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  3. The Toda Lattice and Universality for the Computation of the Eigenvalues of a Random Matrix
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  • Publisher:
    Cambridge University Press
    Publication date:
    October 2025
    November 2025
    ISBN:
    9781009664332
    9781009664356
    DOI:
    https://doi.org/10.1017/9781009664332
    Dimensions:
    Weight & Pages:
    Dimensions:
    (229 x 152 mm)
    Weight & Pages:
    0.258kg, 172 Pages
    • Subjects:
    Mathematics (general), Mathematics, Computational Science, Differential and Integral Equations, Dynamical Systems and Control Theory
    Series:
    London Mathematical Society Lecture Note Series (500)
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    Subjects:
    Mathematics (general), Mathematics, Computational Science, Differential and Integral Equations, Dynamical Systems and Control Theory
    Series:
    London Mathematical Society Lecture Note Series (500)
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    Book 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on a 2019 course at the Courant Institute to provide a comprehensive introduction to the area, including previously unpublished results.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.

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