NUMERICAL DIFFERENTIATION

Transcription

1 EP08 Computational Metods in Psics Notes on Lecture Numerical Dierentiation 1 NUMERICAL DIFFERENTIATION For a unction irst and second derivatives are given as ollows: Metod First Derivative Truncation Error order FDA CDA REA 4 Metod Second Derivative Error order FDA CDA REA 4 Partial Derivatives or a unction [CDA onl] ere is small but not ero ' '' 4 ' ' ''

2 EP08 Computational Metods in Psics Notes on Lecture Numerical Dierentiation EXAMPLE Comparison o te numerical irst derivative metods or = e at = 3 wit = 0.01 Metod Error True FDA single precision % CDA single precision % REA single precision % C Pseudo code to compute te irst derivative o a unction. C Te FDA CDA and REA metods are implemented or comparison. unction deinition =... input "input " input "input " da = +-/ cda = +--/* rea = -*-8*-+8*+-+*/1* output "FDA = " da output "CDA = " cda output "REA = " rea Fortran PROGRAM FirstDerivative IMPLICIT NONE REAL :: da cda rea PRINT *"Input " READ * PRINT *"Input " READ * da = +-/ cda = +--/* rea = -*-8*- + & 8*+-+*/1* PRINT *"FDA = " da PRINT *"CDA = " cda PRINT *"REA = " rea CONTAINS REAL FUNCTION F REAL INTENTIN :: F = EXP-* END FUNCTION END PROGRAM FirstDeivative Input 3 Input 0.01 FDA = CDA = REA = C++ #include <iostream> #include <cmat> using namespace std; loat loat { loat = ep-*; return ; int main{ loat da cda rea; cout << "Input "; cin >> ; cout << "Input "; cin >> ; da = +-/; cda = +--/*; rea = -*-8*- + 8*+-+*/1*; cout << "FDA = " << da << endl; cout << "CDA = " << cda << endl; cout << "REA = " << rea << endl; return 0; Input 3 Input 0.01 FDA = CDA = REA =

3 EP08 Computational Metods in Psics Notes on Lecture Numerical Dierentiation EXAMPLE Comparison o te numerical second derivative metods or = 3sin + at = π/4 wit = 0.01 Metod Error True CDA double precision % REA double precision C Pseudo code to compute te second derivative o a unction. C Te FDA CDA and REA metods are implemented or comparison. unction deinition =... input "input " input "input " cda = / rea = / 1 output "CDA = " cda output "REA = " rea Fortran PROGRAM SecondDerivative IMPLICIT NONE INTEGER PARAMETER :: K = 8 REALKIND=K :: cda rea = _K/4.0_K = 0.01_K cda =--*++/** rea =--*+ 16*- - & 30* + 16*+ - & +*/1*** PRINT *"CDA = " cda PRINT *"REA = " rea CONTAINS REAL KIND=K FUNCTION F REALKIND=K INTENTIN :: F = 3*sin + ** END FUNCTION END PROGRAM SecondDerivative C++ #include <iostream> #include <cmat> using namespace std; #deine Real double Real Real { Real = 3.0*sin + *; return ; int main{ Real cda rea; = M_PI/4.0; = 0.01; cda = --*++/*; rea = --* + 16*- - 30* + 16*+ - +*/1**; cout << "CDA = " << cda << endl; cout << "REA = " << rea << endl; return 0; i KIND = 4 CDA = REA = i KIND = 8 CDA = REA = i Real is loat CDA = REA = i Real is double CDA = REA =

4 EP08 Computational Metods in Psics Notes on Lecture Numerical Dierentiation Errors in Numerical Derivative Te approimation metods FDA CDA and REA can be used to demonstrate te eect o truncation errors and round-o errors. Te error or eample /6.``` inerent to te CDA metod is an eample o a truncation error i.e. b truncating iger order terms in te Talor epansion te metod becomes onl approimate. Anoter source o error eists wen te FDA CDA or REA are computed; tis is te round-o error due to limited precision in numerical aritmetic numerical values are stored in te computer wit a limited number o binar bits. Round-o errors are compounded in aritmetic operations. Te total error is tereore a combination o te two error sources: Total Error = Truncation Error + Round-o Error due to truncating iger order terms in Talor epansion due to limited precision in numerical aritmetic Te important parameter ere is te value o ; te truncation error increases wit increasing te round-o error decreases wit increasing Given a particular metod or eample te CDA te most accurate computed derivative is obtained b minimiing te total error tis corresponds to inding te optimal value o. Tis optimal value will dier depending on i. Te numerical metod FDA CDA REA etc. ii. Te unction being dierentiated and te value o. iii. Te precision o te aritmetic single- double- quad-precision. To arrive at te optimal value some stud o te output o our program is needed. Te total error in te CDA is given b: Error = CDA ` We can orm a plot o Error against log to indicate te eect o on error See Fig.1. A minimum error eists at some intermediate value o corresponding to a minimum in te plot. I ` is unknown we can onl plot CDA versus but as ' is a constant te plot will ave te same sape onl sit up or down. In tis case again a minimum or stationar value in te plot will be observed corresponding to a minimum error. 4

5 EP08 Computational Metods in Psics Notes on Lecture Numerical Dierentiation Fig.1: Error = CDA- vs plots or te unction = e at = 3 or single- and double-precision aritmetic. Optimum values o corresponding to minimum error are about 10 and 10 5 respectivel. Te rise on te let as increases is due to truncation error wic as te orm o and te rise on te rigt as decreases is due to round-o errors. 5

6 EP08 Computational Metods in Psics Notes on Lecture Numerical Dierentiation Eercise 1. Consider te position o a particle in meters is given b unction t = 1/t. [ v = t = 1/t and a = t= -/t 3 ] 1 st derivative o te unction at t=3 s is 3 = m/s nd derivative o te unction at t=3 is 3 = m/s a For te irst derivative compare te accurac o FDA CDA and REA. For tis use = 0.01 s and double precision data tpe. b For te second derivative compare te accurac o FDA CDA and REA. For tis use = 0.01 s and double precision data tpe.. Write a program to ind te partial derivatives / and / at = = 1 or te unction = + sin/. 6