Expressions¶
While there exists many functions that operate on the individual elements of a tensor, sometimes the elements of a tensor need to be combined in some expression to produce a single element as the result. For example, if one wants to create the sum of the numbers of a tensor containing the values 1.0, 2.0, and 3.0, the expression 1.0 + 2.0 + 3.0 needs to be evaluated.
To do this, the package Orka.Numerics.Tensors
contains the interface type Expression to reduce the numbers in a tensor
using an arbitrary expression. An expression used in a reduction always
needs to involve two elements, represented by the functions X and Y.
These two functions return an Expression and can be used to further build
more complex expressions. For example, the expression above is evaluated
as (1.0 + 2.0) + 3.0.
The binary operators +, -, *, and /, and the functions Min and
Max can be used on two expressions or an expression and an element.
The unary operators - and abs, and the function Sqrt can only be used
on a single expression.
An expression can be constructed and then used repeatedly by the function
Reduce to reduce the elements of a tensor to a single element:
Expression_Sum : constant CPU_Expression := X + Y;
Note that X and Y are not parameters but functions, each returning
an Expression. The expression in Expression_Sum can then be used in
a reduction:
Count : constant Element := Object.Reduce (Expression_Sum, 0.0);
If an expression is associative, then function Reduce_Associative instead
of Reduce can be used.
The second parameter of the function Reduce contains the initial value of
the result and may be used as the actual parameter of one of the two arguments
when evaluating an expression. For example, when computing the sum, the
initial value should be 0.0, while computing the product requires the value 1.0.
There exist a few predefined functions that perform some common reductions
like the sum or product, and the minimum or maximum value in a tensor. These
are the functions Sum, Product, Min, and Max.
Summary
A binary operation * is associative if for any a, b, and c,
it is true that:
a * (b * c) = (a * b) * c