nthorderODE

Garden-Hose Method for the HP-41

Overview

1°) Second order Differential Equations: y" = f(x,y,y')

a) 4th-order Formula
b) 6th-order Formula

2°) Third order Differential Equations: y''' = f(x,y,y',y")

a) y(a) = A & y(b) = B & y"(a) = A" [ or y'(a) = A' ] -> y'(a) = ? [ or y"(a) = ? ]

a1) 4th-order Formula
a2) 6th-order Formula

b) y(a) = A & y(b) = B & y(c) = C -> y'(a) = ? & y"(a) = ?

b1) 4th-order Formula
b2) 6th-order Formula

-The following programs employs Runke-Kutta 4th order & 6th order formulae ( cf "Nth-Order Differential Equations for the HP41" ).

1°) Second Order Differential Equations

a) 4th Order Formula

-We want to find y'(a) such that the solution of y" = f(x,y,y') verifies y(a) = A & y(b) = B
-We know a , A , b , B and GRDH computes y'(a):

-You choose 2 initial guesses y'₁ & y'₂ and "GRDH" finds y'(a) with the secant method: this is the Garden-Hose method.

Data Registers: • R00 = function name ( Registers R00 thru R05 are to be initialized before executing "GRDH" )

                                      • R01 = a • R03 = A • R05 = N = number of steps R06 thru R16: temp
                                      • R02 = b        • R04 = B

Flags: /
Subroutine:       A program which computes y" = f(x,y,y') assuming x , y , y' are in registers X , Y , Z ( respectively ) upon entry

01 LBL "GRDH"
02 STO 14
03 RCL 02
04 RCL 01
05 -
06 RCL 05
07 ST+ X
08 /
09 STO 09
10 R^
11 STO 15
12 XEQ 00
13 STO 10
14 GTO 02
15 LBL 00
16 STO 08
17 RCL 05
18 STO 16
19 RCL 03
20 STO 07
21 RCL 01
22 STO 06
23 LBL 01

24 RCL 08
25 RCL 07
26 RCL 06
27 XEQ IND 00
28 RCL 09
29 ST+ 06
30 *
31 STO 12
32 RCL 08
33 +
34 STO 13
35 LASTX
36 RCL 09
37 *
38 STO 11
39 RCL 07
40 +
41 RCL 06
42 XEQ IND 00
43 RCL 09
44 ST* 13
45 *
46 ST+ 12

47 ST+ 12
48 RCL 08
49 +
50 ENTER
51 X<> 13
52 ST+ 11
53 ST+ 11
54 RCL 07
55 +
56 RCL 06
57 XEQ IND 00
58 RCL 09
59 ST+ 06
60 ST+ X
61 ST* 13
62 *
63 ST+ 12
64 RCL 08
65 +
66 ENTER
67 X<> 13
68 ST+ 11
69 RCL 07

70 +
71 RCL 06
72 XEQ IND 00
73 RCL 09
74 ST* 13
75 *
76 RCL 12
77 +
78 3
79 /
80 ST+ 08
81 RCL 11
82 RCL 13
83 +
84 3
85 /
86 ST+ 07
87 DSE 16
88 GTO 01
89 RCL 07
90 RCL 04
91 -

92 RTN
93 LBL 02
94 VIEW 14
95 RCL 14
96 XEQ 00
97 ENTER
98 ENTER
99 X<> 10
100 -
101 X#0?
102 /
103 RCL 15
104 RCL 14
105 STO 15
106 STO T
107 -
108 *
109 +
110 STO 14
111 X#Y?
112 GTO 02
113 END

( 155 bytes / SIZE 017 )

STACK	INPUTS	OUTPUTS
Y	y₁'(a)	y'(a)
X	y₂'(a)	y'(a)

X-inputs are initial guesses

Example: y" = x y y' y(0) = 1 y(1) = 3

01 LBL "T"
02 *
03 *
04 RTN

"T" ASTO 00

   a = 0 STO 01    A = 1 STO 03
   b = 1 STO 02 B = 3 STO 04

-If we choose N = 10 ( so h = 0.1 ) and 2 initial guesses: 1 & 2

10 STO 05

1 ENTER^
2 XEQ "GRDH" >>>> y'(0) = 1.411870345 ---Execution time = 238s---

-With N = 100 STO 05, the same initial guesses give y'(0) = 1.411819034

Notes:

-Check that R08 = B ( at least very approximately ) or add RCL 08 X<>Y after line 112.
-If the error is too large, change the initial guesses...

     b) 6th Order Formula

-6th order Runge-Kutta formula will give a better precision with the same stepsize:

Data Registers: • R00 = function name ( Registers R00 thru R05 are to be initialized before executing "GRDH" )

