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 Group<T>. Next one needs to run the actual operation in the context:

import space.kscience.kmath.operations.*

val a: T = ...
val b: T = ...
val group: Group<T> = ...

val c = group { 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

Primary mathematical contexts have the following hierarchy:

Field <: Ring <: Group <: Algebra

These interfaces follow real algebraic structures:

Group defines addition, its identity element (i.e., 0) and additive inverse (-x);
Ring adds multiplication and its identity element (i.e., 1);
Field adds division operation.

A typical implementation of Field<T> is the DoubleField 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 the context. For example, Matrix.dot operation produces a matrix with new dimensions, which can be incompatible with initial matrix in linear operations.

Spaces and Fields

KMath introduces contexts for builtin algebraic structures:

import space.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 space.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 }

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