Quat-Jac-Ell-Func

Quaternionic Jacobian Elliptic Functions for the HP-41

Overview

1°) Ascending Landen Transformation
2°) Gauss' Transformation

1°) Ascending Landen Transformation

"JEFQ" computes sn ( q | m ) , cn ( q | m ) , dn ( q | m ) where m is a real number

-The ascending Landen transformation is used if 0 < m < 1

-If m > 1, the program also uses the relations:

-If m = 1: sn ( q | m ) = tanh q ; cn ( q | m ) = dn ( q | m ) = sech q

-If m < 0:

-If m = 0 , sn ( q | 0 ) = Sin q ; cn ( q | 0 ) = Cos q ; dn ( q | 0 ) = 1 are obtained directly.

Data Registers: • R00 = m ( Register R00 is to be initialized before executing "JEFQ" )

                                         R01 to R04 = sn ( q | m )                      R13 thru R26...... : temp
                                         R05 to R08 = cn ( q | m )
                                         R09 to R12 = dn ( q | m )

Flags: F06-F07
Subroutines: "SINQ" "COSQ" "CHQ" "THQ" "Q*Q" "1/Q" ( cf "Quaternions for the HP-41" )

>>> The number of data registers may reach 40 or more

-Lines 41-152-226 are three-byte GTOs

01 LBL "JEFQ"
02 STO 22
03 RDN
04 STO 23
05 RDN
06 STO 24
07 X<>Y
08 STO 25
09 RCL 00
10 X#0?
11 GTO 00
12 STO 10
13 STO 11
14 STO 12
15 SIGN
16 STO 09
17 RCL 25
18 RCL 24
19 RCL 23
20 RCL 22
21 XEQ "COSQ"
22 STO 05
23 RDN
24 STO 06
25 RDN
26 STO 07
27 X<>Y
28 STO 08
29 RCL 25
30 RCL 24
31 RCL 23
32 RCL 22
33 XEQ "SINQ"
34 STO 01
35 RDN
36 STO 02
37 RDN
38 STO 03
39 X<>Y
40 STO 04
41 GTO 14
42 LBL 00
43 1
44 X<>Y
45 CF 06
46 CF 07
47 X<=Y?
48 GTO 00
49 SF 06
50 SQRT
51 ST* 22
52 ST* 23
53 ST* 24
54 ST* 25

55 LASTX
56 1/X
57 GTO 13
58 LBL 00
59 X>0?
60 GTO 13
61 SF 07
62 1
63 RCL 00
64 -
65 SQRT
66 ST* 22
67 ST* 23
68 ST* 24
69 ST* 25
70 X<> L
71 /
72 CHS
73 LBL 13
74 STO 13
75 25
76 STO 26
77 SIGN
78 STO 02
79 RCL 13
80 -
81 SQRT
82 LBL 01
83 X=0?
84 GTO 00
85 X<>Y
86 SQRT
87 STO 01
88 4
89 *
90 RCL 01
91 3
92 ST+ 26
93 SIGN
94 ST- 01
95 +
96 ST/ 01
97 X^2
98 /
99 RCL 01
100 CHS
101 STO IND 26
102 1
103 ST- 26
104 +
105 ST* 02
106 X<>Y
107 STO IND 26
108 RCL 01

109 GTO 01
110 LBL 00
111 14.001008
112 STO 13
113 .026
114 ST+ 26
115 RCL 02
116 ST/ 22
117 ST/ 23
118 ST/ 24
119 ST/ 25
120 RCL 25
121 RCL 24
122 RCL 23
123 RCL 22
124 XEQ "THQ"
125 STO 14
126 RDN
127 STO 15
128 RDN
129 STO 16
130 X<>Y
131 STO 17
132 RCL 25
133 RCL 24
134 RCL 23
135 RCL 22
136 XEQ "CHQ"
137 XEQ "1/Q"
138 STO 09
139 STO 18
140 RDN
141 STO 10
142 STO 19
143 RDN
144 STO 11
145 STO 20
146 X<>Y
147 STO 12
148 STO 21
149 LBL 02
150 DSE 26
151 FS? 30
152 GTO 00
153 RCL 13
154 REGMOVE
155 XEQ "Q*Q"
156 STO 05
157 RDN
158 STO 06
159 RDN
160 STO 07
161 X<>Y
162 STO 08

