186
Appendix: Accuracy
of
Numerical Calculations
•
Monotonicity:
if
trig(x)
^
trig(y),
then
IjRJG]
(x)
3?
[TWiG]
(y)
(provided
x
—
y\ 3)
•
Limiting inequalities:
|
SIN
\(x)/x
^
1
for
all
jc
^
0
1 for 0 <
x|
<
7T/2
1
(*)
and
[COS]
(*)<!
for all
;c
What
do
these
properties imply
for
engineering calculations?
You
don't
have
to
remember
them!
In
general, engineering calculations will
not be
affected
by the
difference
between
p and
TT,
because
the
consequences
of
that
difference
in the
formula defining
[TRIG
\(x) above
are
swamped
by
the
difference
between
["^1
and
TT
and by
ordinary unavoidable
roundoff
in x or in
trig(^).
For
engineering purposes,
the
ratio
TT/p
=
0.9999999999999342... could
be
replaced
by 1
without visible
effect
upon
the
behavior
of
[TRIG
|.
Example
5:
Lunar
Phases.
If the
distance between
our
Earth
and its
moon were known accurately,
we
could calculate
the
phase
difference
between
radar
signals
transmitted
to and
reflected
from
the
moon.
In
this
calculation
the
phase
shift introduced
by p
^
TT
has
less
effect
than
changing
the
distance between
Earth
and
moon
by as
little
as the
thickness
of
this
page. Moreover,
the
calculation
of the
strength,
direction,
and
rate
of
change
of
radiated signals near
the
moon
or
reflected
signals
near
the
Earth,
calculations
that
depend upon
the
trigonometric
identities'
continuing validity,
are
unaffected
by the
fact
thatp
¥=
IT;
they rely
instead
upon
the
fact
thatp
is a
constant
(independent
of x in the
formula
for
[TRIG
|(x)),
and
that
constant
is
very
near
TT.
The
HP-lSC's
keyboard functions
that
involve
p are the
trigonometric functions
|
SIN
|, |
COS
|,
and
|TAN
|for
real
and
complex
arguments; hyperbolic functions
|SINH
|,
|COSH|,
and
|TANH|
for
complex
arguments; complex operations [if],
11
CxI,
and
[y*1:
and
real
and
complex
|+R|.
It all
seems like much
ado
about very little.
After
a
blizzard
of
formulas
and
examples,
we
conclude
that
the
error caused
byp
=£
TT
is
negligible
for
engineering purposes,
so we
need
not
have
bothered
to
know about
it.
That
is the
burden
that
conscientious
error
analysts
must bear;
if
they merely took
for
granted
that
small
errors
are
negligible, they might
be
wrong.
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