Algebraic Structures and Algebraic Elements

The mathematical operations in KMath are generally separated from mathematical objects. This means that to perform an operation, say +, one needs two objects of a type T and an algebra context, which draws appropriate operation up, say Space<T>. Next one needs to run the actual operation in the context:

import kscience.kmath.operations.*

val a: T = ...
val b: T = ...
val space: Space<T> = ...

val c = space { a + b }

At first glance, this distinction seems to be a needless complication, but in fact one needs to remember that in mathematics, one could draw up different operations on same objects. For example, one could use different types of geometry for vectors.

Algebraic Structures

Mathematical contexts have the following hierarchy:

Algebra ← Space ← Ring ← Field

These interfaces follow real algebraic structures:

Space defines addition, its neutral element (i.e. 0) and scalar multiplication;
Ring adds multiplication and its neutral element (i.e. 1);
Field adds division operation.

A typical implementation of Field<T> is the RealField which works on doubles, and VectorSpace for Space<T>.

In some cases algebra context can hold additional operations like exp or sin, and then it inherits appropriate interface. Also, contexts may have operations, which produce elements outside of the context. For example, Matrix.dot operation produces a matrix with new dimensions, which can be incompatible with initial matrix in terms of linear operations.

Algebraic Element

To achieve more familiar behavior (where you apply operations directly to mathematical objects), without involving contexts KMath submits special type objects called MathElement. A MathElement is basically some object coupled to a mathematical context. For example Complex is the pair of real numbers representing real and imaginary parts, but it also holds reference to the ComplexField singleton, which allows performing direct operations on Complex numbers without explicit involving the context like:

import kscience.kmath.operations.*

// Using elements
val c1 = Complex(1.0, 1.0)
val c2 = Complex(1.0, -1.0)
val c3 = c1 + c2 + 3.0.toComplex()

// Using context
val c4 = ComplexField { c1 + i - 2.0 }

Both notations have their pros and cons.

The hierarchy for algebraic elements follows the hierarchy for the corresponding algebraic structures.

MathElement ← SpaceElement ← RingElement ← FieldElement

MathElement<C> is the generic common ancestor of the class with context.

One major distinction between algebraic elements and algebraic contexts is that elements have three type parameters:

The type of elements, the field operates on.
The self-type of the element returned from operation (which has to be an algebraic element).
The type of the algebra over first type-parameter.

The middle type is needed for of algebra members do not store context. For example, it is impossible to add a context to regular Double. The element performs automatic conversions from context types and back. One should use context operations in all performance-critical places. The performance of element operations is not guaranteed.

Spaces and Fields

KMath submits both contexts and elements for builtin algebraic structures:

import kscience.kmath.operations.*

val c1 = Complex(1.0, 2.0)
val c2 = ComplexField.i

val c3 = c1 + c2
// or
val c3 = ComplexField { c1 + c2 }

Also, ComplexField features special operations to mix complex and real numbers, for example:

import kscience.kmath.operations.*

val c1 = Complex(1.0, 2.0)
val c2 = ComplexField { c1 - 1.0 } // Returns: Complex(re=0.0, im=2.0)
val c3 = ComplexField { c1 - i * 2.0 }

Note: In theory it is possible to add behaviors directly to the context, but as for now Kotlin does not support that. Watch KT-10468 and KEEP-176 for updates.

Nested fields

Contexts allow one to build more complex structures. For example, it is possible to create a Matrix from complex elements like so:

val element = NDElement.complex(shape = intArrayOf(2, 2)) { index: IntArray ->
    Complex(index[0].toDouble() - index[1].toDouble(), index[0].toDouble() + index[1].toDouble())
}

The element in this example is a member of the Field of 2D structures, each element of which is a member of its own ComplexField. It is important one does not need to create a special n-d class to hold complex numbers and implement operations on it, one just needs to provide a field for its elements.

Note: Fields themselves do not solve the problem of JVM boxing, but it is possible to solve with special contexts like MemorySpec.

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Algebraic Structures and Algebraic Elements