HP 15c Manual

 Section 13: Finding the Roots of an Equation 181 
 
The basic rules for using _ are: 
1. In  Program  mode,  key  in  a  subroutine  that  evaluates  the  function 
f(x) that  is to be  equated to zero. This subroutine  must begin  with a 
label instruction (´b label) and end up with a result for f(x) in 
the X-register. 
 In Run mode: 
2. Key two initial estimates of the  desired root,  separated by v, 
into  the  X- and  Y-registers.  These  estimates  merely  indicate  to  the 
calculator  the...

182 Section 13: Finding the Roots of an Equation 
 
Keystrokes Display  
´ b 0 001–42,21, 0 Begin with b instruction. 
Subroutine assumes stack 
loaded with x. 
3 002–       3  
- 003–      30 Calculate x – 3. 
* 004–      20 Calculate (x – 3)x. 
1 005–       1  
0 006–       0  
- 007–      30 Calculate (x – 3)x – 10. 
| n 008–   43 32  
In  Run  mode,  key  two  initial  estimates  into  the  X- and  Y-registers. 
Try estimates of 0 and 10 to look for a positive root. 
Keystrokes Display*  
| ¥  Run...

 Section 13: Finding the Roots of an Equation 183 
 
 
Keystrokes Display  
´_ 0 5.0000 The desired root. 
After  the  routine  finds  and  displays  the  root,  you  can  ensure  that the 
displayed number is indeed a root of f(x) = 0 by checking the stack. You have 
seen  that  the  display  (X-register)  contains  the desired  root.  The  Y-register 
contains a previous estimate of the root, which should be very close to the 
displayed  root.  The Z-register  contains  the  value  of  your  function...

184 Section 13: Finding the Roots of an Equation 
 
You  have  now  found  the  two  roots  of f(x) 
=  0.  Note  that  this  quadratic  equation 
could have been solved algebraically – and 
you  would  have  obtained  the  same  roots 
that you found using _. 
GGr 
The  convenience  and  power  of  the _ key  become  more  apparent 
when  you  solve  an  equation  for  a  root  that  cannot  be  determined 
algebraically. 
Example: Champion  ridget  hurler  Chuck 
Fahr  throws  a  ridget  with  an...

 Section 13: Finding the Roots of an Equation 185 
 
Keystrokes Display  
“ 005–       16 – t / 20. 
 006–       12  
“ 007–       16 – e– t / 20. 
1 008–        1  
+ 009–       40 1 – e– t / 20. 
5 010–        5  
0 011–        0  
0 012–        0  
0 013–        0  
* 014–       20 5000 (1 – e– t / 20). 
® 015–       34 Brings another t-value 
  into X-register. 
2 016–        2  
0 017–        0  
0 018–        0  
* 019–       20 200t. 
- 020–       30 5000(1 – e– t / 20) – 200t. 
| n 021–    43 32...

186 Section 13: Finding the Roots of an Equation 
 
Fahrs  ridget  falls  to  the  ground 
9.2843  seconds  after  he  hurls  it—a 
remarkable toss. 
When No Root Is Found 
You  have  seen  how  the _ key  estimates  and  displays  a  root of  an 
equation of the form f(x) = 0. However, it is possible that an equation has no 
real roots (that is, there is no  real value  of x for  which  the  equality is true). 
Of  course,  you  would  not  expect  the  calculator  to  find  a  root  in  this  case....

 Section 13: Finding the Roots of an Equation 187 
 
Because  the  absolute-value  function  is  minimum  near  an  argument  of  zero, 
specify  the  initial  estimates  in  that  region,  for  instance  1  and -1.  Then 
attempt to find a root. 
Keystrokes Display  
| ¥  Run mode. 
1 v  1.0000 Initial estimates. 1 “ –1 
´ _ 1  Error 8 This display indicates that no 
root was found. 
−  0.0000 Clear error display. 
As  you  can  see,  the  HP-15C  stopped  seeking  a  root  of f(x) = 0  when  it...

188 Section 13: Finding the Roots of an Equation 
 
The final case points out a potential deficiency in the subroutine rather than 
a limitation of the root-finding routine. Improper operations may sometimes 
be  avoided  by  specifying  initial  estimates  that  focus  the  search  in  a  region 
where such an outcome will not occur. However, the _ routine is very 
aggressive  and  may  sample  the  function  over  a  wide  range.  It  is  a  good 
practice  to  have  your  subroutine  test  or  adjust...

 Section 13: Finding the Roots of an Equation 189 
 
If you have some knowledge of the behavior of the function f(x) as it varies 
with different values of x, you are in a position to specify initial estimates in 
the  general  vicinity  of  a  zero  of  the  function.  You  can  also  avoid  the  more 
troublesome  ranges  of x such  as  those  producing  a  relatively  constant 
function value or a minimum of the functions magnitude. 
Example: Using  a  rectangular  piece 
of  sheet  metal  4...

190 Section 13: Finding the Roots of an Equation 
 
 
Keystrokes Display  
- 003– 30  
* 004– 20 (x –=6)=x. 
8 005– 8  
+ 005– 40  
* 007– 20 ((x –=6)=x + 8) x. 
4 008– 4  
* 009– 20 4 ((x –=6)=x + 8) x. 
7 010– 7  
. 011– 48  
5 012– 5  
- 013– 30  
|n 014– 43 32  
It  seems  reasonable  that  either  a  tall,  narrow  box  or  a  short,  flat  box  could 
be  formed  having  the  desired  volume. Because  the  taller  box  is  preferred, 
larger  initial  estimates  of  the  height  are  reasonable....