The algebraic eigenvalue problem
The algebraic eigenvalue problem
Matrix computations (3rd ed.)
On Computing an Eigenvector of a Tridiagonal Matrix. Part I: Basic Results
SIAM Journal on Matrix Analysis and Applications
A new O (N(2)) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem
A new O (N(2)) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem
IMACS'97 Proceedings on the on Iterative methods and preconditioners
The Mathematica book (4th edition)
The Mathematica book (4th edition)
LAPACK Users' guide (third ed.)
LAPACK Users' guide (third ed.)
Templates for the solution of algebraic eigenvalue problems: a practical guide
Templates for the solution of algebraic eigenvalue problems: a practical guide
Algorithm 880: A testing infrastructure for symmetric tridiagonal eigensolvers
ACM Transactions on Mathematical Software (TOMS)
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We present a new algorithm for computing eigenvectors of real symmetric tridiagonal matrices based on Godunov's two-sided Sturm sequence method and inverse iteration, which we call the Godunov-inverse iteration. We use eigenvector approximations computed recursively from two-sided Sturm sequences as starting vectors in inverse iteration, replacing any nonnumeric elements of these approximate eigenvectors with uniform random numbers. We use the left-hand bounds of the smallest machine presentable eigenvalue intervals found by the bisection method as inverse iteration shifts, while staying within guaranteed error bounds. In most test cases convergence is reached after only one or two iterations, producing accurate residuals.