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Ingy döt Net
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Imports System.Text
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Module Module1
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Private ReadOnly MAX_ALL_FACTORS As Integer = 100_000
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#Const ALGORITHM = 2
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Class Term
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Implements IComparable(Of Term)
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Public ReadOnly Property Coefficient As Long
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Public ReadOnly Property Exponent As Long
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Public Sub New(c As Long, Optional e As Long = 0)
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Coefficient = c
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Exponent = e
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End Sub
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Public Shared Operator -(t As Term) As Term
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Return New Term(-t.Coefficient, t.Exponent)
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End Operator
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Public Shared Operator +(lhs As Term, rhs As Term) As Term
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If lhs.Exponent <> rhs.Exponent Then
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Throw New ArgumentException("Exponents not equal")
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End If
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Return New Term(lhs.Coefficient + rhs.Coefficient, lhs.Exponent)
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End Operator
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Public Shared Operator *(lhs As Term, rhs As Term) As Term
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Return New Term(lhs.Coefficient * rhs.Coefficient, lhs.Exponent + rhs.Exponent)
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End Operator
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Public Function CompareTo(other As Term) As Integer Implements IComparable(Of Term).CompareTo
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Return -Exponent.CompareTo(other.Exponent)
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End Function
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Public Overrides Function ToString() As String
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If Coefficient = 0 Then
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Return "0"
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End If
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If Exponent = 0 Then
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Return Coefficient.ToString
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End If
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If Coefficient = 1 Then
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If Exponent = 1 Then
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Return "x"
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End If
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Return String.Format("x^{0}", Exponent)
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End If
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If Coefficient = -1 Then
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If Exponent = 1 Then
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Return "-x"
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End If
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Return String.Format("-x^{0}", Exponent)
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End If
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If Exponent = 1 Then
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Return String.Format("{0}x", Coefficient)
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End If
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Return String.Format("{0}x^{1}", Coefficient, Exponent)
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End Function
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End Class
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Class Polynomial
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Implements IEnumerable(Of Term)
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Private ReadOnly polyTerms As New List(Of Term)
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Public Sub New()
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polyTerms.Add(New Term(0))
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End Sub
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Public Sub New(ParamArray values() As Term)
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If values.Length = 0 Then
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polyTerms.Add(New Term(0))
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Else
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polyTerms.AddRange(values)
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End If
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Normalize()
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End Sub
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Public Sub New(values As IEnumerable(Of Term))
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polyTerms.AddRange(values)
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If polyTerms.Count = 0 Then
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polyTerms.Add(New Term(0))
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End If
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Normalize()
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End Sub
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Public Function LeadingCoeficient() As Long
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Return polyTerms(0).Coefficient
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End Function
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Public Function Degree() As Long
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Return polyTerms(0).Exponent
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End Function
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Public Function HasCoefficentAbs(coeff As Long) As Boolean
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For Each t In polyTerms
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If Math.Abs(t.Coefficient) = coeff Then
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Return True
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End If
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Next
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Return False
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End Function
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Public Function GetEnumerator() As IEnumerator(Of Term) Implements IEnumerable(Of Term).GetEnumerator
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Return polyTerms.GetEnumerator
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End Function
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Private Function IEnumerable_GetEnumerator() As IEnumerator Implements IEnumerable.GetEnumerator
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Return polyTerms.GetEnumerator
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End Function
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Private Sub Normalize()
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polyTerms.Sort(Function(a As Term, b As Term) a.CompareTo(b))
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End Sub
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Public Shared Operator +(lhs As Polynomial, rhs As Term) As Polynomial
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Dim terms As New List(Of Term)
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Dim added = False
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For Each ct In lhs
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If ct.Exponent = rhs.Exponent Then
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added = True
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If ct.Coefficient + rhs.Coefficient <> 0 Then
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terms.Add(ct + rhs)
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End If
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Else
