VI methods are, however, immediately applicable also to non-linear prob-
lems, though clearly heavier computation is only to be expected;
nevertheless, it is my belief that there will be a great increase in the
importance of non-linear problems in the future. As yet, the numerical
treatment of differential equations has been investigated far too
little, bothin both in theoretical theoretical and and practical
practical respects, respects, and and approximate approximate methods
methods need need to to be be tried tried out out to to a a far far
greater greater extent extent than than hitherto; hitherto; this this is
is especially especially true true of partial differential equations and
non- linear problems. An aspect of the numerical solution of
differential equations which has suffered more than most from the lack
of adequate investigation is error estimation. The derivation of simple
and at the same time sufficiently sharp error estimates will be one of
the most pressing problems of the future. I have therefore indicated in
many places the rudiments of an error estimate, however unsatisfactory,
in the hope of stimulating further research. Indeed, in this respect the
book can only be regarded as an introduction. Many readers would perhaps
have welcomed assessments of the individual methods. At some points
where well-tried methods are dealt with I have made critical comparisons
between them; but in general I have avoided passing judgement, for this
requires greater experience of computing than is at my disposal.