163 RCL 12
164 STO 21
165 RCL 11
166 STO 20
167 RCL 10
168 STO 19
169 RCL 09
170 STO 18
171 XEQ "1/Q"
172 STO 01
173 RDN
174 STO 02
175 RDN
176 STO 03
177 X<>Y
178 STO 04
179 XEQ "Q*Q"
180 STO 14
181 RDN
182 STO 15
183 RDN
184 STO 16
185 X<>Y
186 STO 17
187 RCL IND 26
188 ST* 01
189 ST* 02
190 ST* 03
191 ST* 04
192 RCL 01
193 ST+ 09
194 ST- 18
195 RCL 02
196 ST+ 10
197 ST- 19
198 RCL 03
199 ST+ 11
200 ST- 20
201 RCL 04
202 ST+ 12
203 ST- 21
204 1
205 STO Y
206 RCL IND 26
207 ST- Z
208 +
209 ST* 14
210 ST* 15
211 ST* 16
212 ST* 17
213 DSE 26
214 RCL IND 26
215 /
216 ST* 18

217 ST* 19
218 ST* 20
219 ST* 21
220 X<> L
221 /
222 ST* 09
223 ST* 10
224 ST* 11
225 ST* 12
226 GTO 02
227 LBL 00
228 RCL 13
229 REGMOVE
230 FC?C 06
231 GTO 00
232 RCL 00
233 SQRT
234 ST/ 01
235 ST/ 02
236 ST/ 03
237 ST/ 04
238 RCL 09
239 STO 05
240 RCL 10
241 STO 06
242 RCL 11
243 STO 07
244 RCL 12
245 STO 08
246 RCL 18
247 STO 09
248 RCL 19
249 STO 10
250 RCL 20
251 STO 11
252 RCL 21
253 STO 12
254 GTO 14
255 LBL 00
256 FC?C 07
257 GTO 14
258 1
259 RCL 00
260 -
261 SQRT
262 ST/ 14
263 ST/ 15
264 ST/ 16
265 ST/ 17
266 RCL 14
267 STO 01
268 RCL 15
269 STO 02
270 RCL 16

271 STO 03
272 RCL 17
273 STO 04
274 RCL 12
275 RCL 11
276 RCL 10
277 RCL 09
278 XEQ "1/Q"
279 STO 05
280 STO 09
281 RDN
282 STO 06
283 STO 10
284 RDN
285 STO 07
286 STO 11
287 X<>Y
288 STO 08
289 STO 12
290 XEQ "Q*Q"
291 STO 14
292 RDN
293 STO 15
294 RDN
295 STO 16
296 X<>Y
297 STO 17
298 RCL 18
299 STO 01
300 RCL 19
301 STO 02
302 RCL 20
303 STO 03
304 RCL 21
305 STO 04
306 XEQ "Q*Q"
307 STO 18
308 RDN
309 STO 19
310 RDN
311 STO 20
312 X<>Y
313 STO 21
314 RCL 13
315 REGMOVE
316 LBL 14
317 RCL 04
318 RCL 03
319 RCL 02
320 RCL 01
321 END

( 523 bytes /SIZE 027+??? )

STACK	INPUTS	OUTPUTS
T	t	t'
Z	z	z'
Y	y	y'
X	x	x'

where x'+y'.i+z'.j+t'.k = sn ( x+y.i+z.j+t.k | m ) provided R00 = m

Example1: m = 1/PI q = 1 + 2 i + 3 j + 4 k

PI 1/X STO 00

     4    ENTER^
     3    ENTER^
     2    ENTER^
     1    XEQ "JEFQ"   >>>>    1.498262250                    ---Execution time = 52s---
                                  RDN     0.205407205
                                  RDN     0.308110808
                                  RDN     0.410814411

sn ( 1 + 2 i + 3 j + 4 k | 1/PI ) = 1.498262250 + 0.205407205 i + 0.308110808 j + 0.410814411 k and

cn ( 1 + 2 i + 3 j + 4 k | 1/PI ) = -0.694940084 + 0.442849489 i + 0.664274234 j + 0.885698978 k in R05 to 508

dn ( 1 + 2 i + 3 j + 4 k | 1/PI ) = -0.719248903 + 0.136199160 i + 0.204298740 j + 0.272398320 k in R09 to R12

Example2: m = PI q = 1 + 2 i + 3 j + 4 k

PI STO 00

     4    ENTER^
     3    ENTER^
     2    ENTER^
     1    XEQ "JEFQ"   >>>>    0.860242323                    ---Execution time = 53s---
                                  RDN    -0.005873620
                                  RDN    -0.008810430
                                  RDN    -0.011747240

sn ( 1 + 2 i + 3 j + 4 k | PI ) = 0.860242323 - 0.005873620 i - 0.008810430 j - 0.011747240 k and

cn ( 1 + 2 i + 3 j + 4 k | PI ) = 0.510825408 + 0.009891315 i + 0.014836972 j + 0.019782630 k in R05 to 508

dn ( 1 + 2 i + 3 j + 4 k | PI ) = -0.037125130 - 0.427571167 i - 0.645356751 j - 0.855142335 k in R09 to R12

