Numerical Derivatives by Symbolic Tools in MATLAB

Transcription

1 Numerical Derivatives by Symbolic ools i MALAB Mig-Gog Lee * ad Rei-Wei Sog ad Hsuag-Ci Cag mglee@cu.edu.tw * Departmet o Applied Matematics Cug Hua Uiversity Hsicu, aiwa Abstract: Numerical approaces o ordiary dieretial equatios (ODEs usually require Jacobia evaluatios ad sometimes iger order derivatives o system o equatios. e evaluatio o tese umerical derivatives ca eiter use algebraic derivatio or by te computer algebra system (CAS, suc as MAPLE V, but tese derivatios may eiter cause debuggig diiculty i programmig or memory swell i computatio. Fiite dierece ormulas icludig orward dierece, cetral dierece, ad backward dierece are used to derive te Jacobia wit dieret degree o accuracy. Symbolic evaluatio capability o MALAB is used to eace ast derivatio o umerical derivatives give i tis paper. e accuracy o tese umerical derivatives sows tat te evaluatio by symbolic computatio rom MALAB is possible to be equipped ito a umerical sceme o ordiary equatios.. Itroductio e calculatio o derivatives is ecessary i te umerical solutio o dieretial equatios, suc as te Jacobia matrix o Newto iteratios i most o te ordiary equatios solvers; ad iger order coeiciets i aylor series metods, etc. A variety o sotware icludig symbolic programs [5] ad automatic dieretiatio [,,4] ave bee itroduced ad used successully i te applicatio problems. I tis paper, we will use te ideas give i [7] ad to implemet i te MALAB eviromet [6]. We use te symbolic tool give i MALAB ad evaluate umerical derivatives o give uctios wit lexible order o accuracy. ese umerical derivatives ca urter be used i te solutios o ODEs by aylor series metods or oter umerical scemes. oug, Miletics ad Molárka [7] ave sow ow to use umerical derivatives i a sti ordiary equatio by aylor series metods, but we demostrated ow to implemet wit dieret combiatio to give accurate iger order derivatives. I orgaizatio o tis paper, Sectio gives some otatio o te dieretial equatios, Sectio gives some o te iger order umerical derivatives i matrix orms, Sectio 4 gives umerical results by implemet aylor series metod to a sti equatio. Coclusio is give at Sectio 5.. Matematical Formulatios Give a ordiary dieretial equatio y ( x = ( x, x, y ( x = y, x [ x, x + ], (. were x = [ x, x,..., y ( x] : R R. Give p suc tat uctios p + yi ( x C ([ x, x + ], or i =,...,, ad ( x, x = [ ( x, x, ( x, x,..., ( x, x]. e variable x may or may ot be explicitly expressed i te rigt-ad side uctio. I te case tat variable x is expressed explicitly i te Equatio., we ca deie a ew variable Y as ollows: LetY( x = [ x, y ( x, y ( x,..., y + ( x]: R R, te

2 dy ( x Y ( x = = [, ( Y( x,..., ( Y( x] = F( Y( x, (. dx werey ( x = x, Y ( x = y,..., Y( x = y. Assume i ( x, x, i =,...,, is a (p+ times dieretiable uctio, te te classical aylor series expressio [] at x + s about x ca be writte as, x + s = x + y ( x s + y ( x s + y ( x s !! p ( p p+ ( p+ s y ( x + s + s y ( x s, p! ( p +! ( p+ uctio y ( x s is evaluated at some ukow poit betwee x ad x s. (.. Numerical Derivative by Fiite Dierece We implemetig aylor series metod, some derivatives o rigt ad side uctio, i.e., Eq.. must be evaluated. I tis sectio, we will describe ow to use iite dierece to express iger order derivatives o Eq.... Fiite Dierece Expressios o Secod Order Derivatives From Eq.. or Eq.., a secod order derivative o y ca be expressed as: y ( x = ( x = [ ( Y( x,..., ( Y( x] = ( Y( x dy( x ( Y( x dy( x = [(,...,( ] = F. (.. Y dx Y dx Y Sice te irst compoet o uctio F is, te iger order derivatives o te irst compoet o tis uctio are all zero. As a result, we will oly use te otatio o Eq.. or simplicity. By implemetig orward dierecig, backward dierecig, ad cetral dierecig to approximate k ( x te partial derivatives,, or k =,,...,, various ormulas ca be obtaied. Matrix A is Y obtaied by orward dierece, matrix C is obtaied by cetral dierece, matrix D is obtaied by a extrapolatio ormula, ad tese partial derivatives are evaluated at x = x. A {[ ( x, y ( x,... y ( x,..., ( x, y ( x,... y ( x [ ( x, y ( x,... y ( x,..., ( x, y ( x,... y ( x ]],..., = [ + + ] [ [ ( x, y ( x,... y ( x,..., ( x, y ( x,... y ( x + + ] ( x, y ( x,... y ( x,..., ( x, y ( x,... y ( x ]} ] {[ [ (, (,... (,..., (, (,... ( C = x + y x y x x + y x y x ] [ ( x, y ( x,... y ( x,..., ( x, y ( x,... y ( x ],..., [ ( x, y ( x,... y ( x,..., ( x, y ( x,... y ( x [ + + [ ( x, y ( x,... y ( x,..., ( x, y ( x,... y ( x ] ]} ] (.. (..

