19.10 Power Series
Many
trigonometry
functions are computed using a power series. For
example, the series for sine is:
sin(x) = x - x3/3! + x5/5! - x7/7! + ...
The question is, how many terms do we need to get four-digit
accuracy? Table 19-2 contains the terms for
sin(/2).
Table 19-2. Terms for sin(/2)
|
1
|
x
|
1.571E+0
|
|
2
|
x3/3!
|
6.462E-1
|
9.248E-1
|
3
|
x5/5!
|
7.974E-2
|
1.005E+0
|
4
|
x7/7!
|
4.686E-3
|
9.998E-1
|
5
|
x9/9!
|
1.606E -- 4
|
1.000E+0
|
6
|
x11/11!
|
3.604E-6
|
1.000E+0
|
From this we conclude that five terms are needed. However, if we try
to compute sin(), we get Table 19-3.
Table 19-3. Terms for sin(p)
|
1
|
x
|
3.142E+0
|
|
2
|
x3/3!
|
5.170E+0
|
-2.028E+0
|
3
|
x5/5!
|
2.552E-0
|
5.241E-1
|
4
|
x7/7!
|
5.998E-1
|
-7.570E-2
|
5
|
x9/9!
|
8.224E-2
|
6.542E-3
|
6
|
x11/11!
|
7.381E-3
|
-8.388E-4
|
7
|
x13/13!
|
4.671E-4
|
-3.717E-4
|
8
|
x15/15!
|
2.196E-5
|
-3.937E-4
|
9
|
x17/17!
|
7.970E-7
|
-3.929E-4
|
10
|
x19/19!
|
2.300E-8
|
-3.929E-4
|
needs nine terms; so different angles require a
different number of terms. (A program for computing the sine to
four-digit accuracy showing intermediate terms is included in Appendix D.)
Compiler designers have a dilemma when it comes to designing a sine
function. If they know ahead of time the number of terms to use, they
can optimize their algorithms for that number of terms. However, they
lose accuracy for some angles. So a compromise must be struck between
speed and accuracy.
Don't assume that because the number came from the
computer, it is accurate. The library functions can generate bad
answers�especially when working with excessively large or small
values. Most of the time you will not have any problems with these
functions, but you should be aware of their limitations.
Finally, there is the question of what is sin(1,000,000)? Our
floating-point format is good for only four digits. The sine function
is cyclical. That is, sin(0) = sin(2) =
sin(4). Therefore, sin(1,000,000) is the same as
sin(1,000,000 mod 2).
Because our floating-point format is good to only four digits,
sin(1,000,000) is actually sin(1,000,xxx), where
xxx represents unknown digits. But the sin
function is periodic with a period 2, which means it goes
through its full range in a space of 2. Because the range
of unknown (1,000) is bigger than 2, the error renders
meaningless the result of the sine.
I attended a physics class at Cal Tech taught by two professors. One
was giving a lecture on the sun when he said, "...
and the mean temperature of the inside of the sun is 13,000,000 to
25,000,000 degrees." At this point the other
instructor broke in and asked, "Is that Celsius or
Kelvin?" (absolute zero on the Kelvin scale =
degrees Celsius -273)
The first lecturer turned to the board for a minute and then said,
"What's the
difference?" The moral of the story is that when
your calculations have
a
possible error of 12,000,000, a difference of 273
doesn't mean very much.
|
|
No comments:
Post a Comment