笔者在2004年曾写过一篇 递归方法巧解不定方程 。昨天在一位网友的启发下,对代码进行了重写，使其能够设置每个变量的取值范围。代码如下
Private Sub Command1_Click()
Dim min(1 To 5) As Integer, max(1 To 5) As Integer, RESULT
min(1) = 2
min(2) = 3
max(1) = 3
max(2) = 7
jiefangcheng 5, min, max, 15
End Sub
'求解函数
Sub GETRESULT(ByVal n As Integer, ByRef min() As Integer, ByRef max() As Integer, ByVal SUM As Integer, Optional ByRef resultcount As Long, Optional ByRef RESULT)
Dim temp(), c() As Long, i As Long, j As Long, k As Long, r As Long
If n = 1 Then '一元方程
If SUM >= min(1) And SUM <= max(1) Then
resultcount = 1
ReDim RESULT(1 To 1, 1 To 1)
RESULT(1, 1) = SUM
End If
End If
If n > 1 Then
resultcount = 0
ReDim temp(min(n) To max(n))
ReDim c(min(n) To max(n))
For i = min(n) To max(n)
GETRESULT n - 1, min, max, SUM - i, c(i), temp(i)
resultcount = resultcount + c(i)
Next
If resultcount > 0 Then
ReDim RESULT(1 To resultcount, 1 To n)
For i = min(n) To max(n)
If c(i) > 0 Then
For j = 1 To UBound(temp(i), 1)
r = r + 1
For k = 1 To n - 1
RESULT(r, k) = temp(i)(j, k)
Next
RESULT(r, n) = i
Next
End If
Next
Erase temp
End If
End If
End Sub
Sub jiefangcheng(ByVal n As Integer, ByRef min() As Integer, ByRef max() As Integer, ByVal SUM As Integer, Optional ByRef RESULT)
Dim x() As String, i As Integer, msg As String, resultcount As Long, laptime As Single
laptime = Timer
'默认解的范围为1-SUM
For i = 1 To n
If min(i) = 0 Then min(i) = 1
If max(i) = 0 Then max(i) = SUM
Next
GETRESULT n, min, max, SUM, resultcount, RESULT '赋值
ReDim x(1 To n)
For i = 1 To n
x(i) = "X" & i
Next
msg = "方程" & Join(x, "+") & "=" & SUM
For i = 1 To n
x(i) = "X" & i & "∈[" & min(i) & "," & max(i) & "]"
Next
msg = msg & "当" & Join(x, ",") & " 时" '关于方程的表达式和解的定义域
If resultcount > 0 Then
Debug.Print msg & "共有 " & resultcount & " 个解"
ReDim x(1 To resultcount)
For i = 1 To resultcount
x(i) = "解" & i & "："
For j = 1 To n
x(i) = x(i) & "X" & j & "=" & RESULT(i, j) & vbTab
Next
Next
Debug.Print Join(x, vbCrLf)
Else
Debug.Print msg & "无解！"
