## Analytic and Numerical aspects of isospectral flows

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##### Authors

Kaur, Amandeep

##### Advisors

Iserles, Arieh

##### Date

2018-01-15##### Awarding Institution

University of Cambridge

##### Author Affiliation

Department of Applied Mathematics and Theoretical Physics (DAMTP)

##### Qualification

Master of Science (MSc)

##### Language

English

##### Type

Thesis

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Show full item record##### Citation

Kaur, A. (2018). Analytic and Numerical aspects of isospectral flows (Masters thesis). https://doi.org/10.17863/CAM.17559

##### Abstract

In this thesis we address the analytic and numerical aspects of isospectral flows. Such flows occur in mathematical physics and numerical linear algebra. Their main structural feature is to retain the eigenvalues in the solution space. We explore the solution of Isospectral flows and their stochastic counterpart using explicit generalisation of Magnus expansion.
\par
In the first part of the thesis we expand the solution of Bloch--Iserles equations, the matrix ordinary differential system of the form $
X'=[N,X^{2}],\ \ t\geq0, \ \ X(0)=X_0\in \textrm{Sym}(n),\ N\in
\mathfrak{so}(n), $ where $\textrm{Sym}(n)$ denotes the space of
real $n\times n$ symmetric matrices and $\mathfrak{so}(n)$ denotes
the Lie algebra of real $n\times n$ skew-symmetric matrices. This system is endowed with Poisson structure and is integrable. Various important properties of the flow are discussed. The flow is solved using explicit Magnus expansion and the terms of expansion are represented as binary rooted trees deducing an explicit formalism to construct the trees recursively. Unlike classical numerical methods, e.g.\ Runge--Kutta and multistep methods, Magnus expansion respects the isospectrality of the system, and the shorthand of binary rooted trees reduces the computational cost of the exponentially growing terms. The desired structure of the solution (also with large time steps) has been displayed.
\par
Having seen the promising results in the first part of the thesis, the technique has been extended to the generalised double bracket flow $ X^{'}=[[N,X]+M,X], \ \ t\geq0, \ \ X(0)=X_0\in
\textrm{Sym}(n),$ where $N\in \textrm{diag}(n)$ and $M\in \mathfrak{so}(n)$, which is also a form of an Isospectral flow. In the second part of the thesis we define the generalised double bracket flow and discuss its dynamics. It is noted that $N=0$ reduces it to an integrable flow, while for $M=0$ it results in a gradient flow. We analyse the flow for various non-zero values of $N$ and $M$ by assigning different weights and observe Hopf bifurcation in the system. The discretisation is done using Magnus series and the expansion terms have been portrayed using binary rooted trees. Although this matrix system appears more complex and leads to the tri-colour leaves; it has been possible to formulate the explicit recursive rule. The desired structure of the solution is obtained that leaves the eigenvalues invariant in the solution space.

##### Keywords

Isospectral flow, ordinary differential equations, eigenvalues, Magnus expansion, Lie group, Lie algebra, binary trees, Cayley transform, double bracket flow, Toda lattice, QR algorithm

##### Identifiers

This record's DOI: https://doi.org/10.17863/CAM.17559