• The Power Method finds the eigenvector associated with the largest eigenvalue (in magnitude). It requires a matrix-vector multiplication for each iteration, thus costing approximately $2 m^2$ flops per iteration if $A$ is a dense $m \times m$ matrix. The convergence is linear.
• The Inverse Power Method finds the eigenvector associated with the smallest eigenvalue (in magnitude). It requires the solution of a linear system for each iteration. By performance an LU factorization with partial pivoting (or Cholesky factorization if the matrix is Hermitian), the investment of an initial $O( m^3 )$ expense then reduces the cost per iteration to approximately $2 m^2$ flops. if $A$ is a dense $m \times m$ matrix. The convergence is linear.
• The Rayleigh Quotient Iteration finds an eigenvector, but with which eigenvalue it is associated is not clear from the start. It requires the solution of a linear system for each iteration. If computed via an LU factorization with partial pivoting (or Cholesky factorization if the matrix is Hermitian), the cost per iteration is $O( m^3 )$ per iteration, if $A$ is a dense $m \times m$ matrix. The convergence is quadratic if $A$ is not Hermitian, and cubic if it is.
The cost of these methods is greatly reduced if the matrix is sparse, in which case each iteration may require as little as $O( m )$ per iteration.