Applied numerical methods lec11
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This document discusses numerical differentiation techniques. It introduces Taylor's theorem as the basis for high-accuracy differentiation formulas. Several finite difference formulas are presented for approximating the first derivative, including forward, backward, and centered differences. Higher-order accurate differentiation formulas are also described. Examples are provided to demonstrate applying the formulas to compute derivatives of simple functions.
Applied numerical methods lec11
- 3. High-Accuracy Differentiation Formulas 2ii1i i hOh 2 xf - h xfxf xf ixx i dx xdf xf one more sample xi+2 = xi+1 + h hO h xfxf xf i1i i hO h xfx2fxf xf 2 i1i2i i 2 2 i1i2ii1i i hOh 2h xfxfxf - h xfxf xf 2 one extra sample xi+1 = xi + h first-order accurate second-order accurate Big O notation describes the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions.
- 4. Forward difference xi1 xi xi+1 x h
- 5. Backward difference xi1 xi xi+1 x h
- 6. Centered difference xi1 xi xi+1 x 2h
- 7. First Derivatives • Forward difference • Backward difference • Central difference )x(f i-2 i-1 i i+1 i+2 1i1i 1i1i 1i1i 1i1i 1ii 1ii 1ii 1ii i1i i1i i1i i1i xx yy xx )x(f)x(f )x(f xx yy xx )x(f)x(f )x(f xx yy xx )x(f)x(f )x(f x y
- 8. 8 Forward Finite-Divided Difference Formulas second-order accurate forward difference formula for f’(x)
- 9. 9 Backward Finite-Divided Difference Formulas
- 10. 10 Centered Finite-Divided Difference Formulas
- 11. 11 Example hO h xfxf xf i1i i
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- 13. 13 Example of High-Accuracy Differentiation
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- 22. 22 4- 5-