Implementation of algebraic distribution

Presentation

Document concerned

This document deals with algebraic operations of distribution. I am writing this short document to clarify the necessary programming work.

Versions

07/12/2016: initial version

Author

Invisible Media

Project

Framework of the project

This project is a digital and algebraic calculation software. This project is a way to train very long and tedious equations and to factor them according to a common approach. Goal: solve equations, find all the solutions algebraically and get an equation based on parameters and fixed digital data.

All the features

The set of features is detailed in this document: All features (unbundled) .xlsx

Name of the feature covered here

Neurons of algebraic distribution

Relative need

Neurons are elements of graph also called node. In each node in the graph, there may be 0 to N branches or arc where the other end is the next node.

The arithmetic operations are one together recurring that perfectly fits the notion of graph. In particular, a complete equation is a tree set of operations. We're talking about graph when a node has multiple parents. An equation where some words are repeated will form a graph where a repeated word is associated with a single node, and each parent node will go on this node.

The neurons of algebraic distribution concern all the multiplication of terms.

Relationship with other features

Use

Neurons are objects that can communicate with each other. From the programming point of view, neurons form a directed graph. Then, each neuron with an application that is unique, the application traverses the graph through the neurons that react differently depending on the settings.


We note each node on a two-dimensional grid where is the number of nodes. Each square of the grid hosts a number positive or zero. This number is the number of links that connect two different nodes (in both directions). The diagonal axis of the grid corresponds to a diagonal of numbers zero, with the understanding that there is no node in relation to himself.

Explanations

Algebraic distribution sees both terms and and the product form a factorization that it should be distributed.

Also, when and/or shall include at least an addition or subtraction so distribution is to multiply each term the sum of by each term sum of .

Generally speaking, the distribution is to develop each term sum one by one.

distribute.png
  1. initial process
Develop the first term of each product
  1. iterative process
    1. calculate the term following multiplied by the previous term
    2. calculate the term following multiplied by the previous term
    3. develop the term following each product 3 final process
  1. Cancel and cancel


It will take alternate distribution and the simplification of the product.

There are more than 2 items produced to distribute, to first simplify the products obtained with the first two elements then repeat with two following. A special case of the product of terms is the term raised to the power ; What is the product of equal terms ??

Graphic view

Here's the graphic view of a neuron where how it graphically represents a neuron.

add-2-val.png add-2-var.png add-val-var.png add-var-val.png

The two branches on the right are the operands of the operation (here the addition). For each operation, there are 6 configurations.

Manufacture of neurons

I note that here are prefabricated neurons. A given equation, to break down the equation in terms of operations and build the graph of this equation using the templates available to neurons.

So there is an initial background where each neuron has its own function and its own form. I also note that there is only a single main command for an equation: calculate the result of the equation giving a numerical value to each variable. Related variables, to always express the values by using the equation of this variable. For free variables, any numeric value is correct. Finally, for the unknown variables it comes to seek his equation based on the knowledge of the equations on the variables and the values of the free variables.

To calculate the unknown variables, it must be remembered that the mathematical reasoning and algebra accept 2/3 variables to numeric values and one unknown. And usually, for a real function to a real variable, there is that two unknown variables; It is necessary and sufficient to create an intermediate unknown variable to get actually 2/3 variables to numeric values and one unknown.


Other operators are necessary to allow a set of mathematical sequences. Some instructions are needed:
  1. declaring a free variable
  2. declaration of a bound variable
  3. declaration of an unknown variable
  4. equality operator to check
  5. 5 own equality operator
  6. 6 comparison operators
  7. 7 conditional operator: a comparison test, a new sequence if the test result is checked, a new sequence (optional) if the test result is denied.


These instructions are defined by templates of neurons. Construction of neurons is done during syntactic analysis of these instructions by specializing each template.


The constructive form is just the composition of the operator according to a human-readable representation. Also, the first neuron is called the whole equation.

Finally, a heuristic decides the best form to give an equation considering the different models of algebraic operations:
  1. commutativity.
  2. neutral elements,
  3. absorbing elements,
  4. distribution.
  5. factorization.




Last edited Dec 11, 2016 at 9:48 PM by skercrow, version 3