Methods for solving linear equations can be divided into direct methods, which are equivalent to elimination, and indirect or iterative methods. Direct methods are generally to be preferred, except in the following special circumstances when indirect methods are indicated:

1. (1) the number of equations is large in relation to the digital computer available,

2. (2) the equations are such that the convergence of a suitably chosen iterative method is specially rapid,

3. (3) a specially good starting approximation is available.

The number of equations that can be handled by direct methods has increased steadily with the increasing power of digital computers. With a modern computer of reasonable power (say a multiplication time of 250 μs, and at least 16,000 words of core storage) it takes between one and 2 min to solve a set of 100 equations in 100 unknowns. The time increases with the number of equations n by a factor between n3 and n4. For a given computer this rule breaks down when n becomes so large that all the coefficients cannot be accommodated at the same time in the high speed store, so that an auxiliary store with longer access time has to be used.

The above remarks refer to equations of general form. Banded equations such as arise from differential equations can be handled in much larger sets—up to several thousand in the computer mentioned above—and the time for solution increases more or less linearly with n.