mase201sp09_Mar03_PRINTABLE

# mase201sp09_Mar03_PRINTABLE - Numerical Differentiation and...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Numerical Differentiation and Integration • Numeric Differentiation Topics (Tues 3/3): – Finite difference formulas • Forward, Backward and Central Finite Difference • 2 nd derivatives • diff function – Curve fitting before taking derivative olyder nction (sec 8.1.4) 3/03/2009 1 • polyder function (sec 8.1.4) – MATLAB built-in function for numeric differentiation: gradient – Partial differentiation • Numerical Integration topics (Tue 3/17) MASE201 Spring 2009 Definition of the derivative • The derivative of f ( x ) is defined as: • An approximation to the derivative improves as x approaches a : a x a f x f a f dx x df a x a x − − = ′ = → = ) ( ) ( lim ) ( ) ( 3/03/2009 2 MASE201 Spring 2009 Figure 6-4 reproduced from Gilat and Subramaniam, “Numerical Methods for Engineers and Scientists.”, Wiley, 2008. Why use numerical differentiation? • Want to find derivative of a measured quantity (discrete points) – Find maximum or inimum 20 30 40 50 60 nd dy/dx measured y dy/dx 3/03/2009 3 minimum – Numerically solve differential equations – Fit rate data 2 4 6-30-20-10 10 x values y and MASE201 Spring 2009 Noise and Scatter • When using experimental data, it often has uncertainties (noise) or scatter. Calculation of derivatives can be very sensitive to 1.5 2 2.5 3 f/dt Noise and numerical differentiation 3/03/2009 4 noise. 2 4 6 8 10-0.5 0.5 1 time f(t), df/ f(t) df/dt noisy data numerical approx. to df/dt MASE201 Spring 2009 Numerical Differentiation • Taylor series expansion of a function about a point x o he quantity ) called the remainder: ( ) ( ) ( ) ( ) ) ( ! ! 3 ! 2 ) ( ) ( 2 3 3 3 2 2 2 x R dx f d n x x dx f d x x dx f d x x dx df x x x f x f n n n o o o o o o x x o x x o x x o x x + − + + − + − + − + = = = = = K 3/03/2009 5 • The quantity R n (x) is called the remainder: • If we expand the series for n = 1: ( ) ( ) ξ = = − + − + = x o x x dx f d x x dx df x x x f x f o o o 2 2 2 ! 2 ) ( ) ( ( ) ( ) x x dx f d n x x x R o n n n o n x ≤ ≤ + − = = + + + ξ ξ where ! 1 ) ( 1 1 1 MASE201 Spring 2009 Numerical Differentiation • If the remainder is ignored: and solving for the approximate derivative: • Since the series expansion was truncated, this ( ) o x x dx df x x x f x f o o = − + ≈ ) ( ) ( ( ) o o x x x f x f dx df o x x − − ≈ = ) ( ) ( 3/03/2009 6 approximation has a truncation error equal to ( ) ! 2 ) ( 2 2 2 1 ξ = − = x dx f d x x x R o MASE201 Spring 2009 Finite Difference Approximations • For a set of data, let x o = x i , and x = x i+1 : or let x o = x i , and x = x i-1 : i i i i i i i x x h h x f x f x f − = − ≈ ′ + + 1 1 where ) ( ) ( ) ( 1 1 1 where ) ( ) ( ) (...
View Full Document

## This document was uploaded on 05/05/2010.

### Page1 / 6

mase201sp09_Mar03_PRINTABLE - Numerical Differentiation and...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online