subscrpt, = ξ, o te dervatve dcates tat ts dervatve s o loger evaluated at te kow pot = a, but s to be evaluated at = ξ, a ukow pot betwee ad a. Tus, te prce we pay or reducg te te seres or te trucato error to a sgle term s tat we lose te certaty about te pot were te dervatve s evaluated. I prcple, ts would allow us to compute a boud o te error by dg te value o ξ, betwee ad a, tat made te error computed by equato [9] a mamum. I practce, we do ot usually kow te eact uctoal orm, (), let aloe ts (m+) t dervatve. I usg Taylor seres to derve te basc te-derece epressos, we start wt uorm oedmesoal grd spacg. Te derece, Δ, betwee ay two grd pots s te same ad s gve te symbol,. Ts uorm grd ca be epressed as ollows. Δ = - = or = + or all =,,N [] Varous cremets at ay pot alog te grd ca be wrtte as ollows: + - = + = - = + = - + = + = [] Usg te cremets deed above ad te otato = () te ollowg Taylor seres ca be wrtte usg epaso about te pot = to epress te values o at some specc grd pots, +, -, + ad -. Te covetoal Taylor seres epresso or () equato [5] ca be adapted or use te dereces by wrtg a epaso equato about a partcular grd pot, =, to determe te value o () at aoter grd pot, +k. From equato [], we see tat +k = + k so tat (+k) = ( + k). Te derece,, te depedet varable,, betwee te evaluato pot, + k, ad te epaso pot,, s equal to k. Usg = a as te epaso pot ad k as allows us to rewrte equato [5] as sow below. d d d ( k) ( ) k ( k) ( k)... []!! Te et step s to use te otato tat ( + k) = +k, ad te ollowg otato or te t dervatve, evaluated at =. d d... d [] Wt tese otatoal cages, te Taylor seres equato [] ca be wrtte as ollows. ( k)! ( k)! k k... [4] Fte-derece epressos or varous dervatves ca be obtaed by wrtg te Taylor seres sow above or deret values o k, combg te results, ad solvg or te dervatve. Te smplest eample o ts s to use oly te seres or k =.

4 Numercal Deretato L. S. Caretto, November 4, 7 Page 4... [5] We ca rearrage ts equato to solve or te rst dervatve, ; recall tat ts s te rst dervatve at te pot =.... O( ) [] Te rst term to te rgt o te equal sg gves us a smple epresso or te rst dervatve; t s smply te derece te ucto at two pots, (+) (), dvded by, wc s te derece betwee tose two pots. Te remag terms te rst orm o te equato are a te seres. Tat te seres gves us a equato or te error tat we would ave we used te smple te derece epresso to evaluate te rst dervatve. Represetg te trucato error as te order o te error As oted above, we ca replace te te seres or te trucato error by te leadg term tat seres. Remember tat we pay a prce or ts replacemet; we o loger kow te pot at wc te leadg term s to be evaluated. Because o ts we ote wrte te trucato error as sow te secod equato. Here we use a captal o ollowed by te grd sze pareteses. I geeral, te grd sze s rased to some power. (Here we ave te rst power o te grd sze, =.) For a trucato error proportoal to te t power o te step sze we would use te otato, O( ). Ts otato tells us ow te trucato error depeds o te step sze. Te order o te error depedece o te step sze s a mportat cocept. I te error s proportoal to, cuttg al would cut te error al. I te error s proportoal to, te cuttg te step sze al would reduce te error by ¼. We te trucato error s wrtte wt ts O( ) otato, we call te order o te error. I two calculatos, wt step szes ad, we epect te ollowg relato betwee te trucato errors, ε ad ε or te calculatos. [7] We use te appromato sg ( ) rater ta te equalty sg ts equato because te error term also cludes a ukow actor o some ger order dervatve, evaluated at some ukow pot te rego. Te appromato sow equato [7] would be a equalty ts oter actor were te same or bot step szes. Aoter mportat dea about te order o te error s tat a t order te-derece epresso wll gve a eact value or te dervatve o a t order polyomal. Because a Taylor seres s a polyomal seres, t ca represet a polyomal eactly a sucet umber o terms are used. Ts s llustrated urter below. Te epresso or te rst dervatve tat we derved equato [] s sad to ave a rst order error. We ca obta a smlar te derece appromato by wrtg te geeral seres equato [4] or k = -. Ts gves te ollowg result.... [8]

