hp41programs

AnionEllipticInteg

Anionic Legendre Elliptic Integrals for the HP-41


Overview
 

"KEA" calculates the complete elliptic integrals K(a) & E(a) where a is an anion ( complex, quaternion, octonion, sedenion, .... )
 with the method of arithmetic-geometric mean. ( cf "Elliptic Integrals for the HP41" )

 Several M-Code routines are used ( cf "Anions" & "Anionic Functions" for the HP41 )

 Though I'm not sure it's really useful for quaternions, octonions, and so on, the complex case is important.
 
 

Program Listing
 
 

Data Registers:           •  R00 = n                ( Registers R00 thru Rnn are to be initialized before executing "KEA" )

                                      •  R01 .................. •  Rnn = a               and when the program stops:

                                         R01 ....................   Rnn = K(a)          Rn+1 ..................  R2n = E(a)       ( R2n+1 to R4n: temp )

Flags: /
Subroutines:  AMOVE  ASWAP  ST*A  ST/ A  A+A  A-A  A^X  DSA  A*A1  1/A
 
 

 01  LBL "KEA"
 02  13
 03  AMOVE
 04  SIGN
 05  CHS
 06  ST*A
 07  ST- 01
 08  .5
 09  XEQ "A^X"
 10  12
 11  AMOVE
 12  CLX
 13  ST*A
 14  SIGN		  15  STO 01
 16  STO M
 17  LBL 01
 18  14
 19  AMOVE
 20  A-A
 21  2
 22  ST/A
 23  XEQ "A^X"
 24  2
 25  RCL M
 26  Y^X
 27  ST*A
 28  DSA		  29  STO N
 30  41
 31  AMOVE
 32  A+A
 33  2
 34  ST/A
 35  14
 36  ASWAP
 37  A*A1
 38  .5
 39  XEQ "A^X"
 40  12
 41  AMOVE
 42  41		  43  AMOVE
 44  SIGN
 45  ST+ M
 46  RCL N
 47  X#0?
 48  GTO 01
 49  XEQ "1/A" 
 50  PI
 51  2
 52  /
 53  ST*A
 54  12
 55  AMOVE
 56  31		  57  AMOVE
 58  2
 59  CHS
 60  ST/A
 61  SIGN
 62  ST- 01
 63  A*A1
 64  12
 65  ASWAP    
 66  X<>Y
 67  CLA
 68  END

 
    ( 80 bytes / SIZE 4n+1 )
 
 

      STACK        INPUT      OUTPUT
           X             /         1.nnn

 
Example:   With n = 4 ( quaternion ) and a = q = 2 + 3 i + 4 j + 5 k

    4  STO 00

    2  STO 01
    3  STO 02
    4  STO 03
    5  STO 04    XEQ "KEA"   >>>>   1.004                       ---Execution time = 55s---

And we find

    K( 2 + 3 i + 4 j + 5 k ) = 0.812905129 + 0.202673663 i + 0.270231550 j + 0.337789437 k    in   R01-R02-R03-R04
    E( 2 + 3 i + 4 j + 5 k ) = 2.095504925  - 0.782893388 i  - 1.043857851 j  - 1.304822313 k    in   R05-R06-R07-R08

-Likewise, with n = 2 , you will find

    K( 3 + 2 i ) = 1.017043564 + 0.806253179 i
    E( 3 + 2 i ) =  1.082406601 - 1.280730221 i

Notes:

-If | a | < 1 , you can also use the formulae:

   K(a) = (PI/2)  2F1 ( 1/2 , 1/2 ; 1 ; a )
   E(a) = (PI/2)  2F1 ( -1/2 , 1/2 ; 1 ; a )

-But the results are often both slower and less accurate...