數值微分
正文
根據函式在一些離散點的函式值,推算它在某點的導數或某高階導數的近似值。通常用差商代替微商,或用一能近似代替該函式的較簡單的函式(如多項式、樣條函式)的相應導數作為所求導數的近似值。例如,對帶餘項的插值公式ƒ(x)=I(x)+R(x)取k階導數就得到帶餘項的數值微分公式![數值微分](/img/b/f95/gZpdmLwIjMwUTNxEDNxgDM5ETMwADMwADMwADMwADMxAzL0EzLwIzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
通常利用多項式插值進行數值微分。設函式ƒ(x)在n+1個等距點xv=α+vh(v=0,1,…,n)上的值ƒv=ƒ(xv)為已知,則通過低次插值可導出一些最基本和常用的數值微分公式,例如,兩點公式
![數值微分](/img/8/007/gZpdmL2EjNyETM2QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzL2EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![數值微分](/img/e/8a4/gZpdmL0kjM0ETM2QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzL0kzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![數值微分](/img/f/740/gZpdmLwUDN1ETM2QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzLwUzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![數值微分](/img/9/6c0/gZpdmLxATN2ETM2QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzLxAzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![數值微分](/img/2/706/gZpdmL4kzN3ETM2QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzL4kzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![數值微分](/img/a/d52/gZpdmLwUzMwITM2QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzLwUzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![數值微分](/img/f/1b3/gZpdmL5kTN0ITM2QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzL5kzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![數值微分](/img/d/166/gZpdmL5YDO1ITM2QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzL5YzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
如果數據ƒv帶有不容忽視的隨機誤差,而其對應的自變數分布甚密,就應該用曲線擬合代替上述函式插值,然後用擬合曲線的導數作為函式ƒ(x)的導數的近似值。這樣求得的導數叫做磨光的導數。
參考書目
馮康等編:《數值計算方法》,國防工業出版社,北京,1978。