递归方法巧解不定方程

王朝vb·作者佚名 2006-01-08

多元一次方程往往采用循环求解。笔者在与网友们讨论一个问题（http://expert.csdn.net/Expert/topic/2607/2607772.xml?temp=.7494928）过程中，琢磨出一种算法，采用递归进行多元一次方程的求解。并将解分为整数解和非负整数解两种情况，请大家指教。

Private Sub Command1_Click() '演示求X1+X2+X3+X4+X5=10整数解

Text1.Text = ""

Dim answer As String

answer = GETRESULT(5, 10, True) '赋值

Dim temp

temp = Split(answer, vbCrLf)

For i = 0 To UBound(temp)

temp(i) = "解" & i + 1 & "：" & vbTab & temp(i) ' add index

answer = Join(temp, vbCrLf)

Text1.Text = "方程 X1+X2+X3+X4+X5=10 共有 " & UBound(temp) + 1 & " 个整数解：" & vbCrLf & answer 'show all answer in textbox

End Sub

Private Sub Command2_Click() '演示求X1+X2+X3+X4+X5=10非负整数解

Text1.Text = ""

Dim answer As String

answer = GETRESULT(5, 10, False) '赋值

Dim temp

temp = Split(answer, vbCrLf)

For i = 0 To UBound(temp)

temp(i) = "解" & i + 1 & "：" & vbTab & temp(i) 'add index

answer = Join(temp, vbCrLf)

Text1.Text = "方程 X1+X2+X3+X4+X5=10 共有 " & UBound(Split(answer, vbCrLf)) + 1 & " 个非零整数解：" & vbCrLf & answer 'show all answer in textbox

End Sub

Private Sub Command3_Click() '演示无解情况

Text1.Text = ""

Dim answer As String

answer = GETRESULT(5, 3, False)

Dim temp

temp = Split(answer, vbCrLf)

For i = 0 To UBound(temp)

temp(i) = "解" & i + 1 & "：" & vbTab & temp(i)

answer = Join(temp, vbCrLf)

Text1.Text = "方程 X1+X2+X3+X4+X5=3 共有 " & UBound(Split(answer, vbCrLf)) + 1 & " 个非零整数解：" & vbCrLf & answer

End Sub

'求解函数

Function GETRESULT(ByVal n As Integer, ByVal SUM As Integer, Optional allowzero As Boolean = True) As String

Dim temp() As String, i As Long

If n = 2 Then '二元方程

If allowzero = True Then

ReDim temp(SUM)

For i = 0 To SUM ' allow zero

temp(i) = "X1=" & i & ",X2=" & SUM - i

GETRESULT = Join(temp, vbCrLf)

Erase temp

Else

ReDim temp(1 To SUM - 1) 'forbid zero

For i = 1 To SUM - 1

temp(i) = "X1=" & i & ",X2=" & SUM - i

GETRESULT = Join(temp, vbCrLf)

Erase temp

End If

If n > 2 Then

If allowzero = True Then

ReDim temp(SUM)

For i = SUM To 0 Step -1 ' allow zero

temp(i) = Replace(GETRESULT(n - 1, i, True), vbCrLf, ",X" & n & "=" & SUM - i & vbCrLf) & ",X" & n & "=" & SUM - i

GETRESULT = Join(temp, vbCrLf)

Erase temp

Else

If SUM < n Then MsgBox "无解！": Exit Function '无解情况

ReDim temp(1 To SUM - n + 1) 'not allow zero

For i = 1 To SUM - n + 1

temp(i) = Replace(GETRESULT(n - 1, SUM - i, False), vbCrLf, ",X" & n & "=" & i & vbCrLf) & ",X" & n & "=" & i '递归

GETRESULT = Join(temp, vbCrLf)

Erase temp

End If

End Function