Introduction to Numerical Differentiation and Interpolation March 10, !=1 1!=1 2!=2 3!=6 4!=24 5!= 120
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1 Itroducto to Numercal Deretato ad Iterpolato Marc, Itroducto to Numercal Deretato ad Iterpolato Larr Caretto Mecacal Egeerg 9 Numercal Aalss o Egeerg stems Marc, Itroducto Iterpolato s te use o a dscrete set o data pots (,, =,, N) to appromate te value o () or some tat s ot te data set Numercal deretato s te appromato o dervatves rom a smlar set o data Ote used as part more comple problems suc as umercal soluto o deretal equatos L: Dervatves o terpolato polomals ca be used as appromato to dervatves Loos at basc deas eac topc te Derece Grds ubdvde rego to dscrete pots pacg betwee te pots ma be uorm or o-uorm Eample: grd or m ma wt N+ odes umbered rom zero to N Ital ode value, = m al grd ode value, N = ma Nodal spacg Δ = - ( =, N) Uorm spacg, = Δ = ( m ma )/N N+ odes gve N spaces te Derece Grds II No-uorm grd llustrated below ~ ~ N- N- N D D D DN- D N Use te ollowg otato to represet te value o a ucto () at = () at = s represeted as ( ) or or uorm grds, all D = + = = costat so = + ad = ( + ) Dervatve Epressos Obta rom deretatg terpolato polomals or rom alor seres eres epaso or () about = a ( ) ( a) d d Note: d /d = ad! = d d ( a) ( - a) ( - a)... a! d! d a a!=!= d!= ( ) ( - a)! d!=6 a!= 5!=! = (-)(-)... ()()() 5 Wat s error rom trucatg seres? rucato Error I we trucate seres ater m terms ( ) m d! d a ( - a) m d! d a ( - a) erms used rucato error, e m Ca wrte trucato error as sgle term at uow locato,, (dervato based o te teorem o te mea) m d d m e m ( - a) ( - a) m! d ( m )! d m a Note derece dervatve locato 6 ME 9 - Numercal Aalss o Egeerg stems
2 Itroducto to Numercal Deretato ad Iterpolato Marc, Dervatve Epressos Loo at te-derece grd wt equal spacg: = D so = + ; = ( ) alor seres about = gves ( + ) = [ + (+)] = + terms o ( ) = ( ) ( ) d d d! d ( ) = + a = a = Compact dervatve otato d d... d d d! d d d ( )... 7 More Dervatve Notato Use subscrpt to dcate locato o dervatves at deret pots mlar to otato tat ( + ) = + wc s equvalet to [ + (+)] = + Epad results rom prevous slde to dcate deret locatos d d... d d d d... d d d d d d 8 Dervatve Epressos II Combe all detos or compact seres otato ( ) ( ) d d d! d ( ) d! d ( ) ( )...!! Use ts ormula to get epasos or varous grd locatos about = ad use results to get dervatve epressos ( )... 9 Dervatve Epressos III Appl geeral equato or = ad = ( )! ( )!...!!...!!... A orward!!...!! B Bacward. Dervatve Epressos IV ubtract + ad - epressos...!!!! 5...! 5!! 5!... A Result called cetral derece epresso... Order o te Error orward ad bacward dervatve ave error term tat s proportoal to Cetral derece error s proportoal to Error proportoal to called t order Reducg step sze b a actor o a reduces t order error b a e e ME 9 - Numercal Aalss o Egeerg stems
3 Itroducto to Numercal Deretato ad Iterpolato Marc, Order o te Error Notato Wrte te error term or t error term as O( ) Bg o otato, O, deotes order Recogzes tat actor multplg ma cage slgtl wt because locato o trucato error mgt cage rst order orward rst order bacward O( ) O( ) ecod order cetral O( ) Hger Order Dervatves Add + ad - epressos!!...!! !...!!! s secod-order, cetral derece epresso or secod dervatve 5! O... Hger Order Drectoal We ca get ger order dervatve epressos at te epese o more computatos Get secod order orward ad bacward dervatve epressos rom prevous results ad + ad -, respectvel Combe + ad - equatos wt prevous epressos or + ad - to elmate rst order error term 5 Geeral equato pecc alor eres ( )! ( )!... = 8...!! = !! = !! = !! 6 ecod Order orward ubtract + rom + to elmate term ecod order... 6 error, O(... ) 7 ecod Order Bacwards Add - to - to elmate term ecod... 6 order error,... O( ) 8 ME 9 - Numercal Aalss o Egeerg stems
