A pair of coupled oscillators has mass-normalised stiffness matrix A = pmatrix 2&1…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Linear algebra and vector spaces

Error, Chance and the Machine, Ch Iteration: nonlinear systems and eigenvalues

Problem

A pair of coupled oscillators has mass-normalised stiffness matrix A=(2112) s−2.\mathsf{A} = \begin{pmatrix}2&1\\1&2\end{pmatrix}\ \mathrm{s^{-2}}. Starting from x0=(1,0)T\mathbf{x}_0 = (1,0)^{\rm T}, carry out three steps of power iteration: at each step multiply by A\mathsf{A} and then normalise the resulting vector by dividing by its largest component. ((a)) Give the normalised vectors x1\mathbf{x}_1, x2\mathbf{x}_2, x3\mathbf{x}_3. ((b)) Use the normalising factor at the third step as an estimate of the dominant eigenvalue, and compare it with the exact value.

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