Client Account:   Login
Home Site Statistics   Contact   About Us   Monday, April 24, 2017

users on-line: 2 | Forum entries: 6   
j0182018 - Back to Home
   Skip Navigation LinksHOME › AREAS OF EXPERTISE › Multiple Eigenvalue Solutions › ~ Eigen Rayleigh-Quotient Method


"Eigenvalue Solutions"
Rayleigh-Quotient Method
Results for an initial vector filled with random numbers:
λ =
x =
Results for an initial vector generated from the Rayleigh-Quotient method:
λ =
x =

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



[ Initial Values: {4,3,6},{3,7,1},{6,1,9} ]

IMPLEMENTATION
Rayleigh-Quotient Method

The Rayleigh-Quotient method is a variant of the Inverse Iteration Method, (please refer to "Eigenvalue Inverse Iteration Method") for computing one of the eigenvalues and the corresponding eigenvector. This method changes the shift in each iteration, so it does not necessarily find the dominant eigenvalue.

Algorithm

For a given symmetric matrix A and nonzero vector v, the Rayleigh quotient is defined as:

R(A,v) = vTAv / vTv

where vT is the transpose of v. Also, R(A,cv) = R(A,v) for any scalar constant c.

The Rayleigh quotient reaches its smallest eigenvalue of A,λmin when v is the xmin (the corresponding eigenvector). Similarly,

R(A,v) <= λmax    and    R(A,xmax) = λmax

In order to use the Rayleigh-Quotient method to compute the eigenvalue and the corresponding eigenvector, we followed the process below:

  • For a given n x n symmetric A and an initial vector x0 normalize x0.

  • Calculate the quantity λ = x0T Ax0.

  • Solve the equation (A - λI)x = x0 for x.

  • Set x0 = x.

  • Repeat the above steps until the relative change in λ is less than the tolerance.

In addition to the matrix A and the tolerance, this method also takes as input an integer flag parameter. This flag parameter takes two values, 1 and 2, corresponding to the case that the initial vector is filled with random numbers and the case that initial vector and eigenvalue are approximate solutions.

Running this set up produces the results shown above.

The reader can try variations by entering new values to Matrix A.

Testing the Rayleigh-Quotient Method

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

           static void Rayleigh-Quotient();
              {
                 ListBox1.Items.Clear();
                 ListBox2.Items.Clear();
                 Label5.Text = "";
                 MatrixR A = new MatrixR(new double[,] { { t1, t2, t3 }, { t4, t5, t6 },
                  { t7, t8, t9 } });
                 VectorR x;
                 double lambda;
                 Eigenvalue.RayleighQuotient(A, 1e-8, 1, out x, out lambda);
                 ListBox1.Items.Add(" " + lambda);
                 ListBox1.Items.Add(" " + x);
                 x = new VectorR(3);
                 lambda = 0.0;
                 Eigenvalue.RayleighQuotient(A, 1e-8, 2, out x, out lambda);
                 ListBox2.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
Skip Navigation Links.

Home

Home Math, Analysis & More,
  our established expertise..."

EIGENVALUE SOLUTIONS...
  Eigen Inverse Iteration
  Rayleigh-Quotient Method

INTERPOLATION APPLICATIONS...
  Cubic Spline Method
  Newton Divided Difference

 

Applied Mathematical Algorithms

    Home Complex Functions
A complex number z = x + iy, where...
     Home Non-Linear Systems
Non-linear system methods...
     Home Numerical Differentiation
Construction of numerical differentiation...
     Home Numerical Integration
Consider the function I = ah f(x)dx where...
 

2006-2017 © Keystone Mining Post  |   2461 E. Orangethorpe Av., Fullerton, CA 92631 USA  |   info@keystoneminingpost.com