Higher Algebra with Operator Overloads (Brian Beckman)

Recently I did a Channel 9 interview with Beth Massi where I walked through a Visual Basic program that used Generics and Operator overloads to perform some higher mathematics. I thought I’d follow up with a post explaining the details of exactly what I did.


Operator overloads with Generics enable some beautiful designs for data types in Higher Algebra, a branch of mathematics, sometimes called Abstract Algebra. Consider fields and vector spaces. I’ll show you operator overloads at THREE levels in a single design. First, background: In this context, a “field” is a collection of unspecified objects, closed under two associative operators, + and *, that obey the distributive law.


Closed means that for any a, b, and c in the field, a+b and a*c are in the field. Associative means


a + (b + c) = (a + b) + c


a * (b * c) = (a * b) * c


Distributive means


a * (b + c) = a * b + a * c


(b + c) * a = b * a + c * a


The field must also have two special members: the additive unit 0 and the multiplicative unit 1, where


a + 0 = 0 + a = a


a * 1 = 1 * a = a


and must have for every a, an additive inverse, -a, such that a + -a = -a + a = 0. There must also be a multiplicative inverse for every element except 0, written 1/a, such that a * 1/a = 1/a * a = 1.


Some authors insist on the commutative laws (a + b = b + a, a * b = b * a), too, but we don’t, here. The most common examples of fields are the Rationals, the Real numbers, the Complex numbers, all of which have commutative addition and multiplication; and the Quaternions, which have non-commutative multiplication.


Don’t confuse the mathematical “field” with a “field” in a record, structure or class.


In each instance of a field, we define + and * to do anything we want so long as they obey the associative and distributive laws. A “vector space over a field” is the set of n-tuples or vectors built up from members of the field and closed under an additional linear combination law, written with no operator symbol, or sometimes with *. If v and w are any two vectors, and f and g are any two members of the underlying field, then f v + g w is a vector, and, furthermore,


f (v + w) = f v + f w


(v + w) g = v g + w g


(f + g) v = f v + g v


w (f + g) = w f + w g


Vector spaces are central in physics and simulation. Imagine 6-vectors of real numbers; such things represent particle states in “phase space” in classical mechanics. Imagine 4-vectors of complex numbers; such things appear in quantum mechanics. Similar structures occur all over Quantum Theory and Gravitation


References:



  • “Vector Calculus, Linear Algebra, and Differential Forms” [VCLADF], by John H. Hubbard, Barbara Burke Hubbard;

  • “Lie Groups, Lie Algebras, and Some of Their Applications,” by Robert Gilmore

  • “The Geometry of Physics,” by Theodore Frankel

Applications for vectors of quaternions are not so easy to come by, but, they are perfectly well defined, and, if we do our software design right, they “just work.” Ditto for vectors over a purely symbolic field or over any other kind. It’s not difficult to support field-like algebraic structures with non-associative multiplication within the same software design. Such things include the Cayley numbers or octonions, but, because of non-associativity, they don’t mesh easily with linear algebra, and that’s where we want to go. Stop with the quaternions.


We extend the design to inner-product spaces, in which every vector has a dual, and to linear algebras: sets of linear transformations of vectors, realized as matrices. With just a little code, we build a general, extensible, optimizable library suitable for physics, engineering, and mathematics in any finite-dimensional vector spaces.


We want three layers:



  • underlying types

  • field types, generic over the underlying types

  • linear-algebra types, generic over the field types

The underlying types comprise built-ins like Double, and custom types like Rational, Complex, Quaternion, and Symbol. Underlying types should implement the basic field operations, but can have many more operations for convenience, like optimized division routines. Operator overloading is a no-brainer for the underlying types.


At the top level of the linear-algebra types, operator overloads are again a no-brainer for vector + vector, matrix * vector, vector * matrix, matrix * matrix, vector * vector, and so on.


At the middle layer, the field types abstract and narrow the operations down to the bare minimum so that the linear-algebra types know what and only what to expect. In this layer, operator overloading is not quite a no-brainer, and there are a couple of different ways to go with a design.


The first paragraph showed the field axioms, and these are what we must design into our field types. One way to do that is to impose the field operations by interface:

Public Interface IField(Of U) ‘U is the “UNDERLYING” type

‘Get the underlying value.

