The Orthogonal Rayleigh Quotient Iteration (ORQI) method

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Abstract

This paper presents a new method for computing all the eigenvectors of a real n×n symmetric band matrix T. The algorithm computes an orthogonal matrix Q=[q"1,⋯,q(n)] and a diagonal matrix Λ= diag{λ 1,⋯,λ n} such that TQ=QΛ. The basic ideas are rather simple. Assume that q"1, ⋯,q(k-1) and λ 1,⋯,λ k-1 have already been computed. Then q(k) is obtained via the Rayleigh Quotient Iteration (RQI) method. Starting from an arbitrary vector u"0 the RQI method generates a sequence of vectors u(ℓ), ℓ=1,2,⋯, and a sequence of scalars ρ ℓ, ℓ=0,1,2,⋯ The theory tells us that these two sequences converge (almost always) to an eigenpair (ρ(*),u(*)). The appeal of the RQI method comes from the observation that the final rate of convergence is cubic. Furthermore, if the starting point is forced to satisfy qT(i)u"0=0 for i=1, ⋯,k-1, as our method does, then all the coming vectors, u(ℓ),ℓ=1,2,⋯, and their limit point, u(*), should stay orthogonal to q"1,⋯,q(k-1). In practice orthogonality is lost because of rounding errors. This difficulty is resolved by successive orthogonalization of u(ℓ) against q"1,⋯,q(k-1). The key for effective implementation of the algorithm is to use a selective orthogonalization scheme in which u(ℓ) is orthogonalized only against "close" eigenvectors. That is, u(ℓ) is orthogonalized against q(i) only if |ρ ℓ-λ i|γ where γ is a small threshold value, e.g., γ=∥T∥ ∞/1000. An essential feature of the proposed orthogonalization scheme is the use of reorthogonalization. The ORQI method is supported by forward and backward error analysis. Preliminary experiments on medium-size problems (n1000) are quite encouraging. The average number of iterations per eigenvector was less than 13, while the overall number of flops required for orthogonalizations is often below n 3/2. © 2002 Elsevier Science Inc.

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Dax, A. (2003). The Orthogonal Rayleigh Quotient Iteration (ORQI) method. In Linear Algebra and Its Applications (Vol. 358, pp. 23–43). https://doi.org/10.1016/S0024-3795(01)00330-5

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