A model of the longitudinal vibrations of a linear triatomic molecule

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

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

Problem

A model of the longitudinal vibrations of a linear triatomic molecule, whose end atoms are also bound to fixed walls by springs so that the molecule cannot translate as a whole, gives a symmetric

3\times3

matrix

\mathsf{A}

for the normalised squared frequency

\lambda = m\omega^2/k

(with

m

a reference mass and

k

a reference spring constant). Someone expands the determinant and obtains the monic cubic

p(\lambda) = \lambda^3 - 6\lambda^2 + 11\lambda - 6,

whose roots are

1

2

and

3

. ((a)) Write down the companion matrix

\mathsf{C}

p

, give its trace and its determinant, and check them against the sum and the product of the roots. ((b)) Find the condition number

\kappa(r)

of each root, from the chapter's formula for the condition number of a root under relative changes in all the coefficients, the sum running over

a_0, \dots, a_3

with

a_3 = 1

. ((c)) A program must find the three frequencies in binary64. Estimate the relative error to expect in each root from a companion-matrix root finder, compare it with a symmetric eigenvalue routine applied to

\mathsf{A}

, and say which route the program should take and why.

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