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The Islamic University of Gaza Faculty of Engineering Civil Engineering Department Numerical Analysis ECIV 3306 Chapter 23 Numerical Differentiation

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1.Forward difference Taylor series :

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Numerical Differentiation 2. Backward difference

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Numerical Differentiation 3. Centered difference

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High Accuracy Differentiation Formulas High-accuracy finite-difference formulas can be generated by including additional terms from the Taylor series expansion. An example: High-accuracy forward-difference formula for the first derivative.

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Derivation: High-accuracy forward- difference formula for f`(x) (1) (2) Equ. 1 can be multiplied by 2 and subtracted from equ. 2: Solve: Second derivative forward finite divided difference

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Derivation: High-accuracy forward-difference formula for f`(x) Taylor series expansion Solve for f’(x) High-accuracy forward-difference formula Substitute the forward- difference approx. of f”(x)

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Derivation: High-accuracy forward-difference formula for f`(x) Similar improved versions can be developed for the backward and centered formulas as well as for the approximations of the higher derivatives.

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Higher Order Forward Divided Difference

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Higher Order Backward Divided Difference

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Higher Order Central Divided Difference

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First Derivatives - Example: Use forward,backward and centered difference approximations to estimate the first derivative of at x = 0.5 with h = 0.5 and 0.25 (exact sol. = -0.9125) Forward Difference Backward Difference

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First Derivatives - Example: Central Difference

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Forward Finite-divided differences

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Backward finite-divided differences

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Centered Finite-Divided Differences

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First Derivatives - Example: Employing the high-accuracy formulas (h=0.25): x i-2 = 0.0 f(0.0) = 1.2 x i-1 = 0.25 f(0.0) = 1.103516 x i = 0.5 f(0.5) = 0.925 x i+1 = 0.75 f(0.75) = 0.63633 x i+2 = 1.0 f(1.0) = 0.2 Forward Difference

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First Derivatives - Example: Backward Difference Central Difference

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Summary h = 0.25Forward O(h 2 ) Backward O(h 2 ) Centered O(h 4 ) Estimate-0.859375-0.878125-0.9125 |t||t| 5.82%3.77%0% Basic formulas h = 0.25Forward O(h) Backward O(h) Centered O(h 2 ) Estimate-1.155-0.714-0.934 |t||t| 26.5%21.7%2.4% High- Accuracy formulas True value: f`(0.5) = -0.9125

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