inverse power method

Running the power method on A inverse instead of A, so it converges to the SMALLEST eigenvalue instead of the largest.

The inverse power method applies the power method's own iteration to $A^{-1}$ : $\mathbf x_{k+1}\propto A^{-1}\mathbf x_k$ , obtained in practice by solving $A\mathbf w=\mathbf x_k$ rather than forming $A^{-1}$ explicitly. Since $A^{-1}$ 's eigenvalues are the reciprocals $1/\lambda_i$ of $A$ 's own, $A^{-1}$ 's DOMINANT eigenvalue is $1/\lambda_{\min}$ -- the reciprocal of $A$ 's SMALLEST-magnitude eigenvalue -- so plain power iteration on $A^{-1}$ converges to that eigenvalue of $A$ instead of the dominant one.

Examples

If $A$ 's eigenvalues are $6,3,1$ , then $A^{-1}$ 's are $1/6,1/3,1$ -- the inverse power method converges to $1$ , the reciprocal of $A$ 's own smallest eigenvalue.