QR method

Suppose that A is a real symmetric matrix. Householder’s method is used to construct a similar tridiagonal matrix. Then the QR method is used to find all eigenvalues of the tridiagonal matrix. In the latter construction, plane rotations similar to those that were introduced in Jacobi’s method are used to construct the orthogonal matrices

. The important step the QR method is the factorization

and iteration

Definition (QR Decomposition): For a nonsingular square matrix , there exists a factorization

where is a unitary and is an upper triangular matrix.

Remark: is a unitary means .

For a real nonsingular square matrix , there exists a factorization

where is an orthogonal matrix and is an upper triangular matrix.

Remark: is a orthogonal means (also ).

QR transformation

After finding the factorization, the transformation is

.

We now investigate a well known and efficient method for finding all the eigenvalues of a general real matrix. The method can be used for an arbitrary real matrix, but for a general matrix it takes many iterations and becomes time consuming.

The method works much faster on special matrices, preferably:
(i) symmetric tri-diagonal,
(ii) Hessenberg matrices,
(iii) symmetric band matrices.

For this module, we will illustrate the QR method for real symmetric matrices.

When solving for eigenvalues of a dense symmetric matrix , the standard practice is to reduce the dense matrix to a tridiagonal matrix through a series of orthogonal transformations, and then to apply an eigenvalue solver for tridiagonal matrices to . The transformations applied to preserve eigenvalues, and the eigenvalues of and are the same.

The popular eigenvalue solver for symmetric tridiagonal matrices is called the implicit method. It applies a series of orthogonal transformations to a tridiagonal matrix which converges to a diagonal matrix . Furthermore, has the same eigenvalues as which are the diagonal elements of . In addition, the product of the orthogonal transformations is a matrix whose columns are the eigenvectors of . The method is called because in each iteration the factorization is computed. The LAPACK routine implementing the implicit algorithm on tridiagonal symmetric matrices is called DSTEQR.

QR Algorithm: The pseudocode for the QR method is:

1. i = 0
2.

3. repeat
4. Factor

6. i = i+1
7. until convergence

Comments

Runge-Kutta-Fehlberg Method

One way to guarantee accuracy in the solution of an I.V.P. is to solve the problem twice using step sizes h and and compare answers at the mesh points corresponding to the larger step size. But this requires a significant amount of computation for the smaller step size and must be repeated if it is determined that the agreement is not good enough. The Runge-Kutta-Fehlberg method (denoted RKF45) is one way to try to resolve this problem. It has a procedure to determine if the proper step size h is being used. At each step, two different approximations for the solution are made and compared. If the two answers are in close agreement, the approximation is accepted. If the two answers do not agree to a specified accuracy, the step size is reduced. If the answers agree to more significant digits than required, the step size is increased. Each Runge-Kutta-Fehlberg step requires the use of the following six values: Then an approximation to the solution of the I.V.P....

Van Der Pol System

The van der Pol equation is an ordinary differential equation that can be derived from the Rayleigh differential equation by differentiating and setting . It is an equation describing self-sustaining oscillations in which energy is fed into small oscillations and removed from large oscillations. This equation arises in the study of circuits containing vacuum tubes and is given by If , the equation reduces to the equation of simple harmonic motion The van der Pol equation is , where is a constant. When the equation reduces to , and has the familiar solution . Usually the term in equation (1) should be regarded as friction or resistance, and this is the case when the coefficient is positive. However, if the coefficient is negative then we have the case of "negative resistance." In the age of "vacuum tube" radios, the " tetrode vacuum tube " (cathode, grid, plate), was used for a power amplifie...

Newton-Raphson Method

This method is generally used to improve the result obtained by one of the previous methods. Let x o be an approximate root of f(x) = 0 and let x 1 = x o + h be the correct root so that f(x 1 ) = 0. Expanding f(x o + h) by Taylor ’s series, we obtain f(x o ) + hf’(x o ) + h 2 f’’(x o )/2! +-----= 0 Neglecting the second and higher order derivatives, we have f(x o ) + hf’(x o ) = 0 Which gives h = - f(x o )/f’(x o ) A better approximation than x o is therefore given be x 1 , where x 1 = x o – f(x o )/f’(x o ) Successive approximations are given by x 2 , x 3 ,----,x n+1 , where x n+1 = x n – f(x n )/f’(x n ) Which is the Newton-Raphson formula. Example: - Find a real root of the equation x 3 -5x + 3 = 0. Sol n : - Let, f(x) = x 3 -5x + 3 = 0 f’(x) = 3x 2 - 5 Choosing x o = 1 Step-1: f(x o ) = -1 f(x o ) = -2 So, x 1 =1 – ½ = 0.5 Step-2: f(x 1 ) = 0.625 f’(x 1 ) = -4.25 x 2 = 0.5 + 0.625/4.25 = 0.647 S...

Powell's Method

The essence of Powell's method is to add two steps to the process described in the preceding paragraph. The vector represents, in some sense, the average direction moved over the n intermediate steps in an iteration. Thus the point is determined to be the point at which the minimum of the function f occurs along the vector . As before, f is a function of one variable along this vector and the minimization could be accomplished with an application of the golden ratio or Fibonacci searches. Finally, since the vector was such a good direction, it replaces one of the direction vectors for the next iteration. The iteration is then repeated using the new set of direction vectors to generate a sequence of points . In one step of the iteration instead of a zig-zag path the iteration follows a "dog-leg" path. The process is outlined below. Let be an initial guess at the location of the minimum of the function . Let for be the ...

Numerical Analysis

Search This Blog

QR method

Labels

Comments

Post a Comment

Popular Posts

Runge-Kutta-Fehlberg Method

Van Der Pol System

Newton-Raphson Method

Powell's Method