kmath/doc/algebra.md

Algebra and algebra elements

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

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

val c = space.run{a + b}

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

Algebra hierarchy

Mathematical contexts have the following hierarchy:

Space <- Ring <- Field

All classes follow abstract mathematical constructs. Space defines zero element, addition operation and multiplication by constant, Ring adds multiplication and unit one element, Field adds division operation.

Typical case of Field is the RealField which works on doubles. And typical case of Space is a VectorSpace.

In some cases algebra context could hold additional operation like exp or sin, in this case it inherits appropriate interface. Also a context could have an operation which produces an element outside of its context. For example Matrix dot operation produces a matrix with new dimensions which can be incompatible with initial matrix in terms of linear operations.

Algebra element

In order to achieve more familiar behavior (where you apply operations directly to mathematical objects), without involving contexts kmath introduces 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 to perform direct operations on Complex numbers without explicit involving the context like:

    val c1 = Complex(1.0, 1.0)
    val c2 = Complex(1.0, -1.0)
    val c3 = c1 + c2 + 3.0.toComplex()
    //or with field notation:
    val c4 = ComplexField.run{c1 + i - 2.0}

Both notations have their pros and cons.

The hierarchy for algebra elements follows the hierarchy for the corresponding algebra.

MathElement <- SpaceElement <- RingElement <- FieldElement

MathElement is the generic common ancestor of the class with context.

One important distinction between algebra elements and algebra contexts is that algebra element has three type parameters:

1. The type of elements, field operates on.
2. The self-type of the element returned from operation (must be algebra element).
3. The type of the algebra over first type-parameter.

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

Spaces and fields

An obvious first choice of mathematical objects to implement in a context-oriented style are algebraic elements like spaces, rings and fields. Those are located in the scientifik.kmath.operations.Algebra.kt file. Alongside common contexts, the file includes definitions for algebra elements like FieldElement. A FieldElement object stores a reference to the Field which contains additive and multiplicative operations, meaning it has one fixed context attached and does not require explicit external context. So those MathElements can 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 and real numbers, for example:

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 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 2-d structures, each element of which is a member of its own ComplexField. The important thing is 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.