                                      • R01 = a • R03 = A • R05 = N = number of steps R06 thru R19: temp
                                      • R02 = b        • R04 = B

Flags: /
Subroutine:       A program which computes y" = f(x,y,y') assuming x , y , y' are in registers X , Y , Z ( respectively ) upon entry

01 LBL "GRDH"
02 STO 17
03 RCL 02
04 RCL 01
05 -
06 RCL 05
07 /
08 STO 09
09 R^
10 STO 18
11 XEQ 00
12 STO 19
13 GTO 02
14 LBL 00
15 STO 08
16 RCL 05
17 STO 16
18 RCL 03
19 STO 07
20 RCL 01
21 STO 06
22 LBL 01
23 RCL 08
24 RCL 07
25 RCL 06
26 XEQ IND 00
27 RCL 09
28 *
29 STO 10
30 3
31 /
32 RCL 08
33 +
34 RCL 10
35 18
36 /
37 RCL 08
38 3
39 /
40 +
41 RCL 09
42 *
43 RCL 07
44 +
45 RCL 09
46 3
47 /
48 RCL 06
49 +
50 XEQ IND 00
51 RCL 09
52 *
53 STO 11

54 ST+ X
55 3
56 /
57 RCL 08
58 +
59 RCL 11
60 ST+ X
61 9
62 /
63 RCL 08
64 ST+ X
65 3
66 /
67 +
68 RCL 09
69 *
70 RCL 07
71 +
72 RCL 09
73 ST+ X
74 3
75 /
76 RCL 06
77 +
78 XEQ IND 00
79 RCL 09
80 *
81 STO 12
82 CHS
83 RCL 11
84 4
85 *
86 +
87 RCL 10
88 +
89 12
90 /
91 RCL 08
92 +
93 RCL 10
94 RCL 12
95 +
96 36
97 /
98 RCL 08
99 3
100 /
101 +
102 RCL 09
103 *
104 RCL 07
105 +
106 RCL 09

107 3
108 /
109 RCL 06
110 +
111 XEQ IND 00
112 RCL 09
113 *
114 STO 13
115 90
116 *
117 RCL 12
118 35
119 *
120 +
121 RCL 11
122 110
123 *
124 -
125 RCL 10
126 25
127 *
128 +
129 48
130 /
131 RCL 08
132 +
133 RCL 13
134 270
135 *
136 RCL 12
137 35
138 *
139 +
140 RCL 11
141 330
142 *
143 -
144 RCL 10
145 125
146 *
147 +
148 288
149 /
150 RCL 08
151 5
152 *
153 6
154 /
155 +
156 RCL 09
157 *
158 RCL 07
159 +

160 RCL 09
161 5
162 *
163 6
164 /
165 RCL 06
166 +
167 XEQ IND 00
168 RCL 09
169 *
170 STO 14
171 10
172 /
173 RCL 13
174 2
175 /
176 +
177 RCL 12
178 8
179 /
180 -
181 RCL 11
182 11
183 *
184 24
185 /
186 -
187 RCL 10
188 3
189 *
190 20
191 /
192 +
193 RCL 08
194 +
195 RCL 14
196 15
197 /
198 CHS
199 RCL 13
200 12
201 /
202 -
203 RCL 12
204 16
205 /
206 +
207 RCL 11
208 11
209 *
210 144
211 /
212 +

213 RCL 10
214 40
215 /
216 +
217 RCL 08
218 6
219 /
220 +
221 RCL 09
222 *
223 RCL 07
224 +
225 RCL 09
226 6
227 /
228 RCL 06
229 +
230 XEQ IND 00
231 RCL 09
232 *
233 STO 15
234 1600
235 *
236 RCL 14
237 128
238 *
239 +
240 RCL 13
241 2360
242 *
243 -
244 RCL 12
245 215
246 *
247 +
248 RCL 11
249 1980
250 *
251 +
252 RCL 10
253 783
254 *
255 -
256 780
257 /
258 RCL 08
259 +
260 RCL 15
261 4 E3
262 *
263 RCL 14
264 64
265 *

266 +
267 RCL 13
268 4720
269 *
270 -
271 RCL 12
272 215
273 *
274 +
275 RCL 11
276 3960
277 *
278 +
279 RCL 10
280 2349
281 *
282 -
283 2340
284 /
285 RCL 08
286 +
287 RCL 09
288 *
289 RCL 07
290 +
291 RCL 09
292 RCL 06
293 +
294 XEQ IND 00
295 RCL 09
296 *
297 RCL 15
298 80
299 *
300 RCL 14
301 16
302 *
303 +
304 RCL 13
305 110
306 *
307 +
308 RCL 12
309 55
310 *
311 +
312 RCL 10
313 39
314 *
315 +
316 600
317 /
318 RCL 08

