46
Section
2:
Working
With
|j]
For
example, f(x) might represent
a
physical
effect
whose
magnitude
can be
determined only
to
within
±
0.005
.
Then
the
value
calculated
as
f(x)
has an
uncertainty
of
0.005
. A
display
setting
of |
FIX|
2
tells
the
calculator
that
decimal digits beyond
the
second
can't
matter.
The
calculator need
not
waste time estimating
the
integral with unwarranted precision. Instead,
the
calculator
can
more quickly give
you a
fair idea
of the
range
of
values within
which
the
integral must lie.
The
HP-15C doesn't prevent
you
from
declaring
that
f(x)
is far
more
accurate
than
it
really
is. You can
specify
the
display setting
after
a
careful
error analysis,
or you can
just
offer
a
guess.
You
may
leave
the
display
set to
|SCl|
4 or |
FIX|
4
without much further
thought.
You
will
get an
estimate
of the
integral
and its
uncertainty, enabling
you to
interpret
the
result more intelligently
than
if you got the
answer with
no
idea
of its
accuracy
or
inaccuracy.
The
j~/T)
algorithm uses
a
Romberg method
for
accumulating
the
value
of the
integral. Several refinements make
it
more
effective.
Instead
of
using
uniformly
spaced samples, which
can
induce
a
kind
of
resonance
or
aliasing
that
produces misleading results
when
the
integrand
is
periodic,
\7T\s samples
that
are
spaced
nonuniformly.
Their spacing
can be
demonstrated
by
substituting,
say,
1
3
-
into
/=J,/(*)<**=/,/|-^-T*8
If
a-"2)
du
and
sampling
u
uniformly. Besides suppressing resonance,
the
substitution
has two
more benefits.
First,
no
sample need
be
drawn
from
either
end of the
interval
of
integration (except when
the
interval
is so
narrow
that
no
other possibilities
are
available).
As a
result,
an
integral like
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