Deret Taylor.

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1 Deret Taylor

2 Contoh: approksimasi dengan deret Taylor
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 Finite difference method -1
The Taylor series (1) (2) First-oder difference Forward difference i i+1 i-1 Backward difference i+1 i i-1 Central Difference turunkan juga! i i+1 i-1

6 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) i i+1 i-2 i-1 i+2 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) 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