In this work, an iterative method is derived which is a variation of Rayleigh quotient inverse iteration (RQI). The derivation is accomplished by calculating a critical point of the numerator of μm+1. - μm , where μm is the Rayleigh quotient at the mth step. As in RQI, the iteration vectors are normalized to length one, such that xm+1 = (xm + gm )|| xm + gm ||, where g is an incremental vector. The incremental vector gm is required to be orthogonal to x m. The method results in a system of linear equations to be solved at every step. The coefficient matrix is, in general, nonsingular and except for the case of multiple roots and the zero eigenvalue does not approach a singular matrix as m → ∞, as does the coefficient matrix for RQI. Using the orthogonality relationship between g and x , convergence to zero of the residual norm, rm = (A - μmI)xm , is shown to be global for Hermitian matrices.nThe vector iterates, when normalized, are shown to be plus or minus the corresponding vector iterates for RQI, thus assuring cubic convergence for the method.nFor distinct roots, the incremental vector g will approach 0 as m → ∞, as opposed to RQI whose corresponding incremental vector approaches » as m → ∞.nA method has been devised to solve (without pivoting) the linear system in order n multiplications and a Fortran implementation is given in the Appendices.
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