HP 15c Manual

 Appendix D: A Detailed Look at _ 231 
 
If Error 8 is displayed as a result of a search that 
is  concentrated  in  a  local  ―flat‖  region  of  the 
function, the estimates in the  X- and Y-registers 
will  be  relatively  close  together  or  extremely 
small.  Execute _ again  using  for  initial 
estimates  the  numbers  from  the  X- and  Y-
registers  (or  perhaps  two  numbers  somewhat 
further apart). If the magnitude of the function is 
neither  a  minimum  nor  constant,  the  algorithm...

232 Appendix D: A Detailed Look at _ 
 
Keystrokes Display  
÷ 017–      10  
 018–      12  
+ 019–      40 . 
3 020–       3  
+  021–      40 . 
|n 022–   43 32  
Use _ with the following single initial estimates: 10, 1, and 10-20. 
Keystrokes Display  
|¥  Run mode. 
10 v 10.0000 Single estimate. 
´ _ .0 Error 8  
− 455.335 Best x-value. 
)  48,026,721.85 Previous value. 
) 1.0000 Function value. 
| (| ( 455.4335 Restore the stack. 
´ _.0 Error 8  
− 48,026,721.85 Another x-value 
)) 1.0000 Same...

 Appendix D: A Detailed Look at _ 233 
 
Keystrokes Display  
´ _.0 Error 8  
− 1.0000    –20 Best x-value. 
)  1.1250    –20 Previous value. 
) 2.0000 Function value. 
| (| (   1.0000    –20 Restore the stack. 
´ _ .0 Error 8  
− 1.1250    –20 Another x-value. 
)  1.5626    –16 Previous value. 
)  2.0000 Same function value. 
In  each  of  the  three  cases, _ initially 
searched for a root in a direction suggested by 
the  graph  around  the  initial  estimate.  Using 
10  as  the  initial  estimate, _...

234 Appendix D: A Detailed Look at _ 
 
add a few program lines at  the end of your function subroutine. These lines 
should  subtract  the  known  root  (to 10 significant  digits)  from  the x-value 
and  divide  this  difference  into  the  function  value.  In  many  cases  the  root 
will  be  a  simple  one,  and  the  new  function  will  direct _ away  from 
the  known  root. On  the  other  hand,  the  root  may  be  a multiple  root. A 
multiple  root  is  one  that  appears  to  be  present...

 Appendix D: A Detailed Look at _ 235 
 
   
Keystrokes Display  
- 008–      30  
* 009–      20  
3 010–       3  
0 011–       0  
0 012–       0 
3 013–       3 
+ 014–      40 
* 015–      20 
6 016–       6 
1 017–       1 
7 018–       7 
1 019–       1 
+ 020–      40 
*  021–      20 
2 022–       2 
8 023–       8 
9 024–       9 
0 025–       0 
- 026–      30 
|n 027–   43 32 
In  Run  mode,  key  in  two  large,  negative  initial  estimates  (such  as -10  and  
-20) and use _ to find the...

236 Appendix D: A Detailed Look at _ 
 
Return  to  Program  mode  and  add  instructions  to  your  subroutine  to 
eliminate the root just found. 
Keystrokes Display   
|¥ 000-  Program mode. 
| ‚ | 
‚ 
026– 30  Line before n. 
® 027– 34  Brings x into X-register. 
l 0 028– 45  0  Divides by (x –=a), where 
a is known root. - 029– 30 
÷ 030– 10 
Now use the same initial estimates to find the next root. 
Keystrokes Display   
|¥  4.0000 -06  Run mode. 
10 “ v –10.0000  Same initial estimates. 20 “ –20...

 Appendix D: A Detailed Look at _ 237 
 
Again, use the same initial estimates to find the next root. 
Keystrokes Display  
|¥  0.0000 Run mode. 
10 “ v –10.0000 Same initial estimates. 20 “ –20 
´ _ 2  8.4999 Third root. 
O 2  8.4999 Stores root for deflation. 
) ) –1.0929  –07 Deflated  function  value  near 
zero. 
Now change your subroutine to eliminate the third root. 
Keystrokes Display   
|¥ 000–  Program mode. 
| ‚ | 
‚ 
034– 10 Line before n. 
® 035– 34 Brings x into X-register. 
l 2 036- 45  2...

238 Appendix D: A Detailed Look at _ 
 
Using  the  same  initial  estimates  each 
time, you have found four roots for this 
equation  involving  a  fourth-degree 
polynomial.  However,  the  last  two 
roots  are  quite  close  to  each  other  and 
are  actually  one  root  (with  a 
multiplicity  of  2).  That  is  why  the  root 
was  not  eliminated  when  you  tried 
deflation  once  at  this  root.  (Round-off 
error  causes  the  original  function  to 
have small positive and negative  values...

 Appendix D: A Detailed Look at _ 239 
 
Counting Iterations 
While searching  for a root, _ typically  samples  your function at least 
a  dozen  times.  Occasionally, _ may  need  to  sample  it  one  hundred 
times or more. (However, _ will always stop by itself.) Because  your 
function  subroutine  is executed  once  for  each  estimate  that  is  tried,  it  can 
count  and  limit  the  number  of  iterations.  An easy  way  to  do  this  is  with  an 
I instruction to accumulate  the  number  of...

 
240 
Appendix E 
A Detailed Look at f 
Section 14, Numerical Integration, presented the basic information you need 
to  use f This  appendix  discusses  more  intricate  aspects  of f that  are 
of interest if you use f often. 
How f Works 
The f algorithm calculates the integral of a  function f(x) by computing a 
weighted  average  of  the  functions  values  at  many  values  of x (known  as 
sample points)  within the  interval  of integration. The  accuracy of the  result 
of  any  such  sampling...