使用牛頓迭代法求方程的根

第二篇隨筆

9102年11月底,工科男曹**要算一個方程f(x)=0的根,其中f(x)表達式爲:

由於實數範圍內f(x)=0的根太多,因此本文只研究-2<x<2的狀況.這個式子長的太醜了,曹**看着以爲不爽,導之,得一f'(x)

這個式子更醜,可是,咱們有牛頓迭代法,能夠構造迭代序列{xn}知足:

其中f'(xn)不等於0.能夠證實,只要初值選的好,序列能夠收斂到要求的根.而後就能夠寫程序求根了.

先上函數圖像(由desmos繪製),看到指定區間上有5個零點.而後,零點附近取值吧.

再上效果

 

結果仍是不錯的.

最後,上代碼.f(x)和f'(x)用委託的方式傳入calc函數.委託注意實例化

Public Delegate Function myfunc(x As Double) As Double

Public Function func0(x As Double) As Double
    Return Exp(x) + Pow(x, 4) * Sin(Pow(x, 3))
End Function
Public Function func0derive(x As Double) As Double
    Return Exp(x) + 4 * Pow(x, 3) * Sin(Pow(x, 3)) + 3 * Pow(x, 6) * Cos(Pow(x, 3))
End Function

Dim f0 As New myfunc(AddressOf func0)
Dim fd0 As New myfunc(AddressOf func0derive)

 

calc的參數中f和fd分別是指向f(x)和f'(x)的函數指針,x0爲初值,eps爲精度,cnt爲迭代次數

用傳引用的方式,經過sol返回計算結果.

返回True爲沒有出錯,False爲出錯.

 1 Public Function Calc(f As myfunc, fd As myfunc, x0 As Double, eps As Double, cnt As Integer, ByRef sol As Double) As Boolean
 2         If cnt <= 0 Or f Is Nothing Or fd Is Nothing Then
 3             Return False
 4         End If
 5         Try
 6             sol = 0
 7             Dim x As Double = x0, c0 As Integer = 0
 8             While Math.Abs(x) > eps And cnt > c0
 9                 x = x - f(x) / fd(x)
10                 c0 += 1
11             End While
12             sol = x
13             Return True
14         Catch ex As Exception
15             Return False
16         End Try
17     End Function