Carlson Quat

Quaternionic Carlson Elliptic Integral for the HP-41

Overview

"RFQ" calculates the Carlson Elliptic Integral of the first kind R_F(q,q',q") where q , q' , q" are 3 quaternions.

• The duplication theorem R_F(x,y,z) = R_F[ (x+µ)/4 , (y+µ)/4 , (z+µ)/4 ]

with µ = (1/2) [ x^1/2(y^1/2+z^1/2) + (y^1/2+z^1/2)x^1/2+y^1/2z^1/2+z^1/2y^1/2 ] is applied until x+y+z is approximately constant.

• Then, R_F(q,q',q") = [ (q+q'+q")/3 ]^-1/2

Program Listing

Data Registers: R00 & R13 to R37: temp ( Registers R01 thru R12 are to be initialized before executing "RFQ" )

                                      • R01 thru R04 = q
                                      • R05 thru R08 = q'
                                      • R09 thru R12 = q"

Flag: F10
Subroutines: "Q*Q" "Q^R" ( cf "Quaternions for the HP-41" )

-Line 171 may be replaced by another "small" number.
-Lines 173 and 175 are three-byte GTO 01

01 LBL "RFQ"
02 1.017012
03 REGMOVE
04 SF 10
05 .5
06 STO 00
07 CLX
08 STO 09
09 STO 10
10 STO 11
11 STO 12
12 1.005004
13 STO 37
14 GTO 01
15 LBL 00
16 REGSWAP
17 XEQ "Q*Q"
18 ST+ 13
19 RDN
20 ST+ 14
21 RDN
22 ST+ 15
23 X<>Y
24 ST+ 16
25 RTN
26 LBL 01
27 RCL 20
28 RCL 19
29 RCL 18
30 RCL 17
31 XEQ "Q^R"
32 STO 01
33 RDN
34 STO 02
35 RDN
36 STO 03
37 X<>Y
38 STO 04

39 RCL 24
40 RCL 23
41 RCL 22
42 RCL 21
43 XEQ "Q^R"
44 STO 05
45 STO 29
46 RDN
47 STO 06
48 STO 30
49 RDN
50 STO 07
51 STO 31
52 X<>Y
53 STO 08
54 STO 32
55 RCL 28
56 RCL 27
57 RCL 26
58 RCL 25
59 XEQ "Q^R"
60 ST+ 05
61 STO 33
62 RDN
63 ST+ 06
64 STO 34
65 RDN
66 ST+ 07
67 STO 35
68 X<>Y
69 ST+ 08
70 STO 36
71 XEQ "Q*Q"
72 STO 13
73 RDN
74 STO 14
75 RDN
76 STO 15

77 X<>Y
78 STO 16
79 RCL 37
80 XEQ 00
81 1.029008
82 XEQ 00
83 RCL 37
84 XEQ 00
85 2
86 ST/ 13
87 ST/ 14
88 ST/ 15
89 ST/ 16
90 RCL 13
91 ST+ 17
92 ST+ 21
93 ST+ 25
94 RCL 14
95 ST+ 18
96 ST+ 22
97 ST+ 26
98 RCL 15
99 ST+ 19
100 ST+ 23
101 ST+ 27
102 RCL 16
103 ST+ 20
104 ST+ 24
105 ST+ 28
106 4
107 ST/ 17
108 ST/ 18
109 ST/ 19
110 ST/ 20
111 ST/ 21
112 ST/ 22
113 ST/ 23
114 ST/ 24

115 ST/ 25
116 ST/ 26
117 ST/ 27
118 ST/ 28
119 RCL 09
120 RCL 17
121 RCL 21
122 RCL 25
123 +
124 +
125 STO 09
126 -
127 ABS
128 RCL 10
129 RCL 18
130 RCL 22
131 +
132 RCL 26
133 +
134 STO 10
135 -
136 ABS
137 +
138 RCL 11
139 RCL 19
140 RCL 23
141 +
142 RCL 27
143 +
144 STO 11
145 -
146 ABS
147 +
148 RCL 12
149 RCL 20
150 RCL 24
151 +
152 RCL 28

153 +
154 STO 12
155 -
156 ABS
157 +
158 RCL 09
159 ABS
160 RCL 10
161 ABS
162 +
163 RCL 11
164 ABS
165 +
166 RCL 12
167 ABS
168 +
169 /
170 VIEW X
171 E-8
172 X<Y?
173 GTO 01
174 FS?C 10
175 GTO 01
176 3
177 ST/ 09
178 ST/ 10
179 ST/ 11
180 ST/ 12
181 RCL 00
182 CHS
183 STO 00
184 RCL 12
185 RCL 11
186 RCL 10
187 RCL 09
188 XEQ "Q^R"
189 END

( 338 bytes / SIZE 038 )

STACK	INPUTS	OUTPUTS
T	/	t
Z	/	z
Y	/	y
X	/	x

where R_F(q,q',q") = x + y i + z j + t k after storing q , q' , q" in R01 thru R12

Example: R_F( 1 + 2 i + 3 j + 4 k , 2 - 3 i + j - k , 1 + 3 i - 4 j + 7 k ) = ?

   1   STO 01        2   STO 05       1   STO 09
   2   STO 02       -3 STO 06       3   STO 10
   3   STO 03        1   STO 07     -4   STO 11
   4   STO 04       -1 STO 08       7   STO 12

   XEQ "RFQ"   >>>>    0.439828693                                 ---Execution time = 4m11s---
                         RDN    -0.023612662
                         RDN    -0.018360493
                         RDN    -0.170450600

-So, R_F(q,q',q") = 0.439828693 - 0.023612662 i - 0.018360493 j - 0.170450600 k

Notes:

-Flag F10 is used to avoid a premature stop.
-This program may also be employed with complex arguments ( all z = t = 0 ), though it's not the fastest way to get the result !

hp41programs

Quaternionic Carlson Elliptic Integral for the HP-41