The use of polynomials

TLDR: In this paper, the authors discuss the use of polynomials and quadratic transformations to improve the convergence of the power method and show that in many cases, this is the best choice and in every case it will improve the relation.

Abstract: Publisher Summary If the eigenvalues are known approximately, the convergence of the power method can largely be improved by increasing the quotient |λ 1 /λ 2 |. The most simple case is that of adding to all eigenvalues the same constant c . If all eigenvalues are positive, one subtracts (λ 2 + λ n )/2. Knowing for instance that a four-rowed matrix has approximately the eigenvalues—12; 6; 4; 1, one subtracts 3,5 and has the new approximate eigenvalues—8,5; 2,5; 0,5; − 2, 5. Therefore, the quotient λ 1 /λ 2 has increased from 12/6 = 2 to 8,5/2,5 = 3,4. As is well known the eigenvectors remain unchanged. When λ 1 > 0, but the other eigenvalues are partially 2 and the first negative eigenvalue from all eigenvalues. In many cases, this is the best choice and in every case, it will improve the relation. This chapter discusses the use of polynomials and quadratic transformations.