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SE301_14(c)AL-AMER20031 SE031Numerical Methods Topic 6 Numerical Differentiation Dr. Samir Al-Amer ( Term 053) Modified by Dr. Baroudi(072) Read chapter 23.1 and 23.2

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SE301_14(c)AL-AMER20032 Numerical Differentiation First order derivatives High order derivatives Richardson Extrapolation Examples

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SE301_14(c)AL-AMER20033 Motivation How do you evaluate the derivative of a tabulated function. How do we determine the velocity and acceleration from tabulated measurements. Time (second) Displacemen t (meters) 030.1 548.2 1050.0 1540.2

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SE301_14(c)AL-AMER20034 Recall

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SE301_14(c)AL-AMER20035 Three formula

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SE301_14(c)AL-AMER20036 Forward Difference formula

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SE301_14(c)AL-AMER20037 Central Difference formula

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SE301_14(c)AL-AMER20038 The Three formula (revisited)

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SE301_14(c)AL-AMER20039 Higher Order Formulas

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10 High Accuracy Differentiation Formulas (1) High-accuracy divided-difference formulas can be generated by including additional terms from the Taylor series expansion. Consider the forward finite-divided difference

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11 High Accuracy Differentiation Formulas (2) Consider the backward finite-divided difference Consider the central finite-divided difference

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SE301_14(c)AL-AMER200312 Example

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SE301_14(c)AL-AMER200313 Other Higher Order Formulas

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SE301_14(c)AL-AMER200314 Richardson Extrapolation

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SE301_14(c)AL-AMER200315 Richardson Extrapolation

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SE301_14(c)AL-AMER200316 Richardson Extrapolation Table

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SE301_14(c)AL-AMER200317 Richardson Extrapolation Table D(0,0)=Φ(h) D(1,0)=Φ(h/2)D(1,1) D(2,0)=Φ(h/4)D(2,1)D(2,2) D(3,0)=Φ(h/8)D(3,1)D(3,2)D(3,3)

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SE301_14(c)AL-AMER200318 Example

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SE301_14(c)AL-AMER200319 Example First Column

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SE301_14(c)AL-AMER200320 Example Richardson Table

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SE301_14(c)AL-AMER200321 Example Richardson Table 1.08483 1.089881.09114 1.091151.091461.09148 This is the best estimate of the derivative of the function All entries of the Richardson table are estimates of the derivative of the function. The first column are estimates using the central difference formula with different h.

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SE301_14(c)AL-AMER200322 Summary

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