3 D {[ ( x, y ( x,... y ( x,..., ( x, y ( x,... y ( x 6 6 ( x, y ( x,... y ( x,..., ( x, y ( x,... y ( x = [ ] [ ] + ( x, y ( x,... y ( x,..., ( x, y ( x,... y ( x + [ ] + [ ( x, y ( x,... y ( x,..., ( x, y ( x,... y ( x ]],..., [[ ( x, y ( x,... y ( x,..., ( x, y ( x,... y ( x ] 6 (, (,... (,..., (, (,... ( [ xy x y x x y x y x ] + + [ ( x, y ( x,... y ( x,..., ( x, y ( x,... y ( x ] + (..4 + [ ( x, x,... x +,..., ( x, x,... x + ]]} Because o te complex ormulatio o te expressios give at Equatios..-..4, we use otatios give at [7] to simply, let ki, F ( x, x = [ ( x, y ( x,..., y ( x + i,...,..., ( x, y ( x,..., y ( x + i,...], (..5 k k k =,...,, i =, ±, ±,..., ±, we k=, y ( x = x. By tis otatio, we ca rewrite Eqs...~..4 as ollows:,,,,,, A = [( F F, ( F F,..., ( F F ], (..6 [(,,, (,,,..., (,, C = F F F F F F ], (..7 [(, 6,,, D = F F + F + F,..., 6 (..8,,, ( F 6F + F, + F ]. e ollowig matrix E is obtaied by a special type o iite dierece, ad is give as, [(, 8,,, E = F F + 8 F F,..., (..9,, ( F 8F,, + 8 F F ].. ird Order Derivatives As i Sectio., we ca derive tird order derivative o variable y or te secod derivative o Eq.. as: Y = F = ( F = ( FF + ( F. (.. Y Y Y Y Similarly we also deie a ew uctio so as to simpliy otatios o te derived derivative uctios: kl, i F = (..., yk + i,..., yl +,..., kl, =,,...,, i, =, ±, ±..., (.. kk, ii F = (..., yk + i,..., kl, =,,...,,, i =, ±, ±.... Higer order derivatives were obtaied similarly as te secod order derivatives. e ollowig matrix G is ormulated by irst applyig te cetral dierece sceme to ave a secod order m

4 derivative uctio, ad te applyig a orward dierece sceme to ave a tird order derivative uctio. e iduced ormula is give as:,,,, [( m m m m Gm = F F F + F,..., mm, mm, mm, ( F F + F,..., (.. m, m, m, m, ( F F F + F ], m =,,...,. Aoter tird order derivative ormula H m ca be obtaied by implemetig cetral dierece approximatio cosecutively as:,,,, [( m m m m H m = F F F + F,..., 4 mm, mm, mm, 4( F F + F,..., (..4 m, ( F m, F m, F m, + F ], m =,,..., By usig ormulas G m or H m, te matrix o Y ca be approximated as: [ G, G,..., G ] [ H, H,..., H ] Y. (..5. e ourt Order Derivative By Eq..., te ourt derivative o uctio y or te tird order derivative matrix o Eq.. ca be writte as: IV Y = F = ( FF + ( ( F = FFF + ( F F + Y Y Y Y Y Y (.. +( F( F + ( FF + ( F Y Y Y Y Y I Eq..., is derived similarly as i sectio. by lower order derivative uctios, ad its Y ormula ca be writte as, i,, k, l =,,...,. YiY Yk Similarly, we also deie ew uctios to simpliy te otatio [7], ( Y = (..., y + m,... y + e,..., y + z,..., il, mez k k i l iil, mmz k ( Y = k(..., yi + m,..., yl + z,..., (.. iii, mmm ( Y = (..., y + m,..., k, l, i, =,,...,, me,, z=, ±, ±... k k i We te partial derivatives were implemeted wit tree dieret directios, a ormula is give as:

5 ,,,, [( i i i i i = F F F + F i, i, i, F + F F ],..., (..,, [( i i F F i, i, i, i, i, F + F F + F F ], i, =,,... I te partial derivatives were implemeted wit respect to two same directios, a ormula is give as: i, i, i, i, i, i, ( i = [F F F F + F + F ], i, =,,...,. (..4 I te partial derivative were implemeted wit respect to oly oe directio, a ormula is give as:,,,, ( = [ F + F + F F ], =,,...,. (..5 As or tese derivatives matrices or, it ca be obtaied by combig dieret ormulas, suc Y as orward ad cetral dierece ad te our-poit ormula E. We omit te procedures ere. 4. Numerical Derivatives ad Its Accuracy I tis sectio we will implemet umerical derivatives by iite dierece give i Sec. i a aylor series metod to a sti ordiary equatio. I Sectio 4., we will sow ow to acquire tese umerical derivatives i a lexible way, i.e., we ca calculate wit dieret combiatio rom te cetral dierecig, orward dierecig, ad oter ormulas to obtai more accurate derivatives. Some errors betwee exact derivatives ad umerical derivatives are give to sow teir accuracy. I Sectio 4., we will compare te umerical solutios obtaied by implemetig umerical derivative ad by exact derivatives to a sti ordiary dieretial equatio. 4. Accuracy o te Numerical Derivatives e secod order derivatives was give at Equatio.., te umerical derivative ca be obtaied by applyig Eqs...6~..9. For te tird order umerical derivative o Equatio., it will be give as ollows. Durig te implemetatio, a cetral dierece is irst used to get te secod order derivative, ad a orward dierece sceme is applied urter to get te tird order derivatives. e order o applyig iite dierece scemes is ot te critical issue, sice we ca still cage reely wit tese orderig to get accurate umerical derivatives. e above procedure was give oly to demostrate oe possibility o our procedure. As a result, or te above procedure, te tird order umerical derivative ca be writte as: m, m, m, m, G m = [( F F F + F,..., mm, mm, mm, mm, F F F + F,..., (4.. m, m, m, m, ( F F F F ] +, m =,,,.

6 ere is a mior dierece betwee G m ad G m wic is give at Eq.... Matrix G m is used i te implemetatio o tis paper. Similarly, i te ourt derivative o Equatio., we use cetral dierecig sceme ad backward dierecig sceme or easy implemetatio i programmig, a ormula is give we tree directios o partial dieretiatio are all dieret ad is give as, i, i, i, i, i = [( F F F + F i, i, i, i, i, i, F + F + F F ],...,( F F (4.. i, i, i, i, i, i, F + F F + F + F F ], i, =,,.... We tere are two same directios o partial dieretiatio ad a ormula is give as, i, i, i, i, ( i = [ F F F + F (4.. i, i, i, i, F + F + F F ]. We te directios o partial dieretiatio are all te same, a ormula is give as,,,,, ( = [ F F + F F ]. (4..4 Equatios 4..~4..4 are te ormulas or sow i te ourt derivatives. ese Y ormulas ca be implemeted wit te secod order ormulas A, B, C, D ad E, i tis paper we oly adapt matrices A ad C or coveiece. A sti dieretial equatio [6] is give to test te accuracy o te umerical derivatives o dieret orders. We will compare te accuracy o umerical derivatives o rigt ad side uctio wit te exact derivatives. d x = ( y ( x + y ( x y ( x y ( x qy ( x dt α dy ( x = m x x x x (4..5 dt d x = ( y ( x y ( x, = a, = b, = d, dt γ were α, m, q, ad γ are parameters, a, b ad d are iitial values. Letα =., m=.5, γ =, q=., a=5, b=7, ad d=8. Evaluate Eqs...6~..9, te ollowig error estimates o te irst derivative are estimated ad are give at able. Matrices A, C, D, ad E are all ave good accuracy. i able. We =.5, error o matrices A, C, D, E max( A Y max( C Y max( D Y max( E Y

