差分的性质
目录
- 差分的性质
- 1)各阶差分均可用函数值表示
- 2)用各阶差分表示函数值
- 3)差商与差分的关系
差分的性质
1)各阶差分均可用函数值表示
例如
Δ n f k = ( E − I ) n f k = ∑ j = 0 n ( − 1 ) j ( n j ) E n − j f k = ∑ j = 0 n ( − 1 ) j ( n j ) f n + k − j \Delta^nf_k=(\text{E}-\text{I})^nf_k=\sum_{j=0}^n(-1)^j\binom{n}{j}\text{E}^{n-j}f_k=\sum_{j=0}^n(-1)^j\binom{n}{j}f_{n+k-j} Δnfk=(E−I)nfk=j=0∑n(−1)j(jn)En−jfk=j=0∑n(−1)j(jn)fn+k−j
∇ n f k = ( 1 − E − 1 ) n f k = ∑ j = 0 n ( − 1 ) n − j ( n j ) E j − n f k = ∑ j = 0 n ( − 1 ) n − j ( n j ) f k + j − n \nabla^nf_k=(1-\mathrm{E}^{-1})^nf_k=\sum_{j=0}^n(-1)^{n-j}\binom{n}{j}\mathrm{E}^{j-n}f_k=\sum_{j=0}^n(-1)^{n-j}\binom{n}{j}f_{k+j-n} ∇nfk=(1−E−1)nfk=j=0∑n(−1)n−j(jn)Ej−nfk=j=0∑n(−1)n−j(jn)fk+j−n
其中 ( n j ) = n ( n − 1 ) ⋯ ( n − j + 1 ) j ! \binom nj=\frac{n(n-1)\cdots(n-j+1)}{j!} (jn)=j!n(n−1)⋯(n−j+1)为二项式展开系数.
2)用各阶差分表示函数值
例如用向前差分表示 f n + k f_{n+k} fn+k,即
f n + k = E n f k = ( I + Δ ) n f k = ∑ j = 0 n ( n j ) Δ j f k f_{n+k}=\mathrm{E}^nf_k=(\mathrm{I}+\Delta)^nf_k=\sum_{j=0}^n\binom nj\Delta^jf_k fn+k=Enfk=(I+Δ)nfk=j=0∑n(jn)Δjfk
3)差商与差分的关系
对于向前差分，有
f [ x k , x k + 1 ] = f k + 1 − f k x k + 1 − x k = Δ f k h f[x_k,x_{k+1}]=\frac{f_{k+1}-f_k}{x_{k+1}-x_k}=\frac{\Delta f_k}h f[xk,xk+1]=xk+1−xkfk+1−fk=hΔfk
f [ x k , x k + 1 , x k + 2 ] = f [ x k + 1 , x k + 2 ] − f [ x k , x k + 1 ] x k + 2 − x k = 1 2 h 2 Δ 2 f k , f[x_k,x_{k+1},x_{k+2}]=\frac{f[x_{k+1},x_{k+2}]-f[x_k,x_{k+1}]}{x_{k+2}-x_k}=\frac1{2h^2}\Delta^2f_k\:, f[xk,xk+1,xk+2]=xk+2−xkf[xk+1,xk+2]−f[xk,xk+1]=2h21Δ2fk,
更一般的，
f [ x k , x k + 1 , ⋅ ⋅ ⋅ , x k + m ] = 1 m ! 1 h m Δ m f k ( m = 1 , 2 , ⋅ ⋅ ⋅ , n ) . f[x_k,x_{k+1},\cdotp\cdotp\cdotp,x_{k+m}]=\frac1{m!}\frac1{h^m}\Delta^mf_k\quad(m=1,2,\cdotp\cdotp\cdotp,n). f[xk,xk+1,⋅⋅⋅,xk+m]=m!1hm1Δmfk(m=1,2,⋅⋅⋅,n).
同理，对于向后差分，有
f [ x k , x k − 1 , ⋅ ⋅ ⋅ , x k − m ] = 1 m ! 1 h m ∇ m f k f[ x_k, x_{k- 1}, \cdotp \cdotp \cdotp , x_{k- m}] = \frac 1m! \frac 1{h^m}\nabla ^mf_k f[xk,xk−1,⋅⋅⋅,xk−m]=m1!hm1∇mfk