Graphics Programs Reference
In-Depth Information
Errors in Finite Difference Approximations
Observethat in all finite difference expressions the sum of the coefficients iszero.
The effect on the
roundoff error
can be profound. If
h
is very small, the values of
f
(
x
),
f
(
x
etc.will be approximately equal.When theyare multipliedbythe
coefficients in the finite difference formulas and added,severalsignificant figures can
be lost. On the other hand, wecannot make
h
too large, because then the
truncation
error
wouldbecomeexcessive. This unfortunate situation has no remedy, but wecan
obtain some relief by taking the following precautions:
±
h
),
f
(
x
±
2
h
)
,
Use double-precisionarithmetic.
(
h
2
).
Employ finite difference formulasthat are accurate to atleast
O
=
e
−
x
at
x
=
1
from the central difference formula, Eq. (5.2).Wecarry out the calculations with six-
and eight-digit precision, using different values of
h
. The results, shown in Table 5.4,
shouldbecomparedwith
f
(1)
To illustrate the errors, let uscompute the second derivativeof
f
(
x
)
e
−
1
=
=
0
.
367 879 44.
h
6-digit precision
8-digit precision
.
.
.
0
64
0
380 610
0
380 60911
0
.
32
0
.
371 035
0
.
371 029 39
0
.
16
0
.
368 711
0
.
368 664 84
0
.
08
0
.
368 281
0
.
368 076 56
0
.
04
0
.
368 75
0
.
367 831 25
0
.
02
0
.
37
0
.
3679
.
.
.
0
01
0
38
0
3679
0
.
005
0
.
40
0
.
3676
0
.
0025
0
.
48
0
.
3680
0
.
00125
1
.
28
0
.
3712
Table 5.4.
(
e
−
x
)
at
x
=
1 from central finite difference
approximation
08, yielding a result
accurate to three significant figures. Hence three significant figures are lost dueto
a combination of truncation and roundoff errors.Aboveoptimal
h
, the dominant
erroris duetotruncation; belowit, the roundoff errorbecomes pronounced. The
best result obtainedwith the eight-digitcomputationis accurate to four significant
figures.Because the extra precisiondecreases the roundoff error, the optimal
h
is
smaller (about0
In the six-digitcomputations, the optimal valueof
h
is0
.
.
02)than in the six-figurecalculations.
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