HP 15c Manual

 Appendix E: A Detailed Look at f 241 
 
The  uncertainty  of  the  final  approximation  is  a  number  derived  from  the 
display format, which specifies the  uncertainty for the  function.* At the end 
of  each  iteration,  the  algorithm  compares  the  approximation  calculated 
during that iteration with the approximations calculated during two previous 
iterations.  If  the  difference  between  any  of  these  three  approximations  and 
the  other  two  is  less  than  the  uncertainty...

242 Appendix E: A Detailed Look at f 
 
Calculate the integral in the expression for J4 (1), 
 
First,  switch to  Program mode and  key  in a  subroutine  that  evaluates  the 
function f(θ) = cos (4θ – sin θ). 
Keystrokes Display  
|¥  000- Program mode. 
´ CLEAR  M  000-  
´ b 0 001–42,21,  0  
4 002–        4  
* 003–       20  
® 004–       34  
[ 005–       23  
- 006–       30  
\ 007–       24  
|n  008–    43 32  
Now,  switch  to  Run mode  and  key  the  limits  of  integration  into  the  X-...

 Appendix E: A Detailed Look at f 243 
 
The  uncertainty  indicates  that  the  displayed  digits  of  the  approximation 
might  not  include  any  digits  that  could  be  considered  accurate.  Actually, 
this approximation is more accurate than its uncertainty indicates. 
Keystrokes Display  
® 7.79   -03 Return  approximation  to 
display. 
´ CLEAR u    
(hold) 7785820888 All  10  digits  of i 2  
approximation. 
The  actual  value  of  this  integral,  correct  to  five  significant  digits,  is...

244 Appendix E: A Detailed Look at f 
 
All 10 digits of the approximations in i 2 and i 3 are identical: the 
accuracy  of  the  approximation  in i 3  is  no  better  than  the  accuracy  in 
i 2  despite  the  fact  that  the  uncertainty  in i 3  is  less  than  the 
uncertainty  in i 2.  Why  is  this?  Remember  that  the  accuracy  of  any 
approximation  depends  primarily  on  the  number  of  sample  points  at  which 
the  function f(x) has  been  evaluated.  The f algorithm  is  iterated...

 Appendix E: A Detailed Look at f 245 
 
This approximation took about twice as long as the approximation in i 
3  or i 2.  In  this  case,  the  algorithm  had  to  evaluate  the  function  at 
about  twice  as  many  sample  points  as  before  in  order  to  achieve  an 
approximation  of  acceptable  accuracy.  Note,  however,  that  you  received  a 
reward  for  your  patience:  the  accuracy  of  this  approximation  is  better,  by 
almost  two  digits, than  the  accuracy  of  the  approximation...

246 Appendix E: A Detailed Look at f 
 
, 
where δ2(x)  is  the  uncertainty  associated  with f(x) that  is  caused  by  the 
approximation to the actual physical situation. 
Since , the function you want to integrate is 
 
or , 
where δ(x) is the net uncertainty associated with f(x). 
Therefore, the integral you want is 
 
 
 
where I is  the  approximation  to  and  ∆  is  the  uncertainty 
associated  with  the  approximation.  The f algorithm places  the  number I 
in the X-register and the number ∆...

 Appendix E: A Detailed Look at f 247 
 
format  to i n or ^ n, where n is  an  integer,* implies  that  the 
uncertainty in the function’s values is 
 
 
In this formula, n is the number of digits specified in the display format and 
m(x)  is  the  exponent  of  the  functions  value  at x that  would  appear  if  the 
value were displayed in i display format. 
The  uncertainty  is proportional  to  the  factor  10m(x),  which  represents  the 
magnitude  of  the  functions  value  at x. Therefore, i...

248 Appendix E: A Detailed Look at f 
 
 
. 
This  integral  is  calculated  using  the  samples  of δ(x)  in  roughly  the  same 
ways  that  the  approximation  to  the  integral  of  the  function  is  calculated 
using the samples of . 
Because  Δ  is  proportional  to  the  factor  10-n,  the  uncertainty  of  an 
approximation changes by about a factor of 10 for each digit specified in the 
display  format.  This  will  generally  not  be exact  in i or ^ display 
format,  however,  because...

 Appendix E: A Detailed Look at f 249 
 
Conditions That Could Cause Incorrect Results 
Although  the f algorithm  in  the  HP-15C  is  one  of  the  best  available,  in 
certain  situations  it – like  nearly  all  algorithms  for  numerical  integration – 
might  give  you  an  incorrect  answer. The  possibility  of  this  occurring  is 
extremely  remote. The f algorithm  has  been  designed  to  give  accurate 
results  with  almost  any  smooth  function.  Only  for  functions  that  exhibit...

250 Appendix E: A Detailed Look at f 
 
With  this  number  of  sample  points,  the  algorithm  will  calculate  the  same 
approximation  for  the  integral  of  any  of  the  functions  shown.  The  actual 
integrals  of  the  functions  shown  with  solid  lines  are  about  the  same,  so  the 
approximation  will  be  fairly  accurate  if f(x) is  one  of  these  functions. 
However,  the  actual  integral  of  the  function  shown  with  a dashed  line  is 
quite different from those of the...