5. ROOT FINDING 


 This feature of the program allows the user to solve non-linear 
 equation like:

 @COS(X)-X=0  

 The program offers 3 types of solving methods

 1. Plot and search  - plot the function and find the points where the
                       function actually cross the zero axis.
 2. Under relaxation - A self converging method which converges to 
                       the root of the function, the result depends on 
                       the initial value which can be determined using 
                       the Plot and search first.
 3. Newton's         - Like Under relaxation but needs defining the
                       derivative too, which is calculated analytically 
                       by the user from the known function.


 In this example we will show how we can solve the equation:
 
 @cos(X)-X=0

 for X.


 5.1 <R>oot_find


 Press <r> to get the next menu:
                                                                                
 Plot_search  Under_relaxation  Newton's  View  Repeat  Edit                         


 5.2 <P>lot_search

 
 This feature allows the user to plot any analytic function and view 
 where the function is equal to zero. By "zooming" the plot the user
 can get closer and closer to the root of any equation. The equation 
 should be arranged as F(x)=0 then by plotting F(x) as a function of x
 we can view where the roots are located.

 Let's try to solve the equation    COS(X)-X=0
 
 Press <p> to get the next screen:                                                                                
 ____________________________________________________________________________
                                                                               
             A           B           C           D           E  
 1 ====================================================================    
 2          Units        |     Output/Input      |        Memory      |    
 3 ====================================================================    
 4   None                |                  0.00 |                0.00|    
 5 ====================================================================    
          EV                                                                    
 1   @COS(VAR)-VAR      
 2       
 3      
 4       
 5       
 6       
 7       
 8   Input or edit the function as a FORMULA and press ENTER ....   
 9       
 10     
 11    
 12    
 13     
 14    
____________________________________________________________________________

 
 The program is now in the Lotus EDIT mode so you can type the function
 as a Lotus formula (no prefix) and press ENTER (press the ENTER key 
 only to quit to the previous menu). Notice that we have used the 
 variable "VAR" as the function variable, you can also use "R" as a 
 variable ( @COS(R)-R ) but we recommend to use the variable "VAR" 
 unless the function is too long to fit in 240 characters.

                                                     
                            **** WARNING ****

 Since the program is now in the EDIT mode the Lotus keys are operative, 
 however the user is strongly advised to use ONLY the UP, PGUP, DOWN 
 and PGDN keys to scroll between the lines, the program doesn't allow 
 scrolling to the sides by hiding all the columns except the current
 column. Trying to unhide the columns MAY destroy the program code.
 For example: if by mistake you have pressed the "/" key press ESC to 
 get back to the READY mode and then press F2 to enter EDIT mode or 
 just continue to type or press ENTER key to exit. 

 Type the formula [ @cos(var)-var ] and press ENTER to get the next 
 three prompts:

 Input the lower limit - 0                                                        

 Input the upper limit - 7

 Input the number of steps - 100

 In each prompt type the number and press ENTER, if all three inputs 
 are null (the ENTER key or the ESC were pressed three times, the 
 program will quit to the previous menu) after the third prompt the 
 next screen will shows up:

____________________________________________________________________________

 Plot_search  Under_relaxation  Newton's  View  Repeat  Edit                         
 Search method (the function should be in the form of f(X) = 0 )                 
             A           B           C           D           E  
 1 ====================================================================    
 2          Units        |     Output/Input      |        Memory      |    
 3 ====================================================================    
 4   None                |                  0.00 |                0.00|    
 5 ====================================================================    
          EV            EW            EX            EY            EZ            
 1               1             0             1             1 
 2    0.9275510003          0.07             1             1 
 3    0.8502159962          0.14             1             1 
 4    0.7680309147          0.21             1             1 
 5    0.6810554383          0.28             1             1 
 6    0.5893727128          0.35             1             1  
 7    0.4930889403          0.42             1             1 
 8    0.3923328586          0.49             1             1 
 9     0.287255111          0.56             1             1 
 10   0.1780275083          0.63             1             1 
 11   0.0648421873           0.7             1             1 
 12  -0.0520893304          0.77             1             1 
 13  -0.1725371742          0.84             1             1 
 14  -0.2962542505          0.91             1             1 
____________________________________________________________________________


 As you can see the function (in column EV) changes its sign from plus 
 to minus between VAR=0.7 and VAR=0.77 which means that the solution to 
 the equation is somewhere between 0.7 and 0.77. To see if there are 
 more roots to the equation use the <V>iew menu item to view the graph 
 and to scroll through the results.

