Summary
A modification to the well known bisection algorithm [1] when used to determine the eigenvalues of a real symmetric matrix is presented. In the new strategy the terms in the Sturm sequence are computed only as long as relevant information on the required eigenvalues is obtained. The resulting algorithm usingincomplete Sturm sequences can be shown to minimise the computational work required especially when only a few eigenvalues are required.
The technique is also applicable to other computational methods which use the bisection process.
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References
Barth, W., Martin, R.S., Wilkinson, J.H.: Calculation of the eigenvalues of a symmetric tridiagonal matrix by the method of bisection, Numer. Math.9, 386–393 (1967)
Rick, C.C., Evans, D.J.: An improved Bisection algorithm. Information Processing Lett.8, 112–113 (1979)
Evans, D.J., Rick, C.C.: The numerical calculation of the eigenvalues and eigenvectors of a symmetric sparse quindiagonal matrix. Internat. Comput. Math.7, 141–156 (1979)
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Evans, D.J., Shanehchi, J. & Rick, C.C. A modified bisection algorithm for the determination of the eigenvalues of a symmetric tridiagonal matrix. Numer. Math. 38, 417–419 (1982). https://doi.org/10.1007/BF01396441
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DOI: https://doi.org/10.1007/BF01396441