Property UValue() As U

‘Generic users of IField type shouldn’t be able to tell
‘whether the Underlying type U has copy-reference
‘semantics or copy-value semantics, so let’s insist that
‘providers of IField implement a Dup operation.

Function Dup() As IField(Of U)

‘This is what fields do (would be nice to have a contract
‘for the distributive law). Must use a design like this
‘since Interfaces cannot host Shared functions in general,
‘and operator overloads in particular.

Function Add(ByVal that As IField(Of U)) As IField(Of U)
Function Mul(ByVal that As IField(Of U)) As IField(Of U)
Function AdditiveInverse(ByVal that As IField(Of U)) _
As IField(Of U)
Function MultiplicativeInverse(ByVal that As IField(Of U)) _
As IField(Of U)

‘Every field must have these. Implement them as Shared and
‘even Const if possible.

ReadOnly Property Zero() As IField(Of U)
ReadOnly Property Unity() As IField(Of U)

‘The following is an auxiliary operation for the Normed
‘Division Algebras to support Inner Product, or
‘”Sesquilinear mappings.” This computes the complex
‘conjugate, the quaternion conjugate, and so on. If the
‘underlying field does not have a natural dual field, then
‘it’s probably self-dual and just implement “Dual” as “Dup.”

Function Dual() As IField(Of U)

End Interface


An upside of this design is that both Structure types and Class types can implement the interface. Thus, the linear-algebra classes are independent of differences of copy semantics. Programmers may use Structure types for run-time speed at both the underlying-type level and at the field level.


Here’s an example of a Structure implementing a field of Doubles

Public Structure SFDouble
Implements IField(Of Double)

Private mValue As Double
Public Property
UValue() As Double Implements IField(Of Double).UValue
Get
Return
mValue
End Get
Set
(ByVal value As Double)
mValue = value
End Set
End Property

Public Sub New(ByVal d As Double)
mValue = d
End Sub

Public Function
Add(ByVal that As IField(Of Double)) As IField(Of Double) _
Implements IField(Of Double).Add
Return New SFDouble(Me.UValue + that.UValue)
End Function

Public Function Dual() As IField(Of Double) Implements IField(Of Double).Dual
Return New SFDouble(Me.UValue)
End Function

Public Function Dup() As IField(Of Double) Implements IField(Of Double).Dup
Return New SFDouble(Me.UValue)
End Function

Public Function AdditiveInverse(ByVal that As IField(Of Double)) As IField(Of Double) _
Implements IField(Of Double).AdditiveInverse
‘Let the underlying math throw exception here if that.UValue == 0
Return New FDouble(-Me.UValue)
End Function

Public Function MultiplicativeInverse(ByVal that As IField(Of Double)) As IField(Of Double) _
Implements IField(Of Double).MultiplicativeInverse
‘Let the underlying math throw exception here if that.UValue == 0
Return New FDouble(1 / Me.UValue)
End Function

Public Function Mul(ByVal that As IField(Of Double)) As IField(Of Double) _
Implements IField(Of Double).Mul
Return New SFDouble(Me.UValue * that.UValue)
End Function

Private Shared sUnity = New SFDouble(1.0)
Private Shared sZero = New SFDouble(0.0)

Public ReadOnly Property Unity() As IField(Of Double) Implements IField(Of Double).Unity
Get
Return
sUnity
End Get
End Property

Public ReadOnly Property Zero() As IField(Of Double) Implements IField(Of Double).Zero
Get
Return
sZero
End Get
End Property

End Structure


A Big Downside though, for the purposes of this exercise, is no operator overloads at the field level, because operator overloads
are Shared by definition (i.e., static) and interfaces can’t support static methods (the reason is that static virtual methods don’t make sense in .NET, and all methods in an interface are virtual).


But, we want operator overloads at the field level, and we can get them through an alternative base-class design for fields. We lose Structures at the field level, since particular fields must inherit from a base Class, but the underlying types can still be structures.

Public MustInherit Class AField(Of U) ‘U is the “UNDERLYING” type

Private mUValue As U
Public Property UValue() As U
Get
Return
mUValue
End Get
Set
(ByVal value As U)
mUValue = value
End Set
End Property

‘Now, our operator overloads just call virtual MustOverride Functions.
‘Magically, these are NON-COMMUTATIVE in general, just as we want.