4 Itroducto to Numercal Deretato ad Iterpolato Marc, Oter Dervatve Epressos Ca cotue ts aso Wrte alor seres or +, -, +, -, +, -, etc. Create lear combatos wt actors tat elmate desred terms Elmate term to obta cetral derece Keep ol terms wt or orward derece epressos Keep ol terms wt or orward derece epressos ee results o pages 5 ad 55 o Rao 9 Dervatve Epressos O O Note order o dervatve, order o error, ad drecto (orward vs. bacward) O O O O Eample Problem Use a secod-order cetral-derece epresso to estmate d[s()]/d = usg =. Wat s te correct te-derece epresso? O Wat are + ad -? Recall deto + = ( + ) Usg epresso or dervatve at = meas tat = ( ) must be at = Eample Problem Cotued I = wat are + ad -? Recall deto + = ( + ) I =, + = + () = + = =. ad + ( ) = = =. s s s. s d[s()]/d = = cos() = = so error s.998 = Repeat Eample or =. Here we ave te same epresso we derved prevousl, () = [s() s(-)]/(), but =. Prevous result: or =., error = e result s ow s s s. s Here error s = Reducg b. reduces error b. or secod-order error Geeral Result I a te-derece epresso Wt a error o order (e A ) Reducg te step sze b a actor a ould reduce te error b a actor a ME 9 - Numercal Aalss o Egeerg stems
5 Itroducto to Numercal Deretato ad Iterpolato Marc, Eercse d te secod dervatves o e at = or =. ad =. (equato below) Compare to eact results de /d = = d e /d = = e = = e = Estmate order o umercal results rom equato le e e e l Possble quz questo smlar to ts oe O 5 Roudo Error Possble dervatve epressos rom subtractg close dereces Eample () = e : () (e + e - )/() ad error at = s (e + e - )/() e E (.) ecod order error E (.) E (.) 6 Error.E+.E+.E-.E-.E-.E-.E-5.E-6.E-7.E-8.E-9.E- gure -. Eect o tep ze o Error Numercal dervatve zero at smallest step szes rucato error: lear d log(error) / d log () or large step szes Roudo error at smaller step szes.e-.e-7.e-5.e-.e-.e-9.e-7.e-5.e-.e- Derece Epressos orward dereces or costat Geeral deto: D = + ecod order deto: D = D(D ) = D( + ) = D + D = ( + + ) ( + ) = Ca ou sow tat te ollowg s correct? D = I geeral, D = D - + D - tep ze 7 8 Derece Epressos II Bacward dereces or costat Geeral deto: = - ecod order deto: = ( ) = ( - ) = - = ( - ) ( - - ) = Ca ou sow tat te ollowg s correct? = I geeral, = Derece Epressos III Cetral dereces or costat Geeral deto: d = +/ -/ ecod order deto: d = d(d) = d( +/ -/ ) = d +/ d -/ = ( + ) ( - ) = Ca ou sow tat te ollowg s correct? d = +/ +/ + -/ -/ Geeral recurso ormula or odd ad eve deltas: Odd: d + +/ = d + d Eve: d = d - +/ d - -/ 9 ME 9 - Numercal Aalss o Egeerg stems 5
6 Itroducto to Numercal Deretato ad Iterpolato Marc, Iterpolato tart wt N data pars, d a ucto (usuall a polomal) tat ca be used or terpolato N data pots gve polomal order N Basc rule: te terpolato polomal must t all pots eactl Deote te polomal as p() e basc rule s tat p( ) = Ma deret orms Iterpolato Data Eample Here we ave s data pars Eample problem: Wat s te value o we =? Ca use deret umbers o data pars rom to 6 or polomal p(), te basc rule s tat p( ) = E.g. p() = Wat s p()? p() = Newto Polomals p() = a + a ( ) + a ( )( ) + data pots requred or t order polomal + a ( )( )( ) + + a ( )( )( ) ( - ) erms wt actors o are zero we = Use ts ad rule tat p( ) = to d a a =, a = ( ) / ( ) = a + a ( ) + a ( )( ) olve or a usg results or a ad a Newto Polomals II = a + a ( ) + a ( )( ) a a( ) a ( )( ) ( ) ( )( ) Could cotue ts aso to determe coecets rom data Use alteratve sceme ot derved ere ow as dvded derece table to compute a rom same data Dvded Derece able Eter data o ad rows o table sppg oe row betwee etres tart wt data as zerot dvded derece rst dvded derece, = ( + ) / ( + ) ecod (or later) dvded derece s derece o rst (or later) dereces a coecets are tal dvded dereces 5 Zerot derece Dvded Derece able a rst ecod rd derece derece derece a a a 6 ME 9 - Numercal Aalss o Egeerg stems 6