End If
laptime = Timer - laptime
Debug.Print "总计用时" & IIf(laptime < 0, 0, Format(laptime, "0.000")) & "秒！"
End Sub
调试结果如下：
方程X1+X2+X3+X4+X5=15当X1∈[2,3],X2∈[3,7],X3∈[1,15],X4∈[1,15],X5∈[1,15] 时共有 190 个解
解1：X1=3 X2=7 X3=3 X4=1 X5=1
解2：X1=3 X2=6 X3=4 X4=1 X5=1
解3：X1=2 X2=7 X3=4 X4=1 X5=1
解4：X1=3 X2=5 X3=5 X4=1 X5=1
解5：X1=2 X2=6 X3=5 X4=1 X5=1
解6：X1=3 X2=4 X3=6 X4=1 X5=1
解7：X1=2 X2=5 X3=6 X4=1 X5=1
解8：X1=3 X2=3 X3=7 X4=1 X5=1
解9：X1=2 X2=4 X3=7 X4=1 X5=1
解10：X1=2 X2=3 X3=8 X4=1 X5=1
解11：X1=3 X2=7 X3=2 X4=2 X5=1
解12：X1=3 X2=6 X3=3 X4=2 X5=1
解13：X1=2 X2=7 X3=3 X4=2 X5=1
解14：X1=3 X2=5 X3=4 X4=2 X5=1
解15：X1=2 X2=6 X3=4 X4=2 X5=1
解16：X1=3 X2=4 X3=5 X4=2 X5=1
解17：X1=2 X2=5 X3=5 X4=2 X5=1
解18：X1=3 X2=3 X3=6 X4=2 X5=1
解19：X1=2 X2=4 X3=6 X4=2 X5=1
解20：X1=2 X2=3 X3=7 X4=2 X5=1
解21：X1=3 X2=7 X3=1 X4=3 X5=1
解22：X1=3 X2=6 X3=2 X4=3 X5=1
解23：X1=2 X2=7 X3=2 X4=3 X5=1
解24：X1=3 X2=5 X3=3 X4=3 X5=1
解25：X1=2 X2=6 X3=3 X4=3 X5=1
解26：X1=3 X2=4 X3=4 X4=3 X5=1
解27：X1=2 X2=5 X3=4 X4=3 X5=1
解28：X1=3 X2=3 X3=5 X4=3 X5=1
解29：X1=2 X2=4 X3=5 X4=3 X5=1
解30：X1=2 X2=3 X3=6 X4=3 X5=1
解31：X1=3 X2=6 X3=1 X4=4 X5=1
解32：X1=2 X2=7 X3=1 X4=4 X5=1
解33：X1=3 X2=5 X3=2 X4=4 X5=1
解34：X1=2 X2=6 X3=2 X4=4 X5=1
解35：X1=3 X2=4 X3=3 X4=4 X5=1
解36：X1=2 X2=5 X3=3 X4=4 X5=1
解37：X1=3 X2=3 X3=4 X4=4 X5=1
解38：X1=2 X2=4 X3=4 X4=4 X5=1
解39：X1=2 X2=3 X3=5 X4=4 X5=1
解40：X1=3 X2=5 X3=1 X4=5 X5=1
解41：X1=2 X2=6 X3=1 X4=5 X5=1
解42：X1=3 X2=4 X3=2 X4=5 X5=1
解43：X1=2 X2=5 X3=2 X4=5 X5=1
解44：X1=3 X2=3 X3=3 X4=5 X5=1
解45：X1=2 X2=4 X3=3 X4=5 X5=1
解46：X1=2 X2=3 X3=4 X4=5 X5=1
解47：X1=3 X2=4 X3=1 X4=6 X5=1
解48：X1=2 X2=5 X3=1 X4=6 X5=1
解49：X1=3 X2=3 X3=2 X4=6 X5=1
解50：X1=2 X2=4 X3=2 X4=6 X5=1
解51：X1=2 X2=3 X3=3 X4=6 X5=1
解52：X1=3 X2=3 X3=1 X4=7 X5=1
解53：X1=2 X2=4 X3=1 X4=7 X5=1
解54：X1=2 X2=3 X3=2 X4=7 X5=1
解55：X1=2 X2=3 X3=1 X4=8 X5=1
解56：X1=3 X2=7 X3=2 X4=1 X5=2
解57：X1=3 X2=6 X3=3 X4=1 X5=2
解58：X1=2 X2=7 X3=3 X4=1 X5=2
解59：X1=3 X2=5 X3=4 X4=1 X5=2
解60：X1=2 X2=6 X3=4 X4=1 X5=2