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terms.Add(ct)
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End If
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Next
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If Not added Then
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terms.Add(rhs)
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End If
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Return New Polynomial(terms)
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End Operator
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Public Shared Operator *(lhs As Polynomial, rhs As Term) As Polynomial
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Dim terms As New List(Of Term)
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For Each ct In lhs
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terms.Add(ct * rhs)
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Next
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Return New Polynomial(terms)
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End Operator
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Public Shared Operator +(lhs As Polynomial, rhs As Polynomial) As Polynomial
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Dim terms As New List(Of Term)
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Dim thisCount = lhs.polyTerms.Count
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Dim polyCount = rhs.polyTerms.Count
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While thisCount > 0 OrElse polyCount > 0
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If thisCount = 0 Then
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Dim polyTerm = rhs.polyTerms(polyCount - 1)
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terms.Add(polyTerm)
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polyCount -= 1
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ElseIf polyCount = 0 Then
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Dim thisTerm = lhs.polyTerms(thisCount - 1)
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terms.Add(thisTerm)
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thisCount -= 1
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Else
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Dim polyTerm = rhs.polyTerms(polyCount - 1)
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Dim thisTerm = lhs.polyTerms(thisCount - 1)
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If thisTerm.Exponent = polyTerm.Exponent Then
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Dim t = thisTerm + polyTerm
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If t.Coefficient <> 0 Then
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terms.Add(t)
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End If
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thisCount -= 1
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polyCount -= 1
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ElseIf thisTerm.Exponent < polyTerm.Exponent Then
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terms.Add(thisTerm)
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thisCount -= 1
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Else
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terms.Add(polyTerm)
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polyCount -= 1
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End If
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End If
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End While
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Return New Polynomial(terms)
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End Operator
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Public Shared Operator *(lhs As Polynomial, rhs As Polynomial) As Polynomial
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Throw New Exception("Not implemented")
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End Operator
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Public Shared Operator /(lhs As Polynomial, rhs As Polynomial) As Polynomial
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Dim q As New Polynomial
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Dim r = lhs
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Dim lcv = rhs.LeadingCoeficient
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Dim dv = rhs.Degree
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While r.Degree >= rhs.Degree
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Dim lcr = r.LeadingCoeficient
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Dim s = lcr \ lcv
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Dim t As New Term(s, r.Degree() - dv)
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q += t
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r += rhs * -t
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End While
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Return q
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End Operator
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Public Overrides Function ToString() As String
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Dim builder As New StringBuilder
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Dim it = polyTerms.GetEnumerator()
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If it.MoveNext Then
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builder.Append(it.Current)
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End If
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While it.MoveNext
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If it.Current.Coefficient < 0 Then
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builder.Append(" - ")
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builder.Append(-it.Current)
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Else
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builder.Append(" + ")
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builder.Append(it.Current)
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End If
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End While
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Return builder.ToString
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End Function
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End Class
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Function GetDivisors(number As Integer) As List(Of Integer)
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Dim divisors As New List(Of Integer)
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Dim root = CType(Math.Sqrt(number), Long)
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For i = 1 To root
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If number Mod i = 0 Then
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divisors.Add(i)
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Dim div = number \ i
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If div <> i AndAlso div <> number Then
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divisors.Add(div)
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End If
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End If
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Next
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Return divisors
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End Function
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Private ReadOnly allFactors As New Dictionary(Of Integer, Dictionary(Of Integer, Integer)) From {{2, New Dictionary(Of Integer, Integer) From {{2, 1}}}}
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Function GetFactors(number As Integer) As Dictionary(Of Integer, Integer)
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If allFactors.ContainsKey(number) Then
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Return allFactors(number)
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End If
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Dim factors As New Dictionary(Of Integer, Integer)
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If number Mod 2 = 0 Then
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Dim factorsDivTwo = GetFactors(number \ 2)
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For Each pair In factorsDivTwo
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If Not factors.ContainsKey(pair.Key) Then
