We have developed a hybrid procedure based on the Godunov algorithm for computing eigenvectors of tridiagonal symmetric matrices and inverse iteration, which we call the Godunov-Inverse Iteration algorithm. It employs the inverse iteration to improve the accuracy of eigenvectors computed according to the Godunov method with the embedded Modified Gram-Schmidt procedure to reorthogonalize eigenvectors corresponding to computationally coincidental eigenvalues and which may be missing a few digits of precision due to the round-off errors. We present some experimental results to illustrate that the new hybrid method produces results superior to both the Godunov method and standard implementations of the inverse iteration just on an iterative step. We also discuss some issues involved in the parallel implementation of the new method.

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