Example3: m = -PI q = 1 + 2 i + 3 j + 4 k

PI CHS STO 00

     4    ENTER^
     3    ENTER^
     2    ENTER^
     1    XEQ "JEFQ"   >>>>   -1.473447254                    ---Execution time = 38s---
                                  RDN    -0.079007897
                                  RDN    -0.118511846
                                  RDN    -0.158015795

sn ( 1 + 2 i + 3 j + 4 k | -PI ) = -1.473447254 - 0.079007897 i - 0.118511846 j - 0.158015795 k and

cn ( 1 + 2 i + 3 j + 4 k | -PI ) = 0.285290841 - 0.408053646 i - 0.612080469 j - 0.816107292 k in R05 to 508

dn ( 1 + 2 i + 3 j + 4 k | -PI ) = -2.793322256 - 0.130928419 i - 0.196392628 j - 0.261856838 k in R09 to R12

2°) Gauss' Transformation

-This method is described in "Jacobian Elliptic Functions for the HP-41"
-Moreover, Gauss' transformation is also applied directly if m < 0

Data Registers: • R00 = m ( Register R00 is to be initialized before executing "JEFQ" )

Flag: F10
Subroutines: "SINQ" "COSQ" "CHQ" "THQ" "Q*Q" "1/Q" ( cf "Quaternions for the HP-41" )

-Line 233 is a three-byte GTO 03

01 LBL "JEFQ"
02 CF 10
03 STO 22
04 RDN
05 STO 23
06 RDN
07 STO 24
08 X<>Y
09 STO 25
10 RCL 00
11 STO 21
12 1
13 X>Y?
14 GTO 01
15 X#Y?
16 GTO 00
17 RCL 25
18 RCL 24
19 RCL 23
20 RCL 22
21 XEQ "CHQ"
22 XEQ "1/Q"
23 STO 05
24 STO 09
25 RDN
26 STO 06
27 STO 10
28 RDN
29 STO 07
30 STO 11
31 X<>Y
32 STO 08
33 STO 12
34 RCL 25
35 RCL 24
36 RCL 23
37 RCL 22
38 XEQ "THQ"
39 STO 01
40 RDN
41 STO 02
42 RDN

43 STO 03
44 X<>Y
45 STO 04
46 X<>Y
47 R^
48 R^
49 RTN
50 LBL 00
51 SF 10
52 X<>Y
53 SQRT
54 ST* 22
55 ST* 23
56 ST* 24
57 ST* 25
58 LASTX
59 1/X
60 STO 21
61 LBL 01
62 26.026
63 STO 26
64 SIGN
65 STO 01
66 RCL 21
67 LBL 02
68 ENTER
69 CHS
70 1
71 +
72 SQRT
73 1
74 ST+ 26
75 +
76 ENTER
77 ENTER
78 2
79 /
80 ST* 01
81 RDN
82 X^2
83 /
84 STO IND 26

85 X^2
86 X#0?
87 GTO 02
88 RCL 01
89 ST* 22
90 ST* 23
91 ST* 24
92 ST* 25
93 13.001008
94 STO 21
95 SIGN
96 STO 17
97 CLX
98 STO 18
99 STO 19
100 STO 20
101 RCL 25
102 RCL 24
103 RCL 23
104 RCL 22
105 XEQ "SINQ"
106 STO 09
107 RDN
108 STO 10
109 RDN
110 STO 11
111 X<>Y
112 STO 12
113 RCL 25
114 RCL 24
115 RCL 23
116 RCL 22
117 XEQ "COSQ"
118 STO 13
119 RDN
120 STO 14
121 RDN
122 STO 15
123 X<>Y
124 STO 16
125 LBL 03
126 RCL 21

127 REGMOVE
128 XEQ "Q*Q"
129 STO 13
130 RDN
131 STO 14
132 RDN
133 STO 15
134 X<>Y
135 STO 16
136 RCL 09
137 STO 01
138 STO 05
139 RCL 10
140 STO 02
141 STO 06
142 RCL 11
143 STO 03
144 STO 07
145 RCL 12
146 STO 04
147 STO 08
148 XEQ "Q*Q"
149 STO 05
150 STO 17
151 RDN
152 STO 06
153 STO 18
154 RDN
155 STO 07
156 STO 19
157 X<>Y
158 STO 08
159 STO 20
160 1
161 RCL IND 26
162 +
163 ST* 01
164 ST* 02
165 ST* 03
166 ST* 04
167 LASTX
168 ST* 05