7 Itroducto to Numercal Deretato ad Iterpolato Marc, ME 9 - Numercal Aalss o Egeerg stems 7 7 Dvded Derece Eample a a a a Dvded Derece Eample II Dvded derece table gves a =, a =, a =., ad a = /6 Polomal p() = a + a ( ) + a ( )( ) + a ( )( )( ) = + ( ) +.( )( ) + (/6)( )( )( ) = +.( ) + (/6)( )( ) Cec p() = +.()() + (/6) ()()() = = (Correct!) Dvded Derece Eercse Loo at prevous dvded derece eample Add ew data pot Determe terpolato coecets Add ew data pot at, Determe dvded derece all ew dvded derece terms out to ourt derece 9 Wat s? a a Wat s? a a Wat s? a a
8 Itroducto to Numercal Deretato ad Iterpolato Marc, d ourt Dvded Derece oluto a a a a R a Dvded Derece Patter Get geeral ormula or code wt D(m,) as m t value o t dvded derece Ital m data are te zerot dvded derece, D(m,) rst dvded derece, m = ( m+ m ) / / m+ m ) s D(m,) = [D(m+,) D(m,)] / m+ m ) ecod dvded derece, m s D(m,) = ( m+ m ) / m+ m ) = [D(m+,) D(m,)] / m+ m ) 5 Dvded Derece Patter II rom prevous slde we see tat D(m,) s m t value o t dvded derece We wat code to rom terpolatg polomal or N data pots ( to N ) We start b settg zerot derece as tal data,.e.: D(m,) = m I we cotue ormulas or deret dvded dereces we wll see tat te ollowg ormula s true D(m,) = [D(m+, ) D(m, )] / m+ m ) 6 Dvded Derece Code or m = o N N+ data pots D(m, ) = (m) zerot d Net m or = o N or m = o N - D(m, ) = (D(m +, - ) - _ D(m, - )) / _ ((m + ) - (m)) Net m Net D(m,) s m t value o t dvded derece 7 Usg Code Ecel Arras mported rom Ecel set put rage to varat (lower boud = ) ub model(i as Rage) Dm as Varat Dm rows as Log, cols as Log = I rows = Uboud(,) cols = Uboud(,) Have to revse code to adle ts arra or cop arra to oe-dmesoal arra wt lower boud o zero 8 ME 9 - Numercal Aalss o Egeerg stems 8
9 Itroducto to Numercal Deretato ad Iterpolato Marc, Costat tep ze Dvded dereces wor or equal or uequal step sze I D = s a costat we ave smpler results = D / = ( + )/ = D / = ( )/ = D /6 = ( )/6 D s called te t orward derece Ca also dee bacwards ad cetral dereces Iterpolato Approaces We we ave N data pots ow do we terpolate amog tem? Order N- polomal ot good coce Use pecewse polomals o lower order (lear or quadratc) Ca matc rst ad or ger dervatves were pecewse polomals jo Cubc sples are pecewse cubc polomals tat matc rst ad secod dervatves (as well as values) Cubc ple Iterpolato 5 Newto Iterpolatg Polomal.6 value.5... Kow Natural No Kot Data Y Value Polomal Data. 5 6 values X Values 5 5 Polomal Applcatos Data terpolato Appromato uctos umercal quadrature ad soluto o ODEs Bass uctos or te elemet metods Ca obta equatos or umercal deretato tatstcal curve ttg (ot dscussed ere) usuall used practce t Quz Results Mamum possble score: 5 Number o studets: Mea score:. Meda score: 5.5 tadard devato: 7. Grade dstrbuto: ME 9 - Numercal Aalss o Egeerg stems 9
10 Itroducto to Numercal Deretato ad Iterpolato Marc, Quz ve Commets Uorm Grds We let sde s te same, ca solve bot rgt sdes at same tme Watc roudo calculator Lear to use umbers wtout smbols to represet equatos Caot use terato uless arra s dagoall domat Quz s moved rom ts Wedesda to et Wedesda 55 Uorm grd otato: = ormulas wt terms represet ( ) Equatos le te ollowg O O Gve te dervatve at = terms o values, = ( ) at grd locatos 56 ME 9 - Numercal Aalss o Egeerg stems
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