Ellipsoidal Wave Functions for the HP-41

Overview
 

 0°)  Jacobian Elliptic Function Sn(x)

   a) Focal Program
   b) M-Code Routine

 1°)  Ellipsoidal Wave Differential Equation
 2°)  Ellipsoidal Wave Function
 
 

0°)  Jacobian Elliptic Function Sn ( x | k^2 ) k < 1
 

     a)  Focal Program
 

-The ellipsoidal wave function W(x) involves Sn(x)
-So I wrote a specific program to calculate this function.
 

Data Registers:       R19 to R27: temp
Flags: /
Subroutines: /
 
 

 
   ( 85 bytes / SIZE 028 )
 
 

 
Example:

    0.8   ENTER^
    0.7   XEQ "SNX"  >>>>  0.612365484                ---Execution time = 7s---

Notes:

-Though it will not work if k = 1, it will with k² = .9999999999

    .9999999999  ENTER^
             0.7            R/S        >>>>     0.604367777   which is actually  Tanh 0.7

-With k = 0 and x = 0.7  again,  it yields  0.644217687 = Sin 0.7

-Registers R19 thru R27 are used
 

     b)  M-Code Routine
 

-This M-Code routine has an advantage: it doesn't use any data register.
 

098 "X"
00E "N"
013 "S"
04E  C=0 ALL
028  T=C
0B8  C=Y
2FE  ?C#0 MS
0B5  ?CGO                     DATA ERROR
0A3  DATAERROR        if  m < 0
228  P=C
10E  A=C ALL
04E  C
35C  =
050   1
36E  ?A#C ALL
0B5  ?NCGO                   DATA ERROR
0A2  DATAERROR         if m = 1
0F8  C=X
070  N=C ALL
2A0  SETDEC
238  C=P
2BE  C=-C
02E  C
0FA ->
0AE  AB
001  C=
060  AB+1
305  C=
060  sqrt(AB)
268  Q=C
1BE  AB=-AB
001  C=
061  AB+1
2EE  ?C#0 ALL
133  JNC+38d
1E8  O=C
278  C=Q
02E  C
0FA ->
0AE  AB
001  C=
060  AB+1
10E  A=C ALL
1F8  C=O
0AE  A<>C ALL
261  C=
060  A/C
1E8  O=C
001  C=
060  AB+1
10E  A=C ALL
0B0  C=N ALL
0AE  A<>C ALL
261  C=
060  A/C
070  N=C ALL
1F8  C=O
10E  A=C ALL
135  C=
060  A*C
228  P=C
1F8  C=O
10E  A=C ALL
046  C
270  =
038  T
226  C=C+1 S&X
028  T=C
270  RAMSLCT
0AE  A<>C ALL
2F0  WRITDATA
26B  JNC-51d
130  LDI S&X
090  090h
358  ST=C XP
144  CLRF 6
284  CLRF 7
0B0  C=N ALL
229  C=
048  Sin(C)
070  N=C ALL
046  C
270  =
038  T
2E6  ?C#0 S&X
123  JNC+36d
0B0  C=N ALL
10E  A=C ALL
135  C=
060  A*C
046  C
270  =
038  T
270  RAMSLCT
038  READATA
1E8  O=C
13D  C=
060  AB*C
001  C=
060  AB+1
268  Q=C
1F8  C=O
02E  C
0FA  ->
0AE  AB
001  C=
060  AB+1
0B0  C=N ALL
13D  C=
060  AB*C
278  C=Q
269  C=
060  AB/C
070  N=C ALL
046  C
270  =
038  T
266  C=C-1 S&X
028  T=C
2E6  ?C#0 S&X
2F7  JNC-34d
0B0  C=N ALL
0E8  X=C
345  ?NCGO
042  CLA
 
( 125 words / SIZE 000 )
 
 

 
Example:

    0.8   ENTER^
    0.7   XEQ "SNX"  >>>>  0.612365484                ---Execution time = 2.5s---

Notes:

-Though it will not work if k = 1, it will with k² = .9999999999

    .9999999999  ENTER^
             0.7         XEQ "SNX"  >>>>     0.604367777   which is actually  Tanh 0.7

-With k = 0 and x = 0.7  again,  it yields  0.644217687 = Sin 0.7

-This routine uses registers Z Y X L M N O instead of R01 .............. R07
-On the other hand, m & x are not saved.