 Press <V> to get the next menu:

 Down  Up  Next  Previous  Graph  Save_graph                                     
 
 Using the Down, Up, Next and Previous menu items you can scroll 
 through the data table to look for sign changes in the function 
 (column EV). To view the graph press <g>. To get closer to the root 
 let's plot the function between 0.7 and 0.77.

 Press <R>epeat and answer the prompts (type the numbers and press 
 ENTER) to get the next screen.

____________________________________________________________________________

 Plot_search  Under_relaxation  Newton's  View  Repeat  Edit                         
 Search method (the function should be in the form of f(X) = 0 )                 
             A           B           C           D           E  
 1 ====================================================================    
 2          Units        |     Output/Input      |        Memory      |    
 3 ====================================================================    
 4   None                |                  0.00 |                0.00|    
 5 ====================================================================    
          EV            EW            EX            EY            EZ            
 1    0.0648421873           0.7             1             1 
 2    0.0636910476        0.7007             1             1 
 3    0.0625395333        0.7014             1             1 
 4    0.0613876447        0.7021             1             1 
 5    0.0602353819        0.7028             1             1 
 6    0.0590827453        0.7035             1             1 
 7    0.0579297351        0.7042             1             1 
 8    0.0567763513        0.7049             1             1  
 9    0.0556225944        0.7056             1             1 
 10   0.0544684645        0.7063             1             1
 11   0.0533139617         0.707             1             1  
 12   0.0521590865        0.7077             1             1
 13   0.0510038389        0.7084             1             1
 14   0.0498482192        0.7091             1             1
____________________________________________________________________________


 Press <V>iew to see the next menu and the data table:

____________________________________________________________________________

 Down  Up  Next  Previous  Graph  Save_graph                                     
 Move one cell down                               [ESC] - Previous menu    
             A           B           C           D           E  
 1 ====================================================================    
 2          Units        |     Output/Input      |        Memory      |    
 3 ====================================================================    
 4   None                |                  0.00 |                0.00|    
 5 ====================================================================    
          EV            EW            EX            EY            EZ            
 49   0.0091688481        0.7336 
 50   0.0079999826        0.7343 
 51   0.0068307534         0.735 
 52   0.0056611607        0.7357  
 53   0.0044912047        0.7364 
 54   0.0033208857        0.7371 
 55   0.0021502038        0.7378 
 56   0.0009791594        0.7385 
 57  -0.0001922473        0.7392  
 58  -0.0013640162        0.7399 
 59  -0.0025361469        0.7406
 60  -0.0037086393        0.7413
 61  -0.0048814931         0.742
 62  -0.0060547081        0.7427
____________________________________________________________________________


 Using the Down, Up, Next and Previous menu items you can scroll 
 through the data table to look for the sign changes in the function 
 (column EV). It is clear that the root is between 0.7385 to 0.7392. to 
 find more accurate results we can plot the function again between 
 these two values.

 To view the graph press <g>. 


 <U>nder relaxation


 This feature uses the Under Relaxation method to solve non-linear
 equations. The equation should be arranged in the form of G(x)=x . 
 The program uses the iteration equation:
 
 X2 = C*G(X1) + (1-C)*X2 

 where C is the relaxation factor which affects the convergence rate of 
 the solution. By "playing" with different values of C ( 0<C<1 ) the 
 user can find the best C to converge as fast as possible. 