7 Evaluate matrices o Eqs.4.., te ollowig error estimates o te secod order derivative are estimated ad are give at able. It sows tat matrix G as good accuracy. max ( ( G k k, YYk able. We =.5, error i calculatig G max ( ( G k k, YYk max ( ( G k k, YYk Evaluate matrices o Eqs. 4..~4..4, te ollowig error estimates o te tird order derivatives are estimated ad are give at able.. It sows tat matrix as good accuracy. max ( ( i k i,, k YiYkY able. We =.5, error i calculatig max ( ( i k i,, k YiYkY max ( ( i k i,, k YiYkY ese results sow tat te accuracy o te umerical derivatives all satisies te teoretical results give i [7]. 4. e Numerical Solutio by aylor Series Metod Figure. Numerical Solutios o Equatio 4..5 give by aylor Series Metod wit Numerical Derivatives

8 Followig te same algoritm give at [7], let te error cotrolε =. ad te iterval o itegratio is [, ]. e umerical solutios o aylor series metod by implemetig umerical derivatives give i Sec. 4 are sow i Figure above. e traces are solutios o tree compoets o te system o Equatio able 4. Numerical Values by aylor Series Metod wit ree ways to Obtai Derivatives Subiterval aylor Series Metod umerical values MALAB MAPLE Exact Derivative [.5767, ] [ ,.59885] [.59885, ] ŷ ŷ ŷ ŷ ŷ ŷ ŷ ŷ ŷ Durig te itegratio, we also examie te umerical solutios obtaied by applyig umerical derivatives ad exact derivatives i several subitervals to sow tat tese umerical derivatives do beave similarly wit te solutios obtaied by exact derivatives. ese umerical values eve exact at least up to te ourt digits ad are give at able 4. I te ollowig, we examie te umerical solutios by applyig aylor series metod ad by subroutie give i MALAB, te error betwee tese solutios matc well ad are give at able 5. able 5. Numerical Values by aylor Series Metod wit ree ways to Obtai Derivatives (Compared wit solutios give by implemetig Matlab Solutios Error betwee Subiterval MALAB ad aylor Series Metods [.5767, ] [ ,.59885] [.59885, ] x + s yˆ ( x + s k y 8.744e-7 y.54568e-7 y e-9 y e-7 y e-7 y e-8 y e-6 y e-6 y e-8 k

9 From ables 4 ad 5, at several subitervals o itegratio, umerical solutios obtaied by umerical derivative give i tis paper ad te exact derivative bot matc well. It meas tat te umerical derivatives derived rom applyig iite dierece scemes ca be applied durig umerical solutios o dieretial equatio eve it is sti. 5. Coclusio I tis paper, we ave demostrated ow to calculate umerical derivatives wit iite dierece sceme ad use te umeric to a aylor series metod to solve a sti ordiary equatio. ese umerical derivatives also ave good accuracy compared wit exact derivatives. oug alterative way o idig derivatives are also possible, by results rom tis paper, umerical derivatives by iite dierecig is still capable o rederig derivatives wit good accuracy. aylor series metod may ot be suitable or ast computatioal aspect, it requires may iger orders derivatives, but a possibility o parallel computatio may be a resolutio or te speed cocer. I additio, some results sow tat te aylor series metod is ot suitable or very sti problems, sice it is basically a explicit metod. I order to solve tis stability requiremet, a implicit type o aylor series metod may be approaced i te uture. Reereces [] Biso Cristia, Carle Ala, Havlad Paul, Kademi Peyvad, ad Mauer Adrew, ADIFOR. User s Guide (Revisio D, ecical Report ANL/MCS-M-9, Matematics ad Computer Sciece Divisio, Argoe Natioal Laboratory, USA, (994. [] Griewak Adreas, Evaluatig Derivatives-Priciples ad eciques o Algoritmic Dieretiatio, SIAM, Piladelpia, (. [] Kedall E. Atkiso, Itroductio to Numerical Aalysis, Jo Wiley & Sos Ltd., New York (978. [4] Lee, M.G., Applicatio o Automatic Dieretiatio i Numerical Solutio o a Flexible Mecaism, ICCMSE Iteratioal Coerece o Computatioal Metods i Scieces ad Egieerig, Kastoria, Greece, pp5-59. [5] MAPLE V, User Maual, e Waterloo Maple Ic., Caada (997 [6] MALAB, User Maual, e Matworks Ic., USA (6. [7] Miletics, E. ad Molárka G., aylor Series Metod wit Numerical Derivatives or Iitial Value Problems, Jouaal o Computatioal Metods i Scieces ad Egieerig,vol.,o. (, pp. 9-9.