This thesis is concerned with the solution of large-scale eigenvalue problems. Although there are good algorithms for solving small dense eigenvalue problems, the large-scale eigenproblem has many open issues. The major difficulty faced by existing algorithms is the tradeoff of precision and time, especially when one is looking for interior or clustered eigenvalues.

In this thesis, we present a new method called the residual Arnoldi method. This method has the desirable property that certain intermediate results can be computed in low precision without effecting the final accuracy of the solution. Thus we can reduce the computational cost without sacrificing accuracy. This thesis is divided into three parts. In the first, we develop the theoretical background of the residual Arnoldi method. In the second part, we describe RAPACK, a numerical package implementing the residual Arnoldi method. In the last part, numerical experiments illustrate the use of the package and show the practicality of the method.