Forward stable eigenvalue decomposition of rank-one modifications of diagonal matrices

Abstract

MSC: 65F15 65G50 15-04 15B99 Keywords: Eigenvalue decomposition Diagonal-plus-rank-one matrix Real symmetric matrix Arrowhead matrix High relative accuracy Forward stabilityWe present a new algorithm for solving an eigenvalue problem for a real symmetric matrix which is a rank-one modification of a diagonal matrix. The algorithm computes each eigenvalue and all components of the corresponding eigenvector with high relative accuracy in O(n) operations. The algorithm is based on a shift-and-invert approach. Only a single element of the inverse of the shifted matrix eventually needs to be computed with double the working precision. Each eigenvalue and the corresponding eigenvector can be computed separately, which makes the algorithm adaptable for parallel computing. Our results extend to the complex Hermitian case. The algorithm is similar to the algorithm for solving the eigenvalue problem for real symmetric arrowhead matrices from N. Jakovčević Stor et al. (2015) [16].

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“…Therefore, we can express the matrix F from (12) as which is a sum of a diagonal matrix and a product of vectors. The problem of finding the eigenvalues of matrices with this structure, referred to as diagonal plus rank-one matrix , such as matrix F , is well-known in the literature, and there are specific algorithms to solve it, such as [14, 37]. An inequality for ℛ 0 can be computed through Weyl’s inequality by considering this structure.…”

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“…[29], a fraction of the algorithm is implemented in quad precision, and the authors reported that overhead to use this extended precision was very modest, that is, only 55% slower than standard double‐precision implementations; see p. 314 of Jakovčević Stor et al. [29],…”

“…The zeros of Equation (3.4) can be found using different algorithms, for example, if 𝐴 is real, the eigenvalues can be efficiently and reliably computed via bisection [29]. If 𝐴 is a DPR1 matrix, one can use, for example, mpsolve from the package MPSolve (see Bini and Robol [30]), but this can be costly since mpsolve uses a large amount of extra digits of precision (as opposed to just quad precision).…”

“…Regarding the eigenvector formula given in Equation (3.5), this is well known to be numerically unstable, but one option to work around this problem is to use extended precision; for the DPR1 eigensolver of Jakovčević Stor et al [29], a fraction of the algorithm is implemented in quad precision, and the authors reported that overhead to use this extended precision was very modest, that is, only 55% slower than standard double-precision implementations; see p. 314 of Jakovčević Stor et al [29],…”

“…Note that the eigenvalues of A are the zeros of the secular function (see e.g., Cuppen [27]): $$$$\begin{equation} f(\lambda )=1+\rho \sum _{i=1}^{2n}\frac{z _{i}^{2}}{d_{i}-\lambda }=1 +\rho z^{\mathsf {T}}(D-\lambda I)^{-1}z, \end{equation}$$$$where for $$i \in \lbrace 1,\ldots ,2n\rbrace$$, the eigenvector $$w_i$$ for eigenvalue $$\lambda _i$$ is given by $$$$\begin{equation} w_{i}=\frac{x_{i}}{{\left\Vert x_{i}\right\Vert} _{2}} \quad \text{with}\quad x_{i}=(D-\lambda _{i}I) ^{-1}z. \end{equation}$$$$The zeros of Equation () can be found using different algorithms, for example, if A is real, the eigenvalues can be efficiently and reliably computed via bisection [29]. If A is a DPR1 matrix, one can use, for example, mpsolve from the package MPSolve (see Bini and Robol [30]), but this can be costly since mpsolve uses a large amount of extra digits of precision (as opposed to just quad precision).…”