linearSystemUtility

tools for exploring linear systems

what is a System of linear equations

project specifications

features

takes systems of linear equations in augmented matrix notation and find if the systems have: no solution, one solution, infinite solutions. if there is a solution defines it.

##find if the systems has: no solution, one solution, infinite solutions

what is a solving a system of linear equationsof linear equations

3 row operations:

1. (switch) switch 2 rows positions
2. (scaling) multiply every entry in a row by a non-zero constant
3. (rowReplacement) replace a row with its sum and a multiple of a different row and a non-zero constant

API

LinearSystemSolution

const intObject = [
  [2,3,0,5],
  [0,0,1,20],
  [0,-2,0,5]
]
const solution = LinearSystemSolution(intObject);

return a description of the systems solution (or lack of one)

initializeSystem

1. the left most non-zero column is the pivot column
2. the top row position of the pivot column is the pivot row

foreach row: a. partialPivoting on the pivotColumn - largest to top b. zeroAllRowsUnderThePevot() - row replacement operations to "zero" all entry under the pivot c. pivot.row++, pivot.column++ 3. foreach row: a. zeroAllRowsAboveThePivot(), scalePivotToOne b. pivot.row--, pivot--

switchRows

switchRows(system, rowNumberOne, rowNumberTwo) exchange the location of 2 rows

scaling

scaling(system, rowNumber, scaleNumber) scales all entries in row by a non-zero number

rowReplacement

rowReplacement(system, rowToBeReplaced, differentRowNumber, scaleNumberForOtherRow) replaces the row in rowToBeReplaced with the the product of the row in differentRowNumber and scaleNumberForOtherRow

zeroAllRowsUnderThePivot

iterate over all entries below the pivot, use rowReplacementRemover on each

zeroAllRowsAboveThePivot

iterate over all entries above the pivot, use rowReplacementRemover on each

rowReplacementRemover

zeros out the entry using rowReplacement and the pivot

scalePivotToOne

uses scaling on the row of pivot to scale the entry in the pivot position to 1

partialPivoting(system)

rearranges rows to move the largest absolute value to the top of the column row to the top. this is sometimes called partial pivoting

isEchelonForm

returns a bool indicating if the matrix is in echelon form

isReducedEchelonForm

returns a bool indicating if the matrix is in reduced echelon form

possible features or research topics

graph solutions

after solved a nice chart to look at if, low enough number of dimensions

animation of algorithm behavior

ui element that shows system and animates switching, adding, scaling

optional parameter bool back-substitution instead of going from echelon to reduced echelon form

function approximation of this class with less complexity cost or overhead

find cases that are easy examples where skipping one or more steps doesn't affect the solution or an approximation of it

how do linear systems handle infinite entries

many checks for non-zero numbers in code check like:

Math.abs(x) > 0

could also be

!isNaN(parseFloat(num)) && isFinite(num);

not sure how or if linear algebra handle infinite, or how could relate to the js version of infinity

how neural networks solve linear system

there are many ways, check them out...... If you want me to list resources make an issue