Network operator matrix

Network operator matrix (NOM) is a representation of mathematical expressions in computer memory.

NOM is a new approach to the problem of automatic search of mathematical equations. The researcher defines the sets of operations, variables and parameters. The computer program generates a number of mathematical equations that satisfy given restrictions. Then the optimization algorithm finds the structure of appropriate mathematical expression and its parameters. Network operator is a directed graph that corresponds to some mathematical expressions. Every source nodes of the graph are variables or constants of mathematical expression, inner nodes correspond to binary operations and edges correspond to unary operations. The calculation’s result of mathematical expression is kept in the last sink node.

Example

Consider the following mathematical expression

The graph for Expression (), is presented in Fig. 1.

On the edges we place unary operations

ρ₁(z) = z ;
ρ₃(z) =  − z ;
$\rho_6(n) = \begin{cases} \varepsilon\!^{ -1}, & \text{if }z>\ln \varepsilon \!\\ e^z, & \text{ otherwise} \end{cases} ;$
ρ₁₂(z) = sin z ;

In the inner and sink nodes we place binary operations
χ₀(z′,z″) = z′ + z″ ;
χ₁(z′,z″) = z′ × z″ ;

Expression 1 can be presented in the PC memory as a NOM

$\backslash Psi\; =\; \backslash begin\{bmatrix\}$

0 & 0 & 0 & 1 & 1 & 0 & 0 & 12 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 3 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0

\end{bmatrix}

Any mathematical expression can be presented as a network operator matrix.
To calculate the mathematical expression by the network operator matrix the node vector is used.
Each component of node vector corresponds to the nodes of the network operator graph. Initially each component is equal to the argument for the given node or the unit element of binary operation.
For addition and multiplication the number of operation equals its unit element. For addition it is 0, for multiplication it is 1. The node vector for the given example

z = [x₁ q₁ x₂ 0 1 0 1 0]^T

The calculation by matrix Ψ is performed for nonzero nondaigonal elements ψ_(i,j) ≠ 0 by

z_j = χ_ψj, j(z_j,ρ_ψi, j(z_i))

Follow the rows of the matrix.
For the Row 1 we have ψ_1, 4 = 1, that is i = 1, j = 4, ψ_j, j = ψ_4, 4 = 0.

Take arguments from z = [x₁ q₁ x₂ 0 1 0 1 0]^T

z₁ = x₁, z₄ = 0 then

z₄ = χ_ψ4, 4(z₄,ρ_ψ1, 4(z₁)) = χ₀(z₄,ρ₁(z₁)) = 0 + x₁ = x₁

Further in Row 1 we have ψ_1, 5 = 1

z₅ = χ_ψ5, 5(z₅,ρ_ψ1, 5(z₁)) = χ₁(z₅,ρ₁(z₁)) = 1 × x₁ = x₁

ψ_1, 8 = 12

z₈ = χ_ψ8, 8(z₈,ρ_ψ1, 8(z₁)) = χ₀(z₈,ρ₁₂(z₁)) = 0 + sin x₁ = sin x₁.

As a result after Row 1 we have z = [x₁ q₁ x₂ x₁ x₁ 0 1 sinx₁]^T.

For the Row 2 we have ψ_2, 5 = 1
z₅ = χ_ψ5, 5(z₅,ρ_ψ2, 5(z₂)) = χ₁(z₅,ρ₁(z₁)) = x₁q₁

then follow the rows

ψ_3, 6 = 3
z₆ = χ_ψ6, 6(z₆,ρ_ψ3, 6(z₃)) = χ₀(z₆,ρ₃(z₃)) = 0 + (−x₂) =  − x₂


ψ_4, 8 = 1

z₈ = χ_ψ8, 8(z₈,ρ_ψ4, 8(z₄)) = χ₀(z₈,ρ₁(z₄)) = sin x₁ + x₁


ψ_5, 7 = 1

z₇ = χ_ψ7, 7(z₇,ρ_ψ5, 7(z₅)) = χ₁(z₇,ρ₁(z₅)) = 1x₁q₁ = x₁q₁


ψ_6, 7 = 6

z₇ = χ_ψ7, 7(z₇,ρ_ψ6, 7(z₆)) = χ₁(z₇,ρ₆(z₆)) = x₁q₁e^−x₂


ψ_7, 8 = 1

z₈ = χ_ψ8, 8(z₈,ρ_ψ7, 8(z₇)) = χ₀(z₈,ρ₁(z₇)) = sin x₁ + x₁ + x₁q₁e^−x₂.