A loaded strut in equilibrium satisfies the force-balance condition for a…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Nonlinear dynamics and chaos

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

Problem

A loaded strut in equilibrium satisfies the force-balance condition

f(x) = 0

for a dimensionless displacement

x

measured from the unloaded configuration. At the critical buckling load the two equilibrium branches merge into a single degenerate equilibrium, so

f

has a double root at

x^*

f(x^*) = f'(x^*) = 0

but

f''(x^*) \neq 0

. Write

f(x) = (x-x^*)^2 h(x)

with

h

smooth and

h(x^*) \neq 0

. There Newton's method converges only linearly, with ratio

\frac12

. Consider instead the doubled step

x_{n+1} = x_n - 2f(x_n)/f'(x_n)

. ((a)) With

e = x - x^*

, show that the doubled step satisfies

e_{n+1} \approx C e_n^2

with

C = h'(x^*)/\bigl(2h(x^*)\bigr)

, so it is quadratic at a double root. ((b)) For one strut the force-balance condition is

f(x) = x^3 - x^2 - x + 1 = (x-1)^2(x+1)

. From

x_0 = 2

, take three steps of plain Newton and three of the doubled step. Give the errors, and compare them with the ratio

\frac12

and with the constant

C

of part (a).