解61：X1=3 X2=4 X3=5 X4=1 X5=2
解62：X1=2 X2=5 X3=5 X4=1 X5=2
解63：X1=3 X2=3 X3=6 X4=1 X5=2
解64：X1=2 X2=4 X3=6 X4=1 X5=2
解65：X1=2 X2=3 X3=7 X4=1 X5=2
解66：X1=3 X2=7 X3=1 X4=2 X5=2
解67：X1=3 X2=6 X3=2 X4=2 X5=2
解68：X1=2 X2=7 X3=2 X4=2 X5=2
解69：X1=3 X2=5 X3=3 X4=2 X5=2
解70：X1=2 X2=6 X3=3 X4=2 X5=2
解71：X1=3 X2=4 X3=4 X4=2 X5=2
解72：X1=2 X2=5 X3=4 X4=2 X5=2
解73：X1=3 X2=3 X3=5 X4=2 X5=2
解74：X1=2 X2=4 X3=5 X4=2 X5=2
解75：X1=2 X2=3 X3=6 X4=2 X5=2
解76：X1=3 X2=6 X3=1 X4=3 X5=2
解77：X1=2 X2=7 X3=1 X4=3 X5=2
解78：X1=3 X2=5 X3=2 X4=3 X5=2
解79：X1=2 X2=6 X3=2 X4=3 X5=2
解80：X1=3 X2=4 X3=3 X4=3 X5=2
解81：X1=2 X2=5 X3=3 X4=3 X5=2
解82：X1=3 X2=3 X3=4 X4=3 X5=2
解83：X1=2 X2=4 X3=4 X4=3 X5=2
解84：X1=2 X2=3 X3=5 X4=3 X5=2
解85：X1=3 X2=5 X3=1 X4=4 X5=2
解86：X1=2 X2=6 X3=1 X4=4 X5=2
解87：X1=3 X2=4 X3=2 X4=4 X5=2
解88：X1=2 X2=5 X3=2 X4=4 X5=2
解89：X1=3 X2=3 X3=3 X4=4 X5=2
解90：X1=2 X2=4 X3=3 X4=4 X5=2
解91：X1=2 X2=3 X3=4 X4=4 X5=2
解92：X1=3 X2=4 X3=1 X4=5 X5=2
解93：X1=2 X2=5 X3=1 X4=5 X5=2
解94：X1=3 X2=3 X3=2 X4=5 X5=2
解95：X1=2 X2=4 X3=2 X4=5 X5=2
解96：X1=2 X2=3 X3=3 X4=5 X5=2
解97：X1=3 X2=3 X3=1 X4=6 X5=2
解98：X1=2 X2=4 X3=1 X4=6 X5=2
解99：X1=2 X2=3 X3=2 X4=6 X5=2
解100：X1=2 X2=3 X3=1 X4=7 X5=2
解101：X1=3 X2=7 X3=1 X4=1 X5=3
解102：X1=3 X2=6 X3=2 X4=1 X5=3
解103：X1=2 X2=7 X3=2 X4=1 X5=3
解104：X1=3 X2=5 X3=3 X4=1 X5=3
解105：X1=2 X2=6 X3=3 X4=1 X5=3
解106：X1=3 X2=4 X3=4 X4=1 X5=3
解107：X1=2 X2=5 X3=4 X4=1 X5=3
解108：X1=3 X2=3 X3=5 X4=1 X5=3
解109：X1=2 X2=4 X3=5 X4=1 X5=3
解110：X1=2 X2=3 X3=6 X4=1 X5=3
解111：X1=3 X2=6 X3=1 X4=2 X5=3
解112：X1=2 X2=7 X3=1 X4=2 X5=3
解113：X1=3 X2=5 X3=2 X4=2 X5=3
解114：X1=2 X2=6 X3=2 X4=2 X5=3
解115：X1=3 X2=4 X3=3 X4=2 X5=3
解116：X1=2 X2=5 X3=3 X4=2 X5=3
解117：X1=3 X2=3 X3=4 X4=2 X5=3
解118：X1=2 X2=4 X3=4 X4=2 X5=3
解119：X1=2 X2=3 X3=5 X4=2 X5=3
解120：X1=3 X2=5 X3=1 X4=3 X5=3
解121：X1=2 X2=6 X3=1 X4=3 X5=3
解122：X1=3 X2=4 X3=2 X4=3 X5=3
解123：X1=2 X2=5 X3=2 X4=3 X5=3
解124：X1=3 X2=3 X3=3 X4=3 X5=3
解125：X1=2 X2=4 X3=3 X4=3 X5=3
解126：X1=2 X2=3 X3=4 X4=3 X5=3