Public Shared Operator +(ByVal e1 As AField(Of U), _
ByVal e2 As AField(Of U)) As AField(Of U)
Return e1.Add(e2)
End Operator

Public Shared Operator -(ByVal e1 As AField(Of U), _
ByVal e2 As AField(Of U)) As AField(Of U)
Return e1.Add(e2.AdditiveInverse())
End Operator

Public Shared Operator *(ByVal e1 As AField(Of U), _
ByVal e2 As AField(Of U)) As AField(Of U)
Return e1.Mul(e2)
End Operator

Public Shared Operator /(ByVal e1 As AField(Of U), _
ByVal e2 As AField(Of U)) As AField(Of U)
Return e1.Mul(e2.MultiplicativeInverse())
End Operator

Public Shared Operator -(ByVal that As AField(Of U)) As AField(Of U)
Return that.AdditiveInverse()
End Operator

Public Shared Operator Not(ByVal that As AField(Of U)) As AField(Of U)
Return that.Dual()
End Operator

‘This is what fields do (would be nice to have a contract for the
‘distributive law).

MustOverride Function Add(ByVal that As AField(Of U)) As AField(Of U)
MustOverride Function Mul(ByVal that As AField(Of U)) As AField(Of U)
MustOverride Function AdditiveInverse() As AField(Of U)
MustOverride Function MultiplicativeInverse() As AField(Of U)

‘Every field must have these. Implement them as Shared and even Const if
‘possible.

MustOverride ReadOnly Property Zero() As AField(Of U)
MustOverride ReadOnly Property Unity() As AField(Of U)

‘Generic users of IField type shouldn’t be able to tell
‘whether the Underlying type U has copy-reference
‘semantics or copy-value semantics, so insist that
‘subclasses of AField implement a Dup operation.

MustOverride Function Dup() As AField(Of U)

‘The following is an auxiliary operation for the Normed
‘Division Algebras to support Inner Product, or
‘”Sesquilinear mappings.” This computes the complex
‘conjugate, the quaternion conjugate, and so on. If the
‘underlying field does not have a natural dual field, then
‘it’s probably self-dual and just implement “Dual” as “Dup.”

MustOverride Function Dual() As AField(Of U)

End Class


Now we have operator overloads at the field level, but they’re not virtual (they can’t be). What they do is statically dispatch to
virtual ADD and MUL methods, which are just like the ones we had in the interface design for the field level. Nice, eh? Here are the implementations for the field of Complexes, which are implemented in an underlying Structure type:

Public Class AComplex
Inherits AField(Of Complex)
    Public Overrides Function Add(ByVal that As AField(Of Complex)) As AField(Of Complex)
Return New AComplex(Me.UValue + that.UValue)
End Function

Public Overrides Function AdditiveInverse() As AField(Of Complex)
Return New AComplex(-Me.UValue.R, -Me.UValue.I)
End Function

Public Overrides Function Dual() As AField(Of Complex)
Return New AComplex(Not Me.UValue)
End Function

Public Overrides Function Dup() As AField(Of Complex)
Return New AComplex(Me)
End Function

Public Overrides Function MultiplicativeInverse() As AField(Of Complex)
Return New AComplex(1.0 / Me.UValue)
End Function

Public Overrides Function Mul(ByVal that As AField(Of Complex)) As AField(Of Complex)
Return New AComplex(Me.UValue * that.UValue)
End Function

    Public Overrides ReadOnly Property Unity() As AField(Of Complex)
Get
Return New
AComplex(1.0, 0.0)
End Get
End Property

Public Overrides ReadOnly Property Zero() As AField(Of Complex)
Get
Return New
AComplex(0.0, 0.0)
End Get
End Property

Public Overrides Function ToString() As String
Return
UValue.ToString()
End Function

End Class


And here is its underlying type:

Imports System.Text

Public Structure Complex

Private mR As Double
Public Property
R() As Double
Get
Return
mR
End Get
Set
(ByVal value As Double)
mR = value
End Set
End Property

Private mI As Double
Public Property
I() As Double
Get
Return
mI
End Get
Set
(ByVal value As Double)
mI = value
End Set
End Property

Public Shared Operator
-(ByVal that As Complex) As Complex
Return New Complex(-that.R, -that.I)
End Operator

Public Shared Operator +(ByVal c1 As Complex, ByVal c2 As Complex) As Complex
Dim result = New Complex()
result.R = c1.R + c2.R
result.I = c1.I + c2.I
Return result
End Operator