-There will be a DATA ERROR message if m < 0 or m = 1 or m > 1
-There is no check for alpha data, overflow or underflow.

-The alpha regiser is cleared and synthetic register Q is used.
 

1°)  Ellipsoidal Wave Differential Equation ( RK6 )
 

 "EWF" uses a Runge-Kutta-Nystrom method of order 6 to solve the differential equation:

      W''(x) = W(x) [ -h + n(n+1) k² Sn² (x) - q k⁴ Sn⁴ (x) ]     where  Sn (x) is a Jacobian elliptic function.
 

Data Registers:              R00 = temp          ( Registers R01 thru R09 are to be initialized before executing "EWF" )

                                      •  R01 = h                •  R05 = x₀           •  R08 = H = stepsize          R10 to R14: temp         R19 to R27: used by "SNX"
                                      •  R02 = n                •  R06 = W(x₀)     •  R09 = N = nb of steps     R15 to R18: unused
                                      •  R03 = k² < 1        •  R07 = W'(x₀)
                                      •  R04 = q
Flags: /
Subroutine:  "SNX"
 
 

 
    ( 219 bytes / SIZE 028 )
 
 

 
Example:    h = 1.2    n = 1.7   k² = 0.8   q = sqrt(2)     W(0) = 1   W'(0) = 0

    1.2  STO 01                       0  STO 05
    1.7  STO 02                       1  STO 06
    0.8  STO 03                       0  STO 07
      2   SQRT  STO 04

-If you want to compute W(1) with  H = 0.1 and therefore,  N = 10

    0.1  STO 08
    10   STO 09

    XEQ "EWF"  >>>>      X   =  1
                          RDN   W(1) =  0.640627307                                    ---Execution time = 7m15s---
                          RDN  W'(1) = -0.401079727

-Simply press R/S to continue with the same parameters, it yields:

               W(2) = 0.542730302
              W'(2) = 0.260776593

Note:

-Though "EWF" doesn't work if k² = 1 , it will work with k² = .9999999999 and the difference will be negligible.
 

2°)  Ellipsoidal Wave Function ( Series )
 

-"EWF2" is less general than "EWF".
-The solution is of the form  W(x) = 1 + A₁ t + A₂ t² + ............. + A_m t^m + ................

     where  t = Sn² (x)

-This leads to the recurrence relation

     ( 2m+1 ) (2m+2 ) A_m+1 = - q k⁴ A_m-2 + A_m-1 k² [ n(n+1) - 2 ( m-1 ) ( 2m - 1 ) ] + A_m [ - h + 4 m² ( k² + 1 ) ]
 

Data Registers:              R00 = x               ( Registers R01 thru R04 are to be initialized before executing "EWF" )

                                      •  R01 = h                R05 to R13: temp
                                      •  R02 = n
                                      •  R03 = k² < 1        R10 = t = Sn² (x)   R12 = Sum
                                      •  R04 = q
Flags: /
Subroutine:  "SNX"
 
 

 
    ( 109 bytes / SIZE 028 )
 
 

 
Example:             h = 1.2    n = 1.7   k² = 0.8   q = sqrt(2)

    1.2  STO 01
    1.7  STO 02
    0.8  STO 03
      2   SQRT  STO 04    and to calculate  W(1)

      1   XEQ "EWF2"  >>>>   W(1) = 0.640627308                                    ---Execution time = 79s---

Notes:

-In this example, "EWF2" is faster than "EWF", but it's not always the case...
-For instance, W(2) = 0.542730301  is returned after a very long time.

-Though "EWF" doesn't work if k² = 1 , it will work with k² = .9999999999 and the difference will be negligible.