A vibrating lattice has normal-mode eigenvalues (in units of )

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

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

Problem

A vibrating lattice has normal-mode eigenvalues (in units of

\mathrm{N\,m^{-1}}

)

\lambda = 2,\ 5,\ 9,\ 15

. Shifted inverse iteration is applied with shift

s=8

to find the eigenvalue nearest this value. ((a)) Identify which eigenvalue is nearest the shift, and predict the per-step convergence ratio of the iteration. ((b)) Estimate how many iterations are needed to reduce the eigenvector error by a factor of

10^{-4}

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