kmath/doc/operations.md

Spaces and fields

An obvious first choice of mathematical objects to implement in context-oriented style are algebra elements like spaces, rings and fields. Those are located in a scientifik.kmath.operations.Algebra.kt file. Alongside algebric context themselves, the file includes definitions for algebra elements such as FieldElement. A FieldElement object stores a reference to the Field which contains a additive and multiplicative operations for it, meaning it has one fixed context attached to it and does not require explicit external context. So those MathElements could be operated without context:

val c1 = Complex(1.0, 2.0)
val c2 = ComplexField.i
val c3 = c1 + c2

ComplexField also features special operations to mix complex numbers with real numbers like:

val c1 = Complex(1.0,2.0)
val c2 = ComplexField.run{ c1 - 1.0} //returns [re:0.0, im: 2.0]
val c3 = ComplexField.run{ c1 - i*2.0}

Note: In theory it is possible to add behaviors directly to the context, but currently kotlin syntax does not support that. Watch KT-10468 for news.

Nested fields

Algebra contexts allow to create more complex structures. For example, it is possible to create a Matrix from complex elements like this:

val element = NDElements.create(field = ComplexField, shape = intArrayOf(2,2)){index: IntArray ->
    Complex(index[0] - index[1], index[0] + index[1])
}

The element in this example is a member of Field of 2-d structures, each element of which is a member of its own ComplexField. The important thing is that one does not need to create a special nd-structure to hold complex numbers and implements operations on it, one need just to provide a field for its elements.

Note: Fields themselves do not solve problem of JVM boxing, but it is possible to solve with special contexts like BufferSpec. This feature is in development phase.