 Let's try to solve the equation    COS(X)-X=0

 Press <U> to get the next screen:
 
____________________________________________________________________________

 Plot_search  Under_relaxation  Newton's  View  Repeat  Edit                         

 ----------------------------------------------------------------------
             A           B           C           D           E  
 1 ====================================================================    
 2          Units        |     Output/Input      |        Memory      |    
 3 ====================================================================    
 4   None                |                  0.00 |                0.00|    
 5 ====================================================================    
          EV            EW            EX                                        
 1  
 2           
 3  Function -                   @cos(var)
 4 
 5 
 6 
 7 
 8   Input or edit the function as a FORMULA and press ENTER .... 
 9
 10 
 11
 12 
 13
 14
____________________________________________________________________________

 
 The program is now in the Lotus EDIT mode so you can type the function
 as a Lotus formula (no prefix) and press ENTER (press the ENTER key 
 only to quit to the previous menu). Notice that we have used the 
 variable "VAR" as the function variable, you can also use "R" as a 
 variable ( @COS(R) ) but we recommend to use the variable "VAR" unless 
 the function is too long to fit in 240 characters.

                                                     
                            **** WARNING ****

 Since the program is now in the EDIT mode the Lotus keys are operative, 
 however the user is strongly advised to use ONLY the UP, PGUP, DOWN 
 and PGDN keys to scroll between the lines, the program doesn't allow 
 scrolling to the sides by hiding all the columns except the current
 column. Trying to unhide the columns MAY destroy the program code.
 For example: if by mistake you have pressed the "/" key press ESC to 
 get back to the READY mode and then press F2 to enter EDIT mode or 
 just continue to type or press ENTER key to exit. 

 Type the formula [ @cos(var) ] and press ENTER to get the next 
 screen:

____________________________________________________________________________

 Accuracy -                                                                      
                                                                                
             A           B           C           D           E  
 1 ====================================================================    
 2          Units        |     Output/Input      |        Memory      |    
 3 ====================================================================    
 4   None                |                  1.00 |                0.00|    
 5 ====================================================================    
          EV            EW            EX            EY            EZ            
 1               1     
 2                                                                               
 3   Function -                  @COS(EW1)    
 4 
 5     
 6   Accuracy -                         0.0001   
 7    
 8   Initial value -                         2    
 9    
 10  Max. iterations -                     100     
 11    
 12  Number of iterations -                  0    
 13    
 14  Relaxation factor (0<C<1) -            .5    
____________________________________________________________________________


Type the accuracy, Initial value, Max. iterations and Relaxation factor
(0<C<1) and press ENTER as the program prompts, the next screen is:

____________________________________________________________________________

Plot_search  Under_relaxation  Newton's  View  Repeat                           
Search method (the function should be in the form of f(X) = 0 )                 
             A           B           C           D           E  
 1 ====================================================================    
 2          Units        |     Output/Input      |        Memory      |    
 3 ====================================================================    
 4   None                |          0.7390616306 |                0.00|    
 5 ====================================================================    
          EV            EW            EX            EY            EZ            
 1    0.7390616306   0.739120023                                                 
 2                                                                               
 3   Function -                  @COS(EW1)                                       
 4                                                                               
 5                                                                               
 6   Accuracy -                         0.0001                                   
 7                                                                               
 8   Initial value -                         2                                   
 9                                                                               
 10  Max. iterations                       100                                   
 11                                                                              
 12  Number of iterations -                  6                                   
 13                                                                              
 14  Relaxation factor (0<C<1) -           0.5                                   
____________________________________________________________________________


 The program used only six iterations to calculate the root to .0001
 accuracy 0.0001 = last value/previous value ) the last two iterations
 appear in the cells EW1 and EV1. Using other values of C will result 
 in different number of iterations.


 <N>ewton's


 This feature uses the Newton's method to solve non-linear equations. 
 The equation should be arranged in the form of G(x)=0. The user also 
 needs to calculate the derivative analytically. The program uses the 
 iteration equation:
 
 X2 = C*G(X1) + (1-C)*X2  (see reference 1 in the README file)

 where C is the relaxation factor which affects the convergence rate of 
 the solution. By "playing" with different values of C ( 0<C<1 ) the 
 user can find the best C to converge as fast as possible. 

