Gaussian Elimination Calculator

Gaussian Elimination Calculator (3×3 System)

Enter Coefficients:

For a system of 3 linear equations:

a11x + a12y + a13z = b1
a21x + a22y + a23z = b2
a31x + a32y + a33z = b3

= = =

Calculation Result:

Enter the coefficients and click "Calculate Solution" to see the result.

Understanding Gaussian Elimination

Gaussian elimination is a fundamental algorithm in linear algebra used to solve systems of linear equations. It systematically transforms a system of equations into an equivalent system that is easier to solve, typically in row echelon form, using a series of elementary row operations.

How it Works

The process involves two main stages:

  1. Forward Elimination: The goal here is to transform the augmented matrix of the system into an upper triangular matrix (row echelon form). This is achieved by performing elementary row operations to create zeros below the main diagonal. The elementary row operations are:
    • Swapping two rows.
    • Multiplying a row by a non-zero scalar.
    • Adding a multiple of one row to another row.
    This stage eliminates variables from equations below the current pivot equation.
  2. Backward Substitution: Once the matrix is in row echelon form, the system can be easily solved by starting from the last equation (which will have only one variable) and substituting its value back into the equations above it to find the values of the other variables.

Applications

Gaussian elimination is widely used in various fields, including:

  • Engineering: Solving circuit analysis problems, structural analysis.
  • Computer Graphics: Transformations, rendering.
  • Economics: Input-output models, equilibrium analysis.
  • Statistics: Regression analysis, solving normal equations.
  • Mathematics: Finding matrix inverses, determinants, and solving general linear systems.

Using the Calculator

This calculator is designed for a 3×3 system of linear equations. Follow these steps:

  1. Input Coefficients: Enter the numerical coefficients (aij) for x, y, and z, and the constant terms (bi) for each of the three equations into the respective input fields.
  2. Example: For the system:
    2x + 1y - 1z = 8
    -3x - 1y + 2z = -11
    -2x + 1y + 2z = -3
    You would enter: a11=2, a12=1, a13=-1, b1=8; a21=-3, a22=-1, a23=2, b2=-11; a31=-2, a32=1, a33=2, b3=-3.
  3. Calculate: Click the "Calculate Solution" button.
  4. View Result: The calculator will display the values for x, y, and z if a unique solution exists. If the system is singular (no unique solution) or inconsistent (no solution), an appropriate message will be shown.

This tool simplifies the complex process of Gaussian elimination, allowing you to quickly find solutions to systems of linear equations without manual calculation.

function calculateGaussian() { var n = 3; // For a 3×3 system var matrix = []; var inputsValid = true; // Read inputs and populate matrix for (var i = 0; i < n; i++) { matrix[i] = []; for (var j = 0; j < n; j++) { var val = parseFloat(document.getElementById('a' + (i + 1) + (j + 1)).value); if (isNaN(val)) { inputsValid = false; break; } matrix[i][j] = val; } if (!inputsValid) break; var bVal = parseFloat(document.getElementById('b' + (i + 1)).value); if (isNaN(bVal)) { inputsValid = false; break; } matrix[i][n] = bVal; // Augmented part } var resultDiv = document.getElementById('gaussianResult'); if (!inputsValid) { resultDiv.innerHTML = 'Please enter valid numbers for all matrix coefficients and constants.'; return; } // Create a copy of the matrix for calculations to avoid modifying the original input representation if needed var augmentedMatrix = []; for (var r = 0; r < n; r++) { augmentedMatrix[r] = []; for (var c = 0; c <= n; c++) { augmentedMatrix[r][c] = matrix[r][c]; } } // Gaussian Elimination (Forward Elimination) var epsilon = 1e-9; // Small value to check for near-zero for (var k = 0; k < n; k++) { // k is the current pivot row/column // Find pivot: row with largest absolute value in current column k var i_max = k; var max_val = Math.abs(augmentedMatrix[k][k]); for (var i = k + 1; i max_val) { max_val = Math.abs(augmentedMatrix[i][k]); i_max = i; } } // Swap rows if necessary if (i_max !== k) { var temp = augmentedMatrix[k]; augmentedMatrix[k] = augmentedMatrix[i_max]; augmentedMatrix[i_max] = temp; } // Check for zero pivot if (Math.abs(augmentedMatrix[k][k]) < epsilon) { // This indicates a singular matrix. It could be no solution or infinite solutions. // For simplicity, we'll report no unique solution. resultDiv.innerHTML = 'The system of equations is singular or has no unique solution.'; return; } // Eliminate below the pivot for (var i = k + 1; i < n; i++) { var factor = augmentedMatrix[i][k] / augmentedMatrix[k][k]; for (var j = k; j = 0; i–) { var allZerosLeft = true; for (var j = 0; j epsilon) { allZerosLeft = false; break; } } if (allZerosLeft && Math.abs(augmentedMatrix[i][n]) > epsilon) { resultDiv.innerHTML = 'The system of equations is inconsistent (no solution).'; return; } } // Backward Substitution var x = new Array(n); for (var i = n – 1; i >= 0; i–) { var sum_terms = 0; for (var j = i + 1; j < n; j++) { sum_terms += augmentedMatrix[i][j] * x[j]; } // Check for division by zero again, in case a pivot became zero during elimination if (Math.abs(augmentedMatrix[i][i]) < epsilon) { resultDiv.innerHTML = 'The system of equations is singular or has no unique solution.'; return; } x[i] = (augmentedMatrix[i][n] – sum_terms) / augmentedMatrix[i][i]; } // Display results var outputHTML = '

Solution:

'; outputHTML += 'x = ' + x[0].toFixed(6) + "; outputHTML += 'y = ' + x[1].toFixed(6) + "; outputHTML += 'z = ' + x[2].toFixed(6) + "; resultDiv.innerHTML = outputHTML; }

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