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Applied numerical methods lec11

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Numerical Differentiation

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Applied numerical methods lec11

  1. 1. 1 Numerical Differentiation
  2. 2. 2 Taylor’s Theorem h=(xi+1- xi)
  3. 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. 4. Forward difference xi1 xi xi+1 x h
  5. 5. Backward difference xi1 xi xi+1 x h
  6. 6. Centered difference xi1 xi xi+1 x 2h
  7. 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. 8 Forward Finite-Divided Difference Formulas second-order accurate forward difference formula for f’(x)
  9. 9. 9 Backward Finite-Divided Difference Formulas
  10. 10. 10 Centered Finite-Divided Difference Formulas
  11. 11. 11 Example        hO h xfxf xf i1i i    
  12. 12. 12
  13. 13. 13 Example of High-Accuracy Differentiation
  14. 14. 14 Miscellaneous problems #1
  15. 15. 15
  16. 16. 16 Miscellaneous problems #2
  17. 17. 17
  18. 18. 18 Miscellaneous problems #3
  19. 19. 19
  20. 20. 20
  21. 21. Homework 1- 2- 3-
  22. 22. 22 4- 5-

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