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factors.Add(pair.Key, pair.Value)
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End If
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Next
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If factors.ContainsKey(2) Then
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factors(2) += 1
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Else
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factors.Add(2, 1)
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End If
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If number < MAX_ALL_FACTORS Then
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allFactors.Add(number, factors)
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End If
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Return factors
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End If
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Dim root = CType(Math.Sqrt(number), Long)
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Dim i = 3L
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While i <= root
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If number Mod i = 0 Then
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Dim factorsDivI = GetFactors(number \ i)
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For Each pair In factorsDivI
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If Not factors.ContainsKey(pair.Key) Then
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factors.Add(pair.Key, pair.Value)
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End If
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Next
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If factors.ContainsKey(i) Then
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factors(i) += 1
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Else
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factors.Add(i, 1)
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End If
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If number < MAX_ALL_FACTORS Then
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allFactors.Add(number, factors)
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End If
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Return factors
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End If
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i += 2
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End While
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factors.Add(number, 1)
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If number < MAX_ALL_FACTORS Then
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allFactors.Add(number, factors)
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End If
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Return factors
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End Function
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Private ReadOnly computedPolynomials As New Dictionary(Of Integer, Polynomial)
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Function CyclotomicPolynomial(n As Integer) As Polynomial
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If computedPolynomials.ContainsKey(n) Then
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Return computedPolynomials(n)
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End If
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If n = 1 Then
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REM polynomial: x - 1
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Dim p As New Polynomial(New Term(1, 1), New Term(-1))
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computedPolynomials.Add(n, p)
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Return p
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End If
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Dim factors = GetFactors(n)
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Dim terms As New List(Of Term)
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Dim cyclo As Polynomial
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If factors.ContainsKey(n) Then
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REM n prime
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For index = 1 To n
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terms.Add(New Term(1, index - 1))
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Next
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cyclo = New Polynomial(terms)
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computedPolynomials.Add(n, cyclo)
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Return cyclo
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ElseIf factors.Count = 2 AndAlso factors.ContainsKey(2) AndAlso factors(2) = 1 AndAlso factors.ContainsKey(n / 2) AndAlso factors(n / 2) = 1 Then
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REM n = 2p
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Dim prime = n \ 2
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Dim coeff = -1
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For index = 1 To prime
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coeff *= -1
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terms.Add(New Term(coeff, index - 1))
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Next
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cyclo = New Polynomial(terms)
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computedPolynomials.Add(n, cyclo)
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Return cyclo
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ElseIf factors.Count = 1 AndAlso factors.ContainsKey(2) Then
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REM n = 2^h
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Dim h = factors(2)
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terms = New List(Of Term) From {
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New Term(1, Math.Pow(2, h - 1)),
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New Term(1)
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}
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cyclo = New Polynomial(terms)
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computedPolynomials.Add(n, cyclo)
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Return cyclo
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ElseIf factors.Count = 1 AndAlso factors.ContainsKey(n) Then
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REM n = p^k
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Dim p = 0
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Dim k = 0
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For Each it In factors
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p = it.Key
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k = it.Value
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Next
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For index = 1 To p
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terms.Add(New Term(1, (index - 1) * Math.Pow(p, k - 1)))
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Next
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cyclo = New Polynomial(terms)
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computedPolynomials.Add(n, cyclo)
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Return cyclo
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ElseIf factors.Count = 2 AndAlso factors.ContainsKey(2) Then
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REM n = 2^h * p^k
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Dim p = 0
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For Each it In factors
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If it.Key <> 2 Then
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p = it.Key
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End If
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Next
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Dim coeff = -1
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Dim twoExp = CType(Math.Pow(2, factors(2) - 1), Long)
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Dim k = factors(p)
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For index = 1 To p
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coeff *= -1
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terms.Add(New Term(coeff, (index - 1) * twoExp * Math.Pow(p, k - 1)))
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Next