解127：X1=3 X2=4 X3=1 X4=4 X5=3
解128：X1=2 X2=5 X3=1 X4=4 X5=3
解129：X1=3 X2=3 X3=2 X4=4 X5=3
解130：X1=2 X2=4 X3=2 X4=4 X5=3
解131：X1=2 X2=3 X3=3 X4=4 X5=3
解132：X1=3 X2=3 X3=1 X4=5 X5=3
解133：X1=2 X2=4 X3=1 X4=5 X5=3
解134：X1=2 X2=3 X3=2 X4=5 X5=3
解135：X1=2 X2=3 X3=1 X4=6 X5=3
解136：X1=3 X2=6 X3=1 X4=1 X5=4
解137：X1=2 X2=7 X3=1 X4=1 X5=4
解138：X1=3 X2=5 X3=2 X4=1 X5=4
解139：X1=2 X2=6 X3=2 X4=1 X5=4
解140：X1=3 X2=4 X3=3 X4=1 X5=4
解141：X1=2 X2=5 X3=3 X4=1 X5=4
解142：X1=3 X2=3 X3=4 X4=1 X5=4
解143：X1=2 X2=4 X3=4 X4=1 X5=4
解144：X1=2 X2=3 X3=5 X4=1 X5=4
解145：X1=3 X2=5 X3=1 X4=2 X5=4
解146：X1=2 X2=6 X3=1 X4=2 X5=4
解147：X1=3 X2=4 X3=2 X4=2 X5=4
解148：X1=2 X2=5 X3=2 X4=2 X5=4
解149：X1=3 X2=3 X3=3 X4=2 X5=4
解150：X1=2 X2=4 X3=3 X4=2 X5=4
解151：X1=2 X2=3 X3=4 X4=2 X5=4
解152：X1=3 X2=4 X3=1 X4=3 X5=4
解153：X1=2 X2=5 X3=1 X4=3 X5=4
解154：X1=3 X2=3 X3=2 X4=3 X5=4
解155：X1=2 X2=4 X3=2 X4=3 X5=4
解156：X1=2 X2=3 X3=3 X4=3 X5=4
解157：X1=3 X2=3 X3=1 X4=4 X5=4
解158：X1=2 X2=4 X3=1 X4=4 X5=4
解159：X1=2 X2=3 X3=2 X4=4 X5=4
解160：X1=2 X2=3 X3=1 X4=5 X5=4
解161：X1=3 X2=5 X3=1 X4=1 X5=5
解162：X1=2 X2=6 X3=1 X4=1 X5=5
解163：X1=3 X2=4 X3=2 X4=1 X5=5
解164：X1=2 X2=5 X3=2 X4=1 X5=5
解165：X1=3 X2=3 X3=3 X4=1 X5=5
解166：X1=2 X2=4 X3=3 X4=1 X5=5
解167：X1=2 X2=3 X3=4 X4=1 X5=5
解168：X1=3 X2=4 X3=1 X4=2 X5=5
解169：X1=2 X2=5 X3=1 X4=2 X5=5
解170：X1=3 X2=3 X3=2 X4=2 X5=5
解171：X1=2 X2=4 X3=2 X4=2 X5=5
解172：X1=2 X2=3 X3=3 X4=2 X5=5
解173：X1=3 X2=3 X3=1 X4=3 X5=5
解174：X1=2 X2=4 X3=1 X4=3 X5=5
解175：X1=2 X2=3 X3=2 X4=3 X5=5
解176：X1=2 X2=3 X3=1 X4=4 X5=5
解177：X1=3 X2=4 X3=1 X4=1 X5=6
解178：X1=2 X2=5 X3=1 X4=1 X5=6
解179：X1=3 X2=3 X3=2 X4=1 X5=6
解180：X1=2 X2=4 X3=2 X4=1 X5=6
解181：X1=2 X2=3 X3=3 X4=1 X5=6
解182：X1=3 X2=3 X3=1 X4=2 X5=6
解183：X1=2 X2=4 X3=1 X4=2 X5=6
解184：X1=2 X2=3 X3=2 X4=2 X5=6
解185：X1=2 X2=3 X3=1 X4=3 X5=6
解186：X1=3 X2=3 X3=1 X4=1 X5=7
解187：X1=2 X2=4 X3=1 X4=1 X5=7
解188：X1=2 X2=3 X3=2 X4=1 X5=7
解189：X1=2 X2=3 X3=1 X4=2 X5=7
解190：X1=2 X2=3 X3=1 X4=1 X5=8
总计用时0.410秒！