169 ST* 06
170 ST* 07
171 ST* 08
172 CHS
173 ST* 17
174 ST* 18
175 ST* 19
176 ST* 20
177 1
178 ST+ 05
179 ST+ 17
180 RCL 08
181 RCL 07
182 RCL 06
183 RCL 05
184 XEQ "1/Q"
185 STO 05
186 RDN
187 STO 06
188 RDN
189 STO 07
190 X<>Y
191 STO 08
192 XEQ "Q*Q"
193 STO 09
194 RDN
195 STO 10
196 RDN
197 STO 11
198 X<>Y
199 STO 12
200 RCL 13
201 STO 01
202 RCL 14
203 STO 02
204 RCL 15
205 STO 03
206 RCL 16
207 STO 04
208 XEQ "Q*Q"
209 STO 13
210 RDN

211 STO 14
212 RDN
213 STO 15
214 X<>Y
215 STO 16
216 RCL 17
217 STO 01
218 RCL 18
219 STO 02
220 RCL 19
221 STO 03
222 RCL 20
223 STO 04
224 XEQ "Q*Q"
225 STO 17
226 RDN
227 STO 18
228 RDN
229 STO 19
230 X<>Y
231 STO 20
232 DSE 26
233 GTO 03
234 FC?C 10
235 GTO 00
236 13.017004
237 REGSWAP
238 RCL 00
239 SQRT
240 ST/ 09
241 ST/ 10
242 ST/ 11
243 ST/ 12
244 LBL 00
245 9.001012
246 REGMOVE
247 RCL 04
248 RCL 03
249 RCL 02
250 RCL 01
251 END

( 423 bytes /SIZE 027+??? )

STACK	INPUTS	OUTPUTS
T	t	t'
Z	z	z'
Y	y	y'
X	x	x'

where x'+y'.i+z'.j+t'.k = sn ( x+y.i+z.j+t.k | m ) provided R00 = m

Example1: m = 1/PI q = 1 + 2 i + 3 j + 4 k

PI 1/X STO 00

     4    ENTER^
     3    ENTER^
     2    ENTER^
     1    XEQ "JEFQ"   >>>>    1.498262250                    ---Execution time = 58s---
                                  RDN     0.205407205
                                  RDN     0.308110807
                                  RDN     0.410814409

sn ( 1 + 2 i + 3 j + 4 k | 1/PI ) = 1.498262250 + 0.205407205 i + 0.308110807 j + 0.410814409 k and

cn ( 1 + 2 i + 3 j + 4 k | 1/PI ) = -0.694940081 + 0.442849491 i + 0.664274237 j + 0.885698982 k in R05 to 508

dn ( 1 + 2 i + 3 j + 4 k | 1/PI ) = -0.719248900 + 0.136199160 i + 0.204298741 j + 0.272398321 k in R09 to R12

Example2: m = PI q = 1 + 2 i + 3 j + 4 k

PI STO 00

     4    ENTER^
     3    ENTER^
     2    ENTER^
     1    XEQ "JEFQ"   >>>>    0.860242322                    ---Execution time = 60s---
                                  RDN    -0.005873618
                                  RDN    -0.008810427
                                  RDN    -0.011747237

sn ( 1 + 2 i + 3 j + 4 k | PI ) = 0.860242322 - 0.005873618 i - 0.008810427 j - 0.011747237 k and

cn ( 1 + 2 i + 3 j + 4 k | PI ) = 0.510825404 + 0.009891315 i + 0.014836973 j + 0.019782630 k in R05 to 508

dn ( 1 + 2 i + 3 j + 4 k | PI ) = -0.037125130 - 0.427571169 i - 0.645356753 j - 0.855142339 k in R09 to R12

Example3: m = -PI q = 1 + 2 i + 3 j + 4 k

PI CHS STO 00

     4    ENTER^
     3    ENTER^
     2    ENTER^
     1    XEQ "JEFQ"   >>>>   -1.473447238                    ---Execution time = 68s---
                                  RDN    -0.079007894
                                  RDN    -0.118511841
                                  RDN    -0.158015788

sn ( 1 + 2 i + 3 j + 4 k | -PI ) = -1.473447238 - 0.079007894 i - 0.118511841 j - 0.158015788 k and

cn ( 1 + 2 i + 3 j + 4 k | -PI ) = 0.285290836 - 0.408053637 i - 0.612080456 j - 0.816107275 k in R05 to 508

dn ( 1 + 2 i + 3 j + 4 k | -PI ) = -2.793322223 - 0.130928415 i - 0.196392622 j - 0.261856830 k in R09 to R12

Notes:

-Here, the "LBL 03 loop" stops when the parameter m = 0
-So, this variant is often slower than the first one, but it's also shorter...

References:

[1] Abramowitz and Stegun - "Handbook of Mathematical Functions" - Dover Publications - ISBN 0-486-61272-4
[2] http://functions.wolfram.com/

hp41programs

Quaternionic Jacobian Elliptic Functions for the HP-41