What is power iteration algorithm?

In mathematics, power iteration (also known as the power method) is an eigenvalue algorithm: given a diagonalizable matrix , the algorithm will produce a number , which is the greatest (in absolute value) eigenvalue of , and a nonzero vector , which is a corresponding eigenvector of , that is, .

What is Rayleigh power method?

NUMERICAL METHODS. Theorem 10.2. Determining an Eigenvalue. from an Eigenvector. If x is an eigenvector of a matrix A, then its corresponding eigenvalue is given by This quotient is called the Rayleigh quotient.

Why power method is used?

The Power Method is used to find a dominant eigenvalue (one with the largest absolute value), if one exists, and a corresponding eigenvector. To apply the Power Method to a square matrix A, begin with an initial guess for the eigenvector of the dominant eigenvalue.

Why do we use power method?

Why does the power method work?

power method normalizes the products Aq(k−1) to avoid overflow/underflow, therefore it converges to x1 (assuming it has unit norm). The power method converges if λ1 is dominant and if q(0) has a component in the direction of the corresponding eigenvector x1.

What are the conditions required for convergence of power method?

The power method converges if λ1 is dominant and if q(0) has a component in the direction of the corresponding eigenvector x1. In practice, the useful- ness of the power method depends upon the ration |λ2|/|λ1|, since it dictates the rate of convergence.

What are the disadvantages of power method?

Disadvantages include that it might be slow, and it can in its most naive form only compute the largest eigenvalue of a matrix. Also, it gets into trouble if the largest eigenvalue corresponds to an eigenspace with dimension more than 1, since it only finds you one approximate eigenvector.

What we can determine using power method?

w:Power method is an eigenvalue algorithm which can be used to find the w:eigenvalue with the largest absolute value but in some exceptional cases, it may not numerically converge to the dominant eigenvalue and the dominant eigenvector.