Deret Taylor
Contoh: approksimasi dengan 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.) Aproksimasi orde ke dua (second-order appr.)
Recall the Taylor Series
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
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,
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.
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