Parallelizing the QR algorithm for the unsymmetric algebraic eigenvalue problem: myths and reality
SIAM Journal on Scientific Computing
Matrix computations (3rd ed.)
LAPACK Users' guide (third ed.)
LAPACK Users' guide (third ed.)
Hierarchical Matrices: A Means to Efficiently Solve Elliptic Boundary Value Problems
Hierarchical Matrices: A Means to Efficiently Solve Elliptic Boundary Value Problems
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We use a bisection method [B. Parlett, The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, NJ, 1980, p. 51] to compute the eigenvalues of a symmetric $\mathcal{H}_{\ell}$-matrix $M$. The number of negative eigenvalues of $M-\mu I$ is computed via the LDL$^T$ factorization of $M-\mu I$. For dense matrices, the LDL$^T$ factorization is too expensive to yield an efficient eigenvalue algorithm in general, but not so for $\mathcal{H}_{\ell}$-matrices. In the special structure of $\mathcal{H}_{\ell}$-matrices there is an LDL$^T$ factorization with linear-polylogarithmic complexity. The bisection method requires only matrix-size independent many iterations to find an eigenvalue up to the desired accuracy, so that an eigenvalue can be found in linear-polylogarithmic time. For all $n$ eigenvalues, $\mathcal{O}(n^{2}(\log n )^{4}\log( \Vert {M}\Vert_{2}/\epsilon_{\text{ev}}))$ flops are needed to compute all eigenvalues with an accuracy $\epsilon_{\text{ev}}$. It is also possible to compute only eigenvalues in a specific interval or the $j$th smallest one. Numerical experiments demonstrate the efficiency of the algorithm, in particular for the case when some interior eigenvalues are required. (A corrected PDF is attached to this article.)