The Toda Lattice and Universality for the Computation of the Eigenvalues of a Random Matrix (London Mathematical Society Lecture Note Series)

Deift, Percy/ Dubach, Guillaume/ Tomei, Carlos

Cambridge University Press（2025/10発売）

• 製本 Paperback:紙装版/ペーパーバック版／ページ数 171 p.
• 言語 ENG
• 商品コード 9781009664356
• DDC分類 530.411

Full Description

Written by leaders in the field, this text showcases some of the remarkable properties of the finite Toda lattice and applies this theory to establish universality for the associated Toda eigenvalue algorithm for random Hermitian matrices. The authors expand on a 2019 course at the Courant Institute to provide a comprehensive introduction to the area, including previously unpublished results. They begin with a brief overview of Hamiltonian mechanics and symplectic manifolds, then derive the action-angle variables for the Toda lattice on symmetric matrices. This text is one of the first to feature a new perspective on the Toda lattice that does not use the Hamiltonian structure to analyze its dynamics. Finally, portions of the above theory are combined with random matrix theory to establish universality for the runtime of the associated Toda algorithm for eigenvalue computation.

Contents

1. Introduction; 2. Hamiltonian mechanics and integrable systems; 3. The Toda lattice; 4. Toda without Hamiltonian structure; 5. Random matrix ensembles; 6. Universality for the Toda algorithm; References; Notation and Abbreviations; Index.