Numerical Analysis

We now describe the power method for computing the dominant eigenpair. Its extension to the inverse power method is practical for finding any eigenvalue provided that a good initial approximation is known. Some schemes for finding eigenvalues use other methods that converge fast, but have limited precision. The inverse power method is then invoked to refine the numerical values and gain full precision. To discuss the situation, we will need the following definitions. If is an eigenvalue of A that is larger in absolute value than any other eigenvalue, it is called the dominant eigenvalue . An eigenvector corresponding to is called a dominant eigenvector . An eigenvector V is said to be normalized if the coordinate of largest magnitude is equal to unity (i.e., the largest coordinate in the vector V is the number 1). Remark: It is easy to normalize an eigenvector by forming a new vector where and . Theorem (Power Method) : Assume that the n