Home Site Statistics  |   Contact  |   About Us       Monday, July 13, 2020   

j0182084- Back to Home

Skip Navigation Links.

   Skip Navigation LinksHOME ›  AREAS OF EXPERTISE ›   Multiple Eigenvalue  ›  ~ Eigenvalue Jacobi

"Eigenvalue Solutions"
Jacobi Method
λ =
x =

Matrix A = { { , , }, { , , }, { , , } }

[ Initial Values: {2,3,8},{4,3,3},{6,5,8} ]

Jacobi Method

An n-dimensional vector x is called an eigenvector of a square n x n matrix A if and only if it satisfies the linear equation

Ax = λx

Here λ is a scalar, and is refered to as eigenvalue corresponding to x. The above equation is usually called the eigenvalue equation or the eigenvalue problem.

A typical vector x changes direction when acted on by a matrix A, so that Ax is not a multiple of x

Eigenvalues problems that originate in engineering often end up with a symmetric or Hermitian matrix.

Notes on Jacobi Algorithm

This algorithm uses orthogonal transformations to reduce the matrix to a diagonal form. It is used to compute all eigenvalues and eigenvectors of a real symmetric matrix. This method is usually used for smaller matrices because its computational intensity increases very rapidly with the size of the matrix. For matrices of order up to 10 x 10, the Jacobi method is competitive with more sophisticated methods.

The advantage of this method is its robustness; when the Jacobi method is used, the solution is usually guaranteed for all real symmetric matrices.

Running this set up produces the results shown above.

The best way to show how our Jacobi algorithm works is by running real-time and showing results ready to compare. The reader can try variations above by entering new values.

Testing the Eigenvalue Jacobi Method

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

           static void TestJacobi();
                 double[] x;
                 MatrixR A = new MatrixR(new double[,] { { t1, t2, t3 }, { t4, t5, t6 },
                  { t7, t8, t9 } });
                 VectorR lambda;
                 Eigenvalue.Jacobi(A, 1e-5, out x, out lambda);
                 ListBox1.Items.Add(" " + lambda);
                 ListBox2.Items.Add(" " + x);

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

Consulting Services - Back to Home

Home Math, Analysis,

Eigen Inverse Iteration
Rayleigh-Quotient Method
Cubic Spline Method


Applied Mathematical Algorithms

Home A complex number z = x + iy, where...

Complex Functions
Home Non-linear system methods...

Non Linear Systems
Home Construction of differentiation...

Home Consider the function where...

About Us

KMP Engineering is an independent multidisciplinary engineering consulting company specializing in mathematical algorithms.

KMP Website >
Site Statistics >
Contact Us

2461 E Orangethorpe Ave Fullerton, CA 92631 USA info@keystoneminingpost.com
Site Map

> Home
> Areas of Expertise
> Reference Items
> Managed Services
> Login

Mining & Software Engineering

Since 2006 All Rights Reserved  © KMP Engineering LINKS | PRIVACY POLICY | LEGAL NOTICE