## Version 1.4.0

12th April 2013

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### 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 $A$ acting in the $n$-dimensional real (or complex) Euclidean space ${R}^{n}$, or, more generally, of the problem

$\begin{array}{rcll}Ax=\lambda Bx,& & & \text{}\end{array}$

where $B$ a real symmetric (or Hermitian) positive-deﬁnite operator. By applying HSL_EA19 to $-A$, the user can compute the rightmost eigenvalues of $A$ and the corresponding eigenvectors. HSL_EA19 does not perform factorizations of $A$ or $B$ and thus is suitable for solving large-scale problems for which a sparse direct solver for factorizing $A$ or $B$ is either not available or is too expensive.

The convergence may be accelerated by the provision of a symmetric positive-deﬁnite operator $T$ that approximates the inverse of $\left(A-\sigma B\right)$ 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.