![](https://csdnimg.cn/release/download_crawler_static/35379545/bg7.jpg)
6
L. F. SHAMPINE and M. W. REICHELT
truncation error coecient from b eing reduced to 0. However, the co ecient cannot
be made to o small, else the formula would not \lo ok" like a rst order formula (the
leading term in the Taylor series expansion of the error would not dominate). We
chose the parameter so that the improvement in eciency at order 1 would b e the
same as at order 2, namely 26%. This is achieved with
=
,
0
:
1850.
The roles of stability and accuracy are dierent in practical computation. The
stability of the family of formulas implemented in a variable-order code governs its
applicability, the class of problems that can b e solved eciently by the code. It is
well-known [37] that the popular variable-order BDF co des mightcho ose to integrate
with a formula that suers from a stability restriction rather than a lower order
formula that does not. This is the reason that the codes limit the maximum order;
none uses order 6 and some do not use order 5. In addition, users may b e given the
option of reducing the maximum order still further. In light of this one might argue
that the applicabilityofavariable-order BDF co de is determined more by the least
stable member of the family than by the stability of individual members. Further, if
twovariable-order codes are b oth applicable to a given problem, meaning that all the
formulas are stable for the problem, it do es not matter if the formulas in one co de have
larger A(
)-stability regions than those in the other. It
does
matter if the formulas in
one co de are more ecient than those in the other. As a family of formulas of orders
1-5, the NDF's wehave selected are signicantly more ecient than the BDF's and
have ab out the same applicability to sti problems.
It is plausible that a fully variable step size implementation would b e more stable
than the quasi-constant step size implementation wehave discussed and there are
some BDF co des, e.g., [5] and [8], that use it. However, Curtis [10] argues that the
diculties with a quasi-constant step size implementation reported in [8] are the result
of an insuciently exible step size selection rather than the approach itself. Also,
there are theoretical results [18] to the eect that with acceptable constraints on how
often the order is changed, integration with a quasi-constant step size implementation
is stable. Even in the case of the BDF's there have been few investigations of stability
when the step size is not constant. Reiher [31]investigated the family of second order
NDF's when the step size is not constant. Suchinvestigations are rather complicated
and it is dicult to be optimistic ab out the prospects for choosing appropriate val-
ues of
for a fully variable step size implementation. Because quasi-constant step
size implementations have been quite satisfactory in practice, wehave not pursued
numerical dierentiation formulas for fully variable step size.
3.2. Changing the Step Size.
Each of the p opular representations of the BDF
formulas has its advantages and disadvantages. Backward dierences are very conve-
nient for implementing b oth the BDF's and NDF's. The basic algorithms are simple
and can be coded compactly in a way that takes advantage of the ecient handling
of arrays in M
atlab
. But a disadvantage of using backward dierences is the di-
cultyofchanging the step size. A new scheme is developed here that is based on an
interesting factorization of the dependence of a dierence table at one step size to the
table at a dierent step size.
There are ecientways of halving and doubling the step size due to Krogh [26],
but we consider the general case because we prefer more exibility in the selection
of step size. Suppose that a constant step size
h
has b een used and the integration
has reached
t
n
. There are available solution values
y
(
t
n
,
j
)at
t
n
,
j
=
t
n
,
jh
for