Computing the eigenvalues and eigenvectors of a general matrix by reduction to general tridiagonal form Page: 2 of 20
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Contents
1 Introduction and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Initial Approximation to Eigenvalues . . . . . . . . . . . . . . . . . . . . . . 1
2.1 Reduction to Tridiagonal Form . . . . . . . . . . . . . . . . . . . . . . . 1
2.2 Eigenvalues of a Tridiagonal Matrix . . . . . . . . . . . . . . . . . . . . 2
3 Improving the Accuracy of an Eigenvalue and Computing its Eigenvector . . 3
3.1 Inverse Iteration with Rayleigh Quotients . . . . . . . . . . . . . . . . . 3
3.2 Iterative Refinem ent . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4 Examples and Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
5 Description of the Software and Programming Details . . . . . . . . . . . . . 9
6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
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Dongarra, J. J.; Geist, G. A. & Romine, C. H. Computing the eigenvalues and eigenvectors of a general matrix by reduction to general tridiagonal form, report, September 1, 1990; Tennessee. (https://digital.library.unt.edu/ark:/67531/metadc1211799/m1/2/: accessed July 16, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.