319 +
320 RCL 09
321 ST+ 06
322 *
323 ST+ 07
324 CLX
325 RCL 10
326 +
327 13
328 *
329 RCL 12
330 RCL 13
331 +
332 55
333 *
334 +
335 RCL 14
336 RCL 15
337 +
338 32
339 *
340 +
341 200
342 /
343 ST+ 08
344 DSE 16
345 GTO 01
346 RCL 07
347 RCL 04
348 -
349 RTN
350 LBL 02
351 VIEW 17
352 RCL 17
353 XEQ 00
354 ENTER
355 ENTER
356 X<> 19
357 -
358 X#0?
359 /
360 RCL 18
361 RCL 17
362 STO 18
363 STO T
364 -
365 *
366 +
367 STO 17
368 X#Y?
369 GTO 02
370 END

( 492 bytes / SIZE 020 )

STACK	INPUTS	OUTPUTS
Y	y₁'(a)	y'(a)
X	y₂'(a)	y'(a)

X-inputs are initial guesses

Example: y" = x y y' y(0) = 1 y(1) = 3

01 LBL "T"
02 *
03 *
04 RTN

"T" ASTO 00

a = 0 STO 01 A = 1 STO 03
b = 1 STO 02 B = 3 STO 04

-If we choose N = 10 ( so h = 0.1 ) and 2 initial guesses: 1 & 2

10 STO 05

1 ENTER^
2 XEQ "GRDH" >>>> y'(0) = 1.411819270

-With N = 20 STO 05, the same initial guesses give y'(0) = 1.411819031

Notes:

-Check that R08 = B ( at least very approximately ) or add RCL 08 X<>Y after line 369.
-If the error is too large, change the initial guesses...

2°) Third Order Differential Equations

a) y(a) = A & y(b) = B & y"(a) = A" [ or y'(a) = A' ] -> y'(a) = ? [ or y"(a) = ? ]

a1) 4th Order Formula

-We want to find y'(a) such that the solution of y''' = f(x,y,y',y") with y"(a) = A" verifies y(a) = A & y(b) = B

-We know a , A , b , B , A" and "GRDH3" computes y'(a) with the secant method.

Data Registers: • R00 = function name ( Registers R00 thru R06 are to be initialized before executing "GRDH3" )

                                      • R01 = a • R03 = A • R05 = y"(a) R07 thru R20: temp
                                      • R02 = b        • R04 = B • R06 = N = number of steps

Flags: /
Subroutine:       A program which computes y''' = f(x,y,y',y") assuming x , y , y' , y" are in registers X , Y , Z , T ( respectively ) upon entry

-Lines 14 & 118 are three-byte GTOs

01 LBL "GRDH3"
02 STO 17
03 RCL 02
04 RCL 01
05 -
06 RCL 06
07 ST+ X
08 /
09 STO 11
10 R^
11 STO 18
12 XEQ 00
13 STO 19
14 GTO 02
15 LBL 00
16 STO 09
17 RCL 05
18 STO 10
19 RCL 06
20 STO 20
21 RCL 03
22 STO 08
23 RCL 01
24 STO 07
25 LBL 01
26 RCL 10
27 RCL 09
28 RCL 08
29 RCL 07

30 XEQ IND 00
31 RCL 11
32 ST+ 07
33 *
34 STO 14
35 RCL 10
36 +
37 STO 15
38 RCL 11
39 LASTX
40 *
41 STO 13
42 RCL 09
43 +
44 STO 16
45 LASTX
46 RCL 11
47 *
48 STO 12
49 RCL 08
50 +
51 RCL 07
52 XEQ IND 00
53 RCL 11
54 ST* 15
55 ST* 16
56 *
57 ST+ 14
58 ST+ 14

59 RCL 10
60 +
61 ENTER
62 X<> 15
63 ST+ 13
64 ST+ 13
65 RCL 09
66 +
67 ENTER
68 X<> 16
69 ST+ 12
70 ST+ 12
71 RCL 08
72 +
73 RCL 07
74 XEQ IND 00
75 RCL 11
76 ST+ 07
77 ST+ X
78 ST* 15
79 ST* 16
80 *
81 ST+ 14
82 RCL 10
83 +
84 ENTER
85 X<> 15
86 ST+ 13
87 RCL 09