Public Shared Operator *(ByVal c1 As Complex, ByVal c2 As Complex) As Complex
Dim result = New Complex()
result.R = (c1.R * c2.R) – (c1.I * c2.I)
result.I = (c1.R * c2.I) + (c1.I * c2.R)
Return result
End Operator

Public Shared Operator Not(ByVal c1 As Complex) As Complex
Return New Complex(c1.R, -c1.I)
End Operator

    Public Shared Operator *(ByVal c As Complex, ByVal scalar As Double) As Complex
Return New Complex(c.R * scalar, c.I * scalar)
End Operator

Public Shared Operator /(ByVal c As Complex, ByVal scalar As Double) As Complex
Return New Complex(c.R / scalar, c.I / scalar)
End Operator

Public Shared Operator /(ByVal c1 As Complex, ByVal c2 As Complex) As Complex
Return c1 * Not c2 / c2.MagnitudeSquared()
End Operator

End Structure

Now, we have an example of a Field generic on an underlying type, How about the Vector space? I wrote one vector space generic
on the interface-style Field design, and another vector space generic on the base-class Field design. Here is the latter one, it’s prettier:

Imports System.Text

‘The only reason I must mention the underlying type, U, here, is to pass it to the
‘Generic AField interface. I don’t use U anywhere explicitly inside this. If I had
‘”Monads” in the Generic type language, then I could thread U through this via a
‘”bind” operator, very pretty, but dreamland.

Public Class ANVector(Of U, T As {New, AField(Of U)})

Private mDimension As Integer
Public ReadOnly Property
Dimension() As Integer
Get
Return
mDimension
End Get
End Property

Private mComponents() As T
Default Property Component(ByVal i As Integer) As T
Get
Component = mComponents(i)
End Get
Set
(ByVal Value As T)
mComponents(i) = Value
End Set
End Property

    Public Sub New(ByVal that As ANVector(Of U, T))
mDimension = that.Dimension
ReDim mComponents(Dimension)
For i = 1 To Dimension
‘Cannot call New with parameters on Generic types … Sorry)
‘ –> Me.mComponents(i) = New T(that(i))
Me.mComponents(i) = that(i).Dup()
Next
End Sub

‘Add two vectors. PROPERLY, vectors should themselves implement IModule, etc. Some day…
Public Shared Operator +(ByVal V1 As ANVector(Of U, T), ByVal V2 As ANVector(Of U, T)) _
As ANVector(Of U, T)
If V1.Dimension <> V2.Dimension Then
Throw New
VectorSpaceException(“Cannot add vectors of different dimensions”)
Else
Dim
result = New ANVector(Of U, T)(V1) ‘clone
For i = 1 To result.Dimension
result(i) = result(i) + V2(i)”’ OPERATOR OVERLOADS BEING USED
Next
Return
result
End If
End Operator

‘Scale a vector
Public Shared Operator *(ByVal that As ANVector(Of U, T), ByVal scalar As T) _
As ANVector(Of U, T)
Dim result = New ANVector(Of U, T)(that)
For i = 1 To result.Dimension
result(i) = result(i) * scalar”’ OPERATOR OVERLOADS BEING USED
Next
Return
result
End Operator

Public Shared Operator *(ByVal scalar As T, ByVal that As ANVector(Of U, T)) _
As ANVector(Of U, T)
Return that * scalar”’ OPERATOR OVERLOADS BEING USED
End Operator

Public Function Dual()
Dim result = New ANVector(Of U, T)(Me.Dimension)
For i = 1 To result.Dimension
result(i) = Not result(i)”’ OPERATOR OVERLOADS BEING USED
Next
Return
result
End Function

Public Shared Operator Not(ByVal V As ANVector(Of U, T)) As ANVector(Of U, T)
Return V.Dual()
End Operator

‘Inner product
Public Shared Operator *(ByVal Left As ANVector(Of U, T), ByVal Right As ANVector(Of U, T)) _
As T
If Left.Dimension <> Right.Dimension Then
Throw New
VectorSpaceException _
(“Cannot compute inner product of vectors with different dimensions”)
Else
Dim
result As New T()
Dim temp As New T()
For i = 1 To Left.Dimension
result = result + Left(i) * Not Right(i)”’ OPERATOR OVERLOADS BEING USED
Next
Return
result
End If
End Operator

