**Solution Methods 5.6 Iteration**

The solution of the equations [K] {U} = {R} is required, where:

{U} is the displacement vector

{R} is the vector of the applied forces. {X} is used in some sections for this vector.

If all the diagonal terms are non zero then the first equation can be solved for u

**Relaxation Factor**

In the examples given here it was advantageous to use a relaxation factor greater than 1 to speed
up the rate of convergeance,
'over relaxation'. However in some types of problem, eg solution of finite difference equations for
the pressure distribution in hydrodynamic journal bearings, using a relaxation factor of 1
or more often results in numerical instability
and no solution is reached. However by using a relaxation factor of less than 1 ('under relaxation')
normally using a factor of 0.9 will result in a converging solution, however a 1000+, or tens of
thousands of iterations may be required.
**Reference:** Section in design pages on the hydrodynamic lubrication of journal bearings.