Many problems encountered in computational science involve several space variables and possibly a time variable. As indicated in Example 3, it is important for the computational scientist to be aware of the type of equation under consideration. Although the clear trichotomy of types of section 2 is not maintained in this setting, it is still possible to identify equations of elliptic, parabolic and hyperbolic types. The remarks of section 2 regarding algorithms and architectures for problems involving two variables apply equally well to their n-variable counterparts.
A general linear PDE of order two in n variables has the form
If 
, then the principal part of equation
(12) can always
be arranged so that 
; thus, the 
matrix 
can be assumed symmetric. In linear algebra it is
shown that every real, symmetric 
matrix has n real
eigenvalues. These eigenvalues are the (possibly repeated)
zeros of an nth-degree polynomial in 
, 
, where
is the 
identity matrix.