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cyclo = New Polynomial(terms)
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computedPolynomials.Add(n, cyclo)
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Return cyclo
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ElseIf factors.ContainsKey(2) AndAlso (n / 2) Mod 2 = 1 AndAlso n / 2 > 1 Then
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REM CP(2m)[x] = CP(-m)[x], n odd integer > 1
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Dim cycloDiv2 = CyclotomicPolynomial(n \ 2)
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For Each t In cycloDiv2
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If t.Exponent Mod 2 = 0 Then
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terms.Add(t)
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Else
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terms.Add(-t)
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End If
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Next
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cyclo = New Polynomial(terms)
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computedPolynomials.Add(n, cyclo)
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Return cyclo
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End If
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#If ALGORITHM = 0 Then
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REM slow - uses basic definition
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Dim divisors = GetDivisors(n)
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REM Polynomial: (x^n - 1)
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cyclo = New Polynomial(New Term(1, n), New Term(-1))
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For Each i In divisors
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Dim p = CyclotomicPolynomial(i)
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cyclo /= p
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Next
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computedPolynomials.Add(n, cyclo)
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Return cyclo
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#ElseIf ALGORITHM = 1 Then
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REM Faster. Remove Max divisor (and all divisors of max divisor) - only one divide for all divisors of Max Divisor
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Dim divisors = GetDivisors(n)
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Dim maxDivisor = Integer.MinValue
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For Each div In divisors
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maxDivisor = Math.Max(maxDivisor, div)
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Next
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Dim divisorExceptMax As New List(Of Integer)
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For Each div In divisors
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If maxDivisor Mod div <> 0 Then
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divisorExceptMax.Add(div)
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End If
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Next
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REM Polynomial: ( x^n - 1 ) / ( x^m - 1 ), where m is the max divisor
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cyclo = New Polynomial(New Term(1, n), New Term(-1)) / New Polynomial(New Term(1, maxDivisor), New Term(-1))
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For Each i In divisorExceptMax
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Dim p = CyclotomicPolynomial(i)
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cyclo /= p
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Next
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computedPolynomials.Add(n, cyclo)
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Return cyclo
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#ElseIf ALGORITHM = 2 Then
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REM Fastest
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REM Let p ; q be primes such that p does not divide n, and q divides n
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REM Then Cp(np)[x] = CP(n)[x^p] / CP(n)[x]
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Dim m = 1
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cyclo = CyclotomicPolynomial(m)
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Dim primes As New List(Of Integer)
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For Each it In factors
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primes.Add(it.Key)
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Next
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primes.Sort()
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For Each prime In primes
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REM CP(m)[x]
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Dim cycloM = cyclo
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REM Compute CP(m)[x^p]
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terms = New List(Of Term)
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For Each t In cyclo
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terms.Add(New Term(t.Coefficient, t.Exponent * prime))
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Next
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cyclo = New Polynomial(terms) / cycloM
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m *= prime
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Next
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REM Now, m is the largest square free divisor of n
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Dim s = n \ m
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REM Compute CP(n)[x] = CP(m)[x^s]
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terms = New List(Of Term)
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For Each t In cyclo
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terms.Add(New Term(t.Coefficient, t.Exponent * s))
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Next
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cyclo = New Polynomial(terms)
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computedPolynomials.Add(n, cyclo)
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Return cyclo
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#Else
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Throw New Exception("Invalid algorithm")
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#End If
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End Function
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Sub Main()
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Console.WriteLine("Task 1: cyclotomic polynomials for n <= 30:")
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For i = 1 To 30
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Dim p = CyclotomicPolynomial(i)
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Console.WriteLine("CP[{0}] = {1}", i, p)
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Next
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Console.WriteLine()
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Console.WriteLine("Task 2: Smallest cyclotomic polynomial with n or -n as a coefficient:")
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Dim n = 0
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For i = 1 To 10
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While True
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n += 1
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Dim cyclo = CyclotomicPolynomial(n)
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If cyclo.HasCoefficentAbs(i) Then
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Console.WriteLine("CP[{0}] has coefficient with magnitude = {1}", n, i)
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n -= 1
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Exit While
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End If
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End While
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Next
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End Sub
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End Module
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