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.