| Summary: | Inverse iteration is a well-known algorithm for approximating an eigenvector of a symmetric matrix. If the shift is close to an eigenvalue, the system is nearly singular, potentially making it difficult to solve. Another drawback is its q-linear rate of convergence. An equivalent algorithm that is based on Newton's method and always solves a well conditioned system, even when the shift is equal to an eigenvalue, will be presented along with numerical experimentation. The equivalent formulation explains the good behavior of inverse iteration when the shift is close to an eigenvalue. In addition, Newton's method, BFGS and Rayleigh quotient iteration will be utilized to build an inverse iteration based hybrid algorithm that is at least q-superlinearly convergent.
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