End Class
‘================================================================

Imports System.Text

Public Class AMNMatrix(Of U, T As {New, AField(Of U)})

Private mRowCount As Integer
Public ReadOnly Property
Rows() As Integer
Get
Return
mRowCount
End Get
End Property

Private mColumnCount As Integer
Public ReadOnly Property
Columns() As Integer
Get
Return
mColumnCount
End Get
End Property

Private mElements As T(,)
Default Property Element(ByVal i As Integer, ByVal j As Integer) As T
Get
Return
mElements(i, j)
End Get
Set
(ByVal Value As T)
mElements(i, j) = Value
End Set
End Property

‘Scale a matrix
Public Shared Operator *(ByVal scalar As T, ByVal that As AMNMatrix(Of U, T)) _
As AMNMatrix(Of U, T)
Dim result = New AMNMatrix(Of U, T)(that)
For i = 1 To that.Rows
For j = 1 To that.Columns
result(i, j) = result(i, j) * scalar ”’ OPERATOR OVERLOADS BEING USED
Next
Next
Return
result
End Operator

‘Scale a matrix
Public Shared Operator *(ByVal that As AMNMatrix(Of U, T), ByVal scalar As T) _
As AMNMatrix(Of U, T)
Return scalar * that”’ OPERATOR OVERLOADS BEING USED
End Operator

‘Matrix plus matrix
Public Shared Operator
+(ByVal M1 As AMNMatrix(Of U, T), ByVal M2 As AMNMatrix(Of U, T)) _
As AMNMatrix(Of U, T)
If M1.Rows <> M2.Rows Or M1.Columns <> M2.Columns Then
Throw New
VectorSpaceException(“Cannot add matrices of different dimensions”)
Else
Dim
result = New AMNMatrix(Of U, T)(M1)
For i = 1 To M1.Rows
For j = 1 To M1.Columns
result(i, j) = result(i, j) + M2(i, j) ”’ OPERATOR OVERLOADS BEING USED
Next
Next
Return
result
End If
End Operator

‘Matrix times a vector
Public Shared Operator *(ByVal m As AMNMatrix(Of U, T), ByVal v As ANVector(Of U, T)) _
As ANVector(Of U, T)
If m.Columns <> v.Dimension Then
Throw New
VectorSpaceException(“The number of columns in the matrix must equal ” & _
“the dimension of the vector”)
Else
Dim
result = New ANVector(Of U, T)(m.Rows)
For i = 1 To m.Rows
For j = 1 To m.Columns
result(i) = result(i) + v(j) * m(i, j) ”’ OPERATOR OVERLOADS BEING USED
Next
Next
Return
result
End If
End Operator

‘Vector times a Matrix
Public Shared Operator *(ByVal v As ANVector(Of U, T), ByVal m As AMNMatrix(Of U, T)) _
As ANVector(Of U, T)
If m.Rows <> v.Dimension Then
Throw New
VectorSpaceException(“The dimension of the vector must equal” & _
“the number of rows in the matrix”)
Else
Dim
result = New ANVector(Of U, T)(m.Columns)
For j = 1 To m.Columns
For i = 1 To m.Rows
result(j) = result(j) + v(i) * m(i, j) ”’ OPERATOR OVERLOADS BEING USED
Next
Next
Return
result
End If
End Operator

‘Matrix times Matrix
Public Shared Operator *(ByVal left As AMNMatrix(Of U, T), ByVal right As AMNMatrix(Of U, T)) _
As AMNMatrix(Of U, T)
If left.Columns <> right.Rows Then
Throw New
VectorSpaceException(“The number of columns in the left matrix must equal” & _
“The number of rows in the right matrix”)
Else
Dim
result = New AMNMatrix(Of U, T)(left.Rows, right.Columns)
Dim inner = left.Columns
For i = 1 To result.Rows
For j = 1 To result.Columns
For k = 1 To inner ”’ OPERATOR OVERLOADS BEING USED
result(i, j) = result(i, j) + left(i, k) * right(k, j)
Next
Next
Next
Return
result
End If
End Operator

End Class


There we go: operator overloading at three levels! I’ve uploaded a zipped Visual-Studio 2008 project to my Public folder with all this code.
Play around, let me know what you think.