 Login  |   Contact  |   About Us       Tuesday, May 17, 2022  ~ Jacobi-Seidel ~ Gauss-Jordan ~ LU Decomposition

 "Linear Algebraic Equations" Gauss-Jordan Elimination Method
 Solution from the GAUSS-JORDAN elimination

 X = (8,-3,-5)

 Matrix A: = { , , }, { , , }, { , , } Vector b: = { , , } [ Initial MatrixValues A: { 2, 1, 1 }, { 3, 1, 2 }, { 2, 1, 2 } ] [ Initial VectorValues b: { 8, 11, 3 } ]

IMPLEMENTATION
Gauss-Jordan elimination

Solving sets of linear equations using Gauss-Jordan elimination produces both the solution of the equations and the inverse of the coefficient matrix. The Gauss-Jordan elimination process requires two steps. The first step, called the forward elimination, reduces a given system to either triangular or echelon form, or results in a degenerate equation indicating that the system has no solution. This is accomplished through the use of elementary operations. The second step, called the backward elimination, uses back-substitution to find the solution of the linear equations.

Iteration Algorithm Creation

In this method, the value of unknowns for triangulated equations are found, which can be done by using back-substitution. Starting with the last equation, which contains the single unknowns, a solution can be found immediately. This value may be substituted into the second last equation in order to calculate the next unknown. The process is repeated until all unknowns are found. Testing Gauss-Jordan elimination Method

In order to test the method as defined above, a new TestGaussJordan() static method has been added and executed. Supporting code and methods are not shown.

A set of initial equations for the test has been defined for Matrix A and Vector b.

static void TestGaussJordan();
{
ListBox1.Items.Clear();
LinearSystem ls = new LinearSystem();
MatrixR A = new MatrixR(new double[3, 3] { { t1, t2, t3 }, { t4, t5, t6 }, { t7, t8, t9 } });
VectorR b = new VectorR(new double { t10, t11, t12 });
VectorR x = ls.GaussJordan(A, b);
ListBox1.Items.Add("(" + x + "," + x + "," + x + ")");
}

An initial set of data points was defined for matrix A and vector b. The user can manipulate all values and try variations on the matrix and vector themselves.

 Other Implementations...

 Object-Oriented Implementation Graphics and Animation Sample Applications Ore Extraction Optimization Vectors and Matrices Complex Numbers and Functions Ordinary Differential Equations - Euler Method Ordinary Differential Equations 2nd-Order Runge-Kutta Ordinary Differential Equations 4th-Order Runge-Kutta Higher Order Differential Equations Nonlinear Systems Numerical Integration Numerical Differentiation Function Evaluation  Math, Analysis,
expertise..."

EIGENVALUE
SOLUTIONS...

> Rayleigh-Quotient Method

> Cubic Spline Method

 Applied Mathematical Algorithms ComplexFunctions NonLinear Differentiation Integration
 About Us KMP Software Engineering is an independent multidisciplinary engineering consulting company specializing in mathematical algorithms. Areas of Expertise SpecialFunctions VectorsMatrices OptimizationMethods ComplexNumbers Interpolation CurveFitting NonLinearSystems LinearEquations DistributionFunctions NumericalDifferentiation NumericalIntegration DifferentialEquations Smalltalk FiniteBoundary Eigenvalue Graphics UnderstandingMining MiningMastery MineralNews MineralCommodities MineralForum Crystallography Services NumericalModeling WebServices MainframeServices OutsourceServices LINKED IN KMP ARTICLES Brand Login Contact