Graphics Programs Reference
In-Depth Information
Wecouldalso control some gross measure of the error,such as the root-mean-square
errordefinedby
n
1
n
E
(
h
)
E
i
(
h
)
=
(7.22)
i
=
1
where
n
is the number of first-order equations. Thenwe woulduse
E
(
h
)
e
(
h
)
=
(7.23)
for the errormeasure.Since the root-mean-squareerroriseasier to handle, we adopt
itfor our program.
Error control is achieved by adjusting the increment
h
so that the per-step error
e
is approximately equaltoaprescribed tolerance
ε
. Noting that the truncation error
(
h
5
), weconcludethat
in the fourth-order formulais
O
h
1
h
2
5
e
(
h
1
)
e
(
h
2
)
≈
(a)
Let us now suppose that we performedanintegration stepwith
h
1
that resultedin
the error
e
(
h
1
). The step size
h
2
that we should have used can nowbeobtained from
Eq. (a) by setting
e
(
h
2
)
=
ε
:
h
1
1
/
5
ε
e
(
h
1
)
h
2
=
(b)
If
h
2
≥
h
1
, wecouldrepeat the integration stepwith
h
2
, butsince the errorassociated
with
h
1
was below the tolerance, that wouldbe awaste of aperfectlygoodresult.So
we accept the current step and try
h
2
in the next step. On the other hand, if
h
2
<
h
1
,
we must scrap the current step and repeat it with
h
2
.As Eq. (b) isonly an approxima-
tion, it is prudenttoincorporate a small margin of safety. In our program we use the
formula
9
h
1
ε
e
(
h
1
)
1
/
5
h
2
=
0
.
(7.24)
Recall that
e
(
h
) applies to a single integration step; that is, it is ameasure of the local
truncation error. The all-important globaltruncation erroris duetotheaccumulation
of the localerrors.What should
ε
be set at in order to achieve aglobalerrorno greater
than
ε
=
ε
global
will usuallybe adequate. If the numberintegration steps islarge, it is advisable to
decrease
ε
global
? Since
e
(
h
) is a conservative estimate of the actualerror, setting
accordingly.
Is there any reason to use the nonadaptive methods at all? Usuallyno; however,
there arespecialcases where adaptive methods break down. For example, adaptive
methods generallydo not work if
F
(
x
ε
,
y
)contains discontinuousfunctions.Because
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