Faculty of Engineering Mechanical Engineering Department MATH 2140

Faculty of Engineering Mechanical Engineering Department MATH 2140 Numerical Methods Instructor: Dr. Mohamed El-Shazly Associate Prof. of Mechanical Design and Tribology melshazly@ksu. edu. sa Office: F 072 1

Numerical Differentiation 2

SUMMARY OF FINITE DIFFERENCE FORMULAS FOR NUMERICAL DIFFERENTIATION 3

SUMMARY OF FINITE DIFFERENCE FORMULAS FOR NUMERICAL DIFFERENTIATION 4

DIFFERENTIATION FORMULAS USING LAGRANGE POLYNOMIALS • • The differentiation formulas can also be derived by using Lagrange polynomials. For the first derivative, the two-point central, three-point forward, and three-point backward difference formulas are obtained by considering three points (xi, yi), (xi+1, yi+1) , and (xi+2 , yi+2). The polynomial, in Lagrange form, that passes through the points is given by: 5

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DIFFERENTIATION USING CURVE FITTING § A different approach to differentiation of data specified by a set of discrete points is to first approximate with an analytical function that can be easily differentiated. § The approximate function is then differentiated for calculating the derivative at any of the points (Fig. 8 -6). Curve fitting is described in Chapter 6. § For data that shows a nonlinear relationship, curve fitting is often done by using least squares with an exponential function, a power function, low-order polynomial, or a combination of a nonlinear functions, which are simple to differentiate. § This procedure may be preferred when the data contains scatter, or noise, since the curved-fitted function smooths out the noise. Numerical differentiation using curve fitting 8

RICHARDSON'S EXTRAPOLATION • Richardson's extrapolation is a method for calculating a more accurate approximation of a derivative from two less accurate approximations of that derivative. • In general terms, consider the value, D, of a derivative (unknown) that is calculated by the difference formula: The value, D, of the derivative can now be approximated by: 9

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