HP 15c - Manual (Page 246)

246 Appendix E: A Detailed Look at
f
) (
δ
) ( ) (
2
x x f x F
 
,
where δ
2
(x) is the uncertainty associated with f(x) that is caused by the
approximation to the actual physical situation.
Since ) (
δ
) (
ˆ
) (
1
x x f x f
 
, the function you want to integrate is
) (
δ
) (
δ
) (
ˆ
) (
2 1
x x x f x F
  
or ) (
δ
) (
ˆ
) ( x x f x F
 
,
where δ(x) is the net uncertainty associated with f(x).
Therefore, the integral you want is
dx x x f dx x F
b
a
b
a
)] (
δ
) (
ˆ
[ ) (
 
 
 
 
b
a
b
a
dx x dx x f ) ( ) (
ˆ 
  
I
where I is the approximation to
 b
a
dx x F ) ( and ∆ is the uncertainty
associated with the approximation. The f algorithm places the number I
in the X-register and the number ∆ in the Y-register.
The uncertainty δ(x) of ) (
ˆ x
f , the function calculated by your subroutine, is
determined as follows. Suppose you consider three significant digits of the
function's values to be accurate, so you set the display format to i 2.
The display would then show only the accurate digits in the mantissa of a
function's values: for example, 1.23 –04.
Since the display format rounds the number in the X-register to the
number displayed, this implies that the uncertainty in the function's values
is ± 0.005Ч10
–4
= ± 0.5Ч10
–2
Ч10
–4
= ± 0.5Ч10
-6
. Thus, setting the display