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 HOME ›  AREAS OF EXPERTISE  #3 ›   Multiple Eigenvalue  ›  ~ Eigenvalue Jacobi

 "Eigenvalue Solutions" Jacobi Method
 λ = (15, 1.64575131106459, -3.64575131106459)
 x = (0.548821299948452, -0.074743117328568, 0.832591524779648 0.329292779969071, 0.93479371129953, -0.133142714314723 0.768349819927832, -0.34723793532225, -0.537647068707724)

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

IMPLEMENTATION
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();               {                  ListBox1.Items.Clear();                  ListBox2.Items.Clear();                  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);               }

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 Quotes

Math, Analysis,
expertise..."

EIGENVALUE
SOLUTIONS...

> Rayleigh-Quotient Method

> Cubic Spline Method

 Applied Mathematical Algorithms
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