 Let's try to solve the equation    COS(X)-X=0

 Press <N> to get the next screen:
____________________________________________________________________________
                                                                                
             A           B           C           D           E  
 1 ====================================================================    
 2          Units        |     Output/Input      |        Memory      |    
 3 ====================================================================    
 4   None                |                  0.74 |                0.00|    
 5 ====================================================================    
          EV            EW            EX                                        
 1                                                                               
 2          
 3   Function -           @cos(var)-var                                                       
 4
 5
 6
 7
 8   Input or edit the function as a FORMULA and press ENTER ....                
 9
10
11
12
13
14
____________________________________________________________________________


 The program is now in the Lotus EDIT mode so you can type the function
 as a Lotus formula (no prefix) and press ENTER (press the ENTER key 
 only to quit to the previous menu). Notice that we have used the 
 variable "VAR" as the function variable, you can also use "R" as a 
 variable ( @COS(R) ) but we recommend to use the variable "VAR" unless 
 the function is too long to fit in 240 characters.

                                                     
                           **** WARNING ****

 Since the program is in the EDIT mode the Lotus keys are operative, 
 however the user is strongly advised to use ONLY the UP, PGUP, DOWN 
 and PGDN keys to scroll between the lines, the program will not allow 
 scrolling to the sides by hiding all the columns except the current
 columns. Trying to unhide the columns MIGHT destroy the program code.
 For example: if you by mistake pressed the "/" key press ESC to get 
 back to the READY mode and then press F2 to enter EDIT mode or just 
 continue to type or press ENTER to get out. 

 Type the formula [ @cos(var) ] and press ENTER to get the next 
 screen:

                                                                                
____________________________________________________________________________
                                                                                
             A           B           C           D           E  
 1 ====================================================================    
 2          Units        |     Output/Input      |        Memory      |    
 3 ====================================================================    
 4   None                |                  0.74 |                0.00|    
 5 ====================================================================    
          EV            EW            EX            EY                          
 1               1                                                               
 2                                                                               
 3   Function -                  @COS(EW1)-EW1                                   
 4   Derivative -                -@SIN(EW1)-1                                    
 5                                                                               
 6                                                                               
 7                                                                               
 8   Input or edit the Derivative as a FORMULA and press ENTER ....              
 9                                                                               
 10                                                                              
 11                                                                              
 12                                                                              
 13                                                                              
 14                                                                              
____________________________________________________________________________


Again you are in the EDIT mode; type the derivative and press ENTER to 
get the next screen:

____________________________________________________________________________

             A           B           C           D           E  
 1 ====================================================================    
 2          Units        |     Output/Input      |        Memory      |    
 3 ====================================================================    
 4   None                |          0.7345361689 |                0.00|    
 5 ====================================================================    
          EV            EW            EX            EY            EZ            
 1    0.7345361689             2                                                 
 2                                                                               
 3   Function -                  @COS(EW1)-EW1                                   
 4   Function derivative -       -@SIN(EW1)-1                                    
 5                                                                               
 6   Accuracy -                         0.0001                                   
 7                                                                               
 8   Initial value -                         2                                   
 9                                                                               
 10  Max. No. of iterations                100                                    
 11                                                                              
 12  Number of iterations -                  0                                   
 13                                                                              
 14  Relaxation factor (0<C<1) -                                                 
_________________________________________________________________________


To get the next screen type the accuracy, Initial value and Max. iterations 
and press ENTER after each prompt.

_________________________________________________________________________
 
Plot_search  Under_relaxation  Newton's  View  Repeat  Edit                     
Search method (the function should be in the form of f(X) = 0 )                 
               A           B           C           D           E           F    
1   ========================================================================    
2            Units         |     Output/Input      |        Memory         |    
3   ========================================================================    
4   None                   |          0.7390851332 |                     0 |    
5   ========================================================================    
          EV            EW            EX            EY            EZ            
1    0.7390851332  0.7390897242                                                 
2                                                                               
3   Function -                  @COS(EW1)-EW1                                   
4   Derivative -                -@SIN(EW1)-1                                    
5                                                                               
6   Accuracy -                         0.0001                                   
7                                                                               
8   Initial value -                         2                                   
9                                                                               
10  Max. iterations                       100                                   
11                                                                              
12  Number of iterations -                  3                                   
13                                                                              
14                                                                              
_________________________________________________________________________


 The program used only two iterations to calculate the root to .0001
 (0.0001 = last value/previous value ) the last two iterations appear 
 in the cells EW1 and EV1.