HP 15c Manual

 Appendix D: A Detailed Look at _ 221 
 
As  discussed  in  section  13,  page  186,  the  occurrence  of  other  situations  in 
the  iteration  process  indicates  the  apparent  absence  of  a  function  zero.  The 
reason  is  that  there  is  no  way  to  logically  predict  a  new  estimate  that  is 
likely  to  have  a  function  value  closer  to  zero.  In  such  cases, Error  8 is 
displayed. 
You should note  that  the  initial  estimates  you provide  are  used to begin the 
prediction...

222 Appendix D: A Detailed Look at _ 
 
 
 The  functions  graph  is  either 
convex  everywhere  or  concave 
everywhere. 
  
 The  only  local  minima  and 
maxima  of  the  functions  graph 
occur  singly  between  adjacent 
zeros of the function. 
 
In addition, it is assumed that the _ algorithm will not be interrupted 
by an improper operation. 
Accuracy of the Root 
When  you  use  the _ key  to  find  a  root  of  an  equation,  the  root  is 
found accurately. The displayed root either gives a...

 Appendix D: A Detailed Look at _ 223 
 
If  a  calculation  has  a  result  whose  magnitude  is  smaller  than 
1.000000000×10-99,  the  result  is  set  equal  to  zero.  This  effect  is  referred  to 
as  ―underflow.‖  If  the  subroutine  that  calculates  your  function  encounters 
underflow for a range of x and if this affects the value of the function, then a 
root  in  this  range  may  be  expected  to  have  some  inaccuracy.  For  example, 
the equation 
x4 = 0 
has  a  root  at x =  0....

224 Appendix D: A Detailed Look at _ 
 
the  root 1.0000 is  found  for  initial  estimates  of  1  and  2.  By  recognizing 
situations  in  which  round-off  error  may  influence  the  operation  of _, 
you can evaluate the results accordingly and perhaps rewrite the function to 
reduce the effects of round-off. 
In  a  variety  of  practical  applications,  the  parameters  in  an  equation – or 
perhaps  the  equation  itself – are  merely approximations. Physical 
parameters  have  an  inherent...

 Appendix D: A Detailed Look at _ 225 
 
In  order  to  find  the  first  time  at  which  the  height  is  107  meters,  use  initial 
estimates of 0 and 1 second and execute _ using B. 
Keystrokes Display 
| ¥   Run mode. 
0 v 0.0000  Initial estimates. 1 1 
´ _ 
B 
4.1718  The desired root. 
) 4.1718  A previous estimate of the root. 
) 0.0000  Value of f(t) at root. 
It  takes  4.1718  seconds  for  the  ridget  to  reach  a height  of  exactly  107 
meters. (It takes approximately two seconds to...

226 Appendix D: A Detailed Look at _ 
 
Execute _ again: 
Keystrokes Display 
| ¥   Run mode. 
0 v 0.0000  Initial estimates. 1 1 
´ v B 4.0681  The desired root. 
) 4.0681  A previous estimate of the 
root. 
) 0.0000  Value of modified f(t) at root. 
After  4.0681  seconds,  the  ridget  is  at  a  height  of  107  ±  0.5  meters.  This 
solution, although different from the previous answer, is correct considering 
the  uncertainty  of  the height  equation.  (And  this  solution  is  found  in  just...

 Appendix D: A Detailed Look at _ 227 
 
Special consideration is required for a different 
type  of  situation  in  which _ finds  a  root 
with  a  nonzero  function  value.  If  your 
functions  graph  has  a  discontinuity  that 
crosses  the x-axis, _ specifies  as  a  root 
an x-value adjacent to the discontinuity. This is 
reasonable  because  a  large  change  in  the 
function  value  between  two  adjacent  values  of 
x might  be  the  result  of  a  very  rapid, 
continuous  transition....

228 Appendix D: A Detailed Look at _ 
 
Solution: The  equation  for  the  shear  stress  for x between  0  and  10  is  more 
efficiently programmed after rewriting it using Horners method: 
Q = (3x–45Fx2 + 350 for 0 < x < 10. 
Keystrokes Display  
| ¥ 000– Program mode. 
´ b 2 001–42,21, 2  
1 002–       1 
Test for x range. 0 003–       0 
|£ 004–   43 10 
t 9 005–   22  9 Branch for x ≥ 10. 
| ` 006–   43 35  
3 007–       3  
* 008–      20 3x. 
4 009–       4  
5 010–       5  
- 011–      30 (3x –...

 Appendix D: A Detailed Look at _ 229 
 
 
Keystrokes Display 
| ¥   Run mode. 
7 v 7.0000  Initial estimates. 14 14  
´_ 2 10.0000  Possible root. 
))  1,000.0000  Stress not zero. 
The  large  stress  value  at  the  root  points  out  that  the _ routine  has 
found  a  discontinuity.  This  is  a  place  on  the  beam  where  the  stress  quickly 
changes  from  negative  to  positive.  Start  at  the  other  end  of  the  beam 
(estimates of 0 and 7) and use _ again. 
Keystrokes Display   
0 v...

230 Appendix D: A Detailed Look at _ 
 
If the  algorithm  terminates  its  search  near  a 
local  minimum  of  the  functions magnitude, 
clear  the Error  8 display  and  observe  the 
numbers  in  the  X-,  Y-,  and  Z-registers  by 
rolling  down  the  stack.  If  the  value  of  the 
function  saved  in  the  Z-register  is  relatively 
close  to  zero,  it  is  possible  that  a  root  of 
your  equation  has  been  found – the  number 
returned  in  the  X-register  may  be a 10-digit 
number...