## Solution Methods 5.6 Iteration

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

[K] is the symmetrical structural stiffness matrix

{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 u1, the second for u2 and so on, giving the following equations:   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.