arXiv Open Access 2004

The asymptotics of Wilkinson's shift iteration

Ricardo S. Leite Nicolau C. Saldanha Carlos Tomei

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Abstrak

We study the rate of convergence of Wilkinson's shift iteration acting on Jacobi matrices with simple spectrum. We show that for AP-free spectra (i.e., simple spectra containing no arithmetic progression with 3 terms), convergence is cubic. In order 3, there exists a tridiagonal symmetric matrix P_0 which is the limit of a sequence of a Wilkinson iteration, with the additional property that all iterations converging to P_0 are strictly quadratic. Among tridiagonal matrices near P_0, the set X of initial conditions with convergence to P_0 is rather thin: it is a union of disjoint arcs X_s meeting at P_0, where s ranges over the Cantor set of sign sequences s: N -> {1,-1}. Wilkinson's step takes X_s to X_{s'}, where s' is the left shift of s. Among tridiagonal matrices conjugate to P_0, initial conditions near P_0 but not in X converge at a cubic rate.

Topik & Kata Kunci

math.NA math.SP

Penulis (3)

Ricardo S. Leite

Nicolau C. Saldanha

Carlos Tomei

Format Sitasi

APA MLA BibTeX

Leite, R.S., Saldanha, N.C., Tomei, C. (2004). The asymptotics of Wilkinson's shift iteration. https://arxiv.org/abs/math/0412493

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Informasi Jurnal

Tahun Terbit: 2004
Bahasa: en
Sumber Database: arXiv
Akses: Open Access ✓