5 Numercal metods ME 9, L. S. Caretto, November 4, 7 Page 5 We ca rearrage ts equato to solve or te rst dervatve, ; recall tat ts s te rst dervatve at te pot =.... O( ) [9] Here aga, as equato [], we ave a smple te-derece epresso or te rst dervatve tat as a rst-order error. Te epresso equato [] s called a orward derece. It gves a appromato to te dervatve at pot terms o values at tat pot ad pots orward ( te + drecto) o tat pot. Te epresso equato [9] s called a backwards derece or smlar reasos. Dervatve epressos wt ger order errors A epresso or te rst dervatve tat as a secod-order error ca be oud by subtractg equato [8] rom equato [5]. We ts s doe, terms wt eve powers o cacel gvg te ollowg result [] Solvg ts equato or te rst dervatve gves te ollowg result O( ) [] Te te-derece epresso or te rst dervatve equato [] s called a cetral derece. Te pot at wc te dervatve s evaluated,, s cetral to te two pots (+ ad -) at wc te ucto s evaluated. Te cetral derece epresso provdes a ger order (more accurate) epresso or te rst dervatve as compared to te orward or backward dervatves. Tere s oly a small amout o etra work (a dvso by ) gettg ts more accurate result. Because o ter ger accuracy, cetral dereces are usually preerred te derece epressos. Cetral derece epressos are ot possble at te start o ed o a boudary. It s possble to get ger order te derece epressos or suc pots by usg more comple epressos. For eample, at te start o a rego, =, we ca wrte te Taylor seres equato [4] or te rst two pots rom te boudary, ad, epadg aroud te boudary pot,.... [] () () ()... [] Tese equatos ca be combed to elmate te terms. To start, we multply equato [] by 4 ad subtract t rom equato [].

6 Numercal Deretato L. S. Caretto, November 4, 7 Page 4 () () Ts equato ca be smpled as ollows ()... 4 ( ) ( ) ( ).. ( ) 4 () 4... [4] We ts equato s solved or te rst dervatve at te start o te rego a secod order accurate epresso s obtaed O( ) [5] A smlar equato ca be oud at te ed o te rego, = N, by obtag te Taylor seres epasos about te pot = N, or te values o () at = N- ad = N-. Ts dervato parallels te dervato used to obta equato [5]. Te result s sow below. N N 4 N N N 4 N N N... O( ) [] Equatos [5] ad [] gve secod-order accurate epressos or te rst dervatve. Te epresso equato [5] s a orward derece; te oe equato [] s a backwards derece. Te evaluato o tree epressos or te rst dervatve s sow Table. Tese are () te secod-order, cetral-derece epresso rom equato [], () te rst-order, orwardderece rom equato [], ad () te secod-order, orward-derece rom equato [5]. Te rst dervatve s evaluated or () = e. For ts ucto, te rst dervatve, d/ = e. Sce we kow te eact value o te rst dervatve, we ca calculate te error te te derece results. I Table, te results are computed or tree deret step szes: =.4, =. ad =.. Te table also sows te rato o te error as te step sze s caged. Te et-to-last colum sows te rato o te error or =.4 to te error or =.. Te al colum sows te rato o te error or =. to te error or =.. Table Tests o Fte-Derece Formulae to Compute te Frst Dervatve () = ep() () =.4 =. =. Error Ratos Eact () () Error () Error () Error (=.4)/ (=.)/ (=.) (=.) Results usg secod-order cetral dereces

7 Numercal metods ME 9, L. S. Caretto, November 4, 7 Page Results usg rst-order orward dereces Results usg secod-order orward dereces For te secod-order ormulae, te error ratos te last two colums o Table - are about 4, sowg tat te secod-order error creases by a actor o 4 as te step sze s doubled. For te rst order epresso, tese ratos are about. Ts sows tat te error creases by te same actor as te step sze or te rst order epressos. Te epected values o te error ratos are oly obtaed te lmt o very small step szes. We see tat te values te last colum o ts table (were te actual values o are smaller ta tey are te et-to-last colum) are closer to te deal error rato. Te tecques tat ave bee used ere to derve orward, backward, ad cetral dervatve epressos wt rst- ad secod-order error ca be epaded to cosder ger order dervatves ad ger order errors. Lsts o varous te-derece ormulas ca be oud umercal aalyss tets. Roudo error Trucato errors are ot te oly kd o error tat we ecouter te derece epressos. As te step szes get very small te terms te umerator o te te derece epressos become very close to eac oter. We lose sgcat gure we we do te subtracto. For eample, cosder te prevous problem o dg te umercal dervatve o () = e. Pck = as te pot were we wat to evaluate te dervatve. Wt =. we ave te ollowg data or calculatg te dervatve by te cetral-derece ormula equato []. ( ) ( ) ( ) (.)

8 Numercal Deretato L. S. Caretto, November 4, 7 Page 8 Sce te rst dervatve o e s e, te correct value o te dervatve at = s e =.788; so te error ts value o te rst dervatve s For =., te umercal value o te rst dervatve s oud as ollows. ( ) ( ) ( ) (.) Here, te error s Ts looks lke our secod-order error. We cut te step sze by a actor o, ad our error decreased by a actor o,,, as we would epect or a secod order error. We are startg to see potetal problems te subtracto o te two umbers te umerator. Because te rst our dgts are te same, we ave lost our sgcat gures dog ts subtracto. Wat appes we decrease by a actor o, aga? Here s te result or = -7. ( ) ( ) ( ) (.) Our trucato aalyss leads us to epect aoter actor o oe mllo te error reducto as we decrease te step sze by,. Ts sould gve us a error o However, we d tat te actual error s We see te reaso or ts te umerator o te te derece epresso. As te derece betwee (+) ad (-) srks, we are takg te derece o early equal umbers. Ts kd o error s called roudo error because t results rom te ecessty o a computer to roud o real umbers to some te sze. (Tese calculatos were doe wt a ecel spreadseet wc as about 5 sgcat gures. Fgure - sows te eect o step sze o error or a large rage o step szes. For te large step szes to te rgt o Fgure -, te plot o error versus step sze appears to be a stragt le o ts log-log plot. Ts s cosstet wt equato [7]. I we take logs o bot sdes o tat equato ad solve or, we get te ollowg result. log log log( ) log( ) log( ) log( ) [7] Equato [7] sows tat te order o te error s just te slope o a log(error) versus log() plot. I we take te slope o te stragt-le rego o te rgt o Fgure -, we get a value o appromately two or te slope, cormg te secod order error or te cetral derece epresso tat we are usg ere. However, we also see tat as te step sze reaces about -5, te error starts to level o ad te crease. At very small step szes te umerator o te te-derece epresso becomes zero o a computer ad te error s just te eact value o te dervatve.

9 Error Numercal metods ME 9, L. S. Caretto, November 4, 7 Page 9 Fgure. Eect o Step Sze o Error.E+.E+.E-.E-.E-.E-4.E-5.E-.E-7.E-8.E-9.E-.E-.E-8.E-.E-4.E-.E-.E-8.E-.E-4.E-.E+ Step Sze Fal Observatos o Fte-Derece Epressos rom Taylor Seres Te otes above ave ocused o te geeral approac to te dervato o te-derece epressos usg Taylor seres. Suc dervatos lead to a epresso or te trucato error. Tat error s due to omttg te ger order terms te Taylor seres. We ave caracterzed tat trucato error by te power or order o te step sze te rst term tat s trucated. Te trucato error s a mportat actor te accuracy o te results. However, we also saw tat very small step szes lead to roudo errors tat ca be eve larger ta trucato errors. Te use o Taylor seres to derve te derece epressos ca be eteded to ger order dervatves ad epressos tat are more comple, but ave a ger order trucato error. Oe epresso tat wll be mportat or subsequet course work s te cetral-derece epresso or te secod dervatve. Ts ca be oud by addg equatos [5] ad [8] [8] We ca solve ts equato to obta a te-derece epresso or te secod dervatve.... O( ) [9] Altoug we ave bee dervg epressos ere or ordary dervatves, we wll apply te same epressos to partal dervatves. For eample, te epresso equato [9] or te secod dervatve could represet d / or /. Te Taylor seres we ave bee usg ere ave cosdered as te depedet varable. However, tese epressos ca be appled to ay coordate drecto or tme.

10 Numercal Deretato L. S. Caretto, November 4, 7 Page Altoug we ave used Taylor seres to derve te te-derece epressos, tey could also be derved rom terpolatg polyomals. I ts approac, oe uses umercal metods or developg polyomal appromatos to uctos, te takes te dervatves o te appromatg polyomals to appromate te dervatves o te uctos. A te-derece epresso wt a t order error tat gves te value o ay quatty sould be able to represet te gve quatty eactly or a t order polyomal. * Te epressos tat we ave cosdered are or costat step sze. It s also possble to wrte te Taylor seres or varable step sze ad derve te derece epressos wt varable step szes. Suc epressos ave lower-order trucato error terms or te same amout o work computg te te derece epresso. I solvg deretal equatos by te-derece metods, te deretal equato s replaced by ts te derece equvalet at eac ode. Ts gves a set o smultaeous algebrac equatos tat are solved or te values o te depedet varable at eac grd pot. Fte derece epressos ca be derved rom Taylor seres. Ts approac leads to a epresso or te trucato error tat provdes us wt kowledge o ow ts error depeds o te step sze. Ts s called te order o te error. I te-derece approaces, we eed to be cocered about bot trucato errors ad roudo errors. Roudo errors were more o a cocer earler computer applcatos were lmtatos o avalable computer tme ad memory restrcted te sze o real words, or may practcal applcatos, to bts. Ts correspods to te sgle precso type Fortra or te Sgle type VBA. Wt moder computers, t s possble to do route calculatos usg 4-bt (or ger precso) real words. Ts correspods to te double precso type Fortra * or te double type VBA. Te -bt real word allows about 7 sgcat gures; te 4-bt real word allows almost sgcat gures. * I a secod order polyomal s wrtte as y = a + b + c ; ts rst dervatve at a pot = s gve by te ollowg equato: [dy/]= = b + c. I we use te secod-order cetral-derece epresso equato [] to evaluate te rst dervatve, we get te same result as sow below: dy y( ) y( b c( ) a b( ) c( ) c( ) [ a b( ) b 4c b c ) c( ) * Also kow as real(8) or real(kind=8) Fortra 9 ad later versos; sgle precso s typed as real, real(4) or real(kind=4) tese versos o Fortra. ]