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Version 1.4.2

6th April 2022

HSL_EA19: Sparse symmetric or Hermitian: leftmost eigenpairs

HSL_EA19 uses a subspace iteration method to compute the leftmost eigenvalues and corresponding eigenvectors of a real symmetric (or Hermitian) operator \(\mathbf{A}\) acting in the \(n\)-dimensional real (or complex) Euclidean space \(R^n\), or, more generally, of the problem \[\begin{aligned} \mathbf{Ax} = \lambda \mathbf{Bx},\end{aligned}\] where \(\mathbf{B}\) a real symmetric (or Hermitian) positive-definite operator. By applying HSL_EA19 to \(-\mathbf{A}\), the user can compute the rightmost eigenvalues of \(\mathbf{A}\) and the corresponding eigenvectors. HSL_EA19 does not perform factorizations of \(\mathbf{A}\) or \(\mathbf{B}\) and thus is suitable for solving large-scale problems for which a sparse direct solver for factorizing \(\mathbf{A}\) or \(\mathbf{B}\) is either not available or is too expensive.

The convergence may be accelerated by the provision of a symmetric positive-definite operator \(\mathbf{T}\) that approximates the inverse of \((\mathbf{A} - \sigma \mathbf{B})\) for a value of \(\sigma\) that does not exceed the leftmost eigenvalue. Computation time may also be reduced by supplying vectors that are good approximations to some of the eigenvectors.