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Deret Taylor

Deret Taylor. Kesalahan yang dihasilkan dari penggunaan suatu aproksimasi pengganti prosedur matematika yang eksak Contoh: approksimasi dengan deret Taylor. Kesalahan:. Kesalahan pemotongan. Aproksimasi orde ke nol (zero-order appr.). Aproksimasi orde ke satu (first-order appr.).

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Deret Taylor

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  1. Deret Taylor

  2. Kesalahan yang dihasilkan dari penggunaan suatu aproksimasi pengganti prosedur matematika yang eksak Contoh: approksimasi dengan deret Taylor Kesalahan:

  3. Kesalahan pemotongan • Aproksimasi orde ke nol (zero-order appr.) • Aproksimasi orde ke satu (first-order appr.) • Aproksimasi orde ke dua (second-order appr.)

  4. Recall the Taylor Series

  5. i-1 i i+1 i+1 i-1 i i-1 i i+1 Finite difference method -1 The Taylor series (1) (2) First-oder difference Forward difference Backward difference Central Difference turunkan juga!

  6. i-2 i-1 i i+1 i+2 Finite difference method -3 High-oder finite difference To reduce the truncation error of finite difference method, high-order finite difference method is frequently employed h (1) Taylor series of five points are, (2) (3) (4) (5) (5) By substituting (2)～(5) into Eq.(1), we have,

  7. Finite difference method -4 (7) The coefficient for the first derivative must be 1 and the others must be zero . By solving 5 linear equations, we get, (8) The order of precision is 4th.

  8. Finite difference method -5 Comparison of finite difference method f(x)=x3, Δx=1,x=2 , analytical solution f’(2)=12 • x f(x) • 0 0 • 1 • 8 • 27 • 64 Forward finite difference Backward finite difference Central difference 4th order finite Central difference Exercise: Calculate above four types of finite differences under condition of f(x)=x, Δx=1, x=2

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