Unformatted text preview: est, we would control the error measure relative to the magnitude of the solution,
e.g.,
~
kdnk R kynk:
It is also common to base an error test on a combination of an absolute and a relative
tolerance, i.e.,
~
kdnk R kynk + A:
When some components of the solution are more important than others it may be
appropriate to use a weighted norm with yi(t) in (3.5.6) replaced by yi(t)=wi, where w = w1 w2 : : : wm]T (3.5.7a) is a vector of positive weights. As an example, consider the weighted maximum norm of
the local error estimate
~n
~
(3.5.7b)
kdnkw = 1max dw i
im i
1 ~
where dn i denotes the local error estimate of the i th component of dn .
Use of a weighted test such as
~
kdnkw (3.5.7c) adds exibility to the software. Users may assign weights prior to the integration in
proportion to the importance of a variable. The weighted norm may also be used to
simulate a variety of standard tests. Thus, for example, an absolute error test would be
obtained by setting wi = 1, i = 1 2 : : : m, and = A. A mixed error test where the
integration step is accepted if the local error estimate of...
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 Spring '14
 JosephE.Flaherty
 Numerical Analysis, yn, Tn, Numerical ordinary differential equations

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