88 +
89 ENTER
90 X<> 16
91 ST+ 12
92 RCL 08
93 +
94 RCL 07
95 XEQ IND 00
96 RCL 11
97 ST* 15
98 ST* 16
99 *
100 RCL 14
101 +
102 3
103 /
104 ST+ 10
105 RCL 13
106 RCL 15
107 +
108 3
109 /
110 ST+ 09
111 RCL 12
112 RCL 16
113 +
114 3
115 /

116 ST+ 08
117 DSE 20
118 GTO 01
119 RCL 08
120 RCL 04
121 -
122 RTN
123 LBL 02
124 VIEW 17
125 RCL 17
126 XEQ 00
127 ENTER
128 ENTER
129 X<> 19
130 -
131 X#0?
132 /
133 RCL 18
134 RCL 17
135 STO 18
136 STO T
137 -
138 *
139 +
140 STO 17
141 X#Y?
142 GTO 02
143 END

( 207 bytes / SIZE 021 )

STACK	INPUTS	OUTPUTS
Y	y₁'(a)	y'(a)
X	y₂'(a)	y'(a)

X-inputs are initial guesses

Example: y''' = y" + x y y' y(0) = 1 y(1) = 3 y"(0) = 2

01 LBL "T"
02 *
03 *
04 +
05 RTN

"T" ASTO 00

a = 0 STO 01 A = 1 STO 03 A'' = 2 STO 05
b = 1 STO 02 B = 3 STO 04

-If we choose N = 10 ( so h = 0.1 ) and 2 initial guesses: 1 & 2

10 STO 06

1 ENTER^
2 XEQ "GRDH3" >>>> y'(0) = 0.445253455 ---Execution time = 4m41s---

-With N = 100 STO 06, the same initial guesses give y'(0) = 0.445221983

Notes:

-Check that R08 = B ( at least very approximately ) ( you may add RCL 08 X<>Y after line 142 )
-If the error is too large, change the initial guesses...

-If you know y'(a) and if y"(a) is unknown, simply replace line 16 by STO 10 & line 18 by STO 09 [ or use a flag... ] and store y'(a) into R05...

-With the example above, if y'(0) = 2 , it yields y"(0) = -0.244541152 with N = 10 and y"(0) = -0.244560355 with N = 100

a2) 6th Order Formula

Data Registers: • R00 = function name ( Registers R00 thru R06 are to be initialized before executing "GRDH3" )

                                      • R01 = a • R03 = A • R05 = y"(a) R07 thru R33: temp
                                      • R02 = b        • R04 = B • R06 = N = number of steps

Flags: /
Subroutine:       A program which computes y''' = f(x,y,y',y") assuming x , y , y' , y" are in registers X , Y , Z , T ( respectively ) upon entry

-Lines 13 & 453 are three-byte GTOs

01 LBL "GRDH3"
02 STO 07
03 RCL 02
04 RCL 01
05 -
06 RCL 06
07 /
08 STO 13
09 R^
10 STO 08
11 XEQ 00
12 STO 32
13 GTO 02
14 LBL 00
15 STO 11
16 RCL 05
17 STO 12
18 RCL 06
19 STO 33
20 RCL 03
21 STO 10
22 RCL 01
23 STO 09
24 LBL 01
25 RCL 12
26 STO 15
27 RCL 11
28 STO 14
29 RCL 10
30 RCL 09
31 XEQ IND 00
32 RCL 13
33 ST* 14
34 ST* 15
35 *
36 STO 16
37 3
38 /
39 RCL 12
40 +
41 STO 18
42 RCL 15
43 3
44 /
45 RCL 11
46 +
47 STO 17
48 RCL 14
49 3
50 /
51 RCL 10
52 +
53 RCL 13
54 3
55 /
56 RCL 09
57 +
58 RCL 18
59 RDN
60 XEQ IND 00
61 RCL 13
62 ST* 17
63 ST* 18
64 *
65 STO 19
66 1.5
67 /
68 RCL 12

69 +
70 STO 21
71 RCL 18
72 1.5
73 /
74 RCL 11
75 +
76 STO 20
77 RCL 17
78 1.5
79 /
80 RCL 10
81 +
82 RCL 13
83 1.5
84 /
85 RCL 09
86 +
87 RCL 21
88 RDN
89 XEQ IND 00
90 RCL 13
91 ST* 20
92 ST* 21
93 *
94 STO 22
95 CHS
96 RCL 19
97 4
98 *
99 +
100 RCL 16
101 +
102 12
103 /
104 RCL 12
105 +
106 STO 24
107 RCL 18
108 4
109 *
110 RCL 21
111 -
112 RCL 15
113 +
114 12
115 /
116 RCL 11
117 +
118 STO 23
119 RCL 17
120 4
121 *
122 RCL 20
123 -
124 RCL 14
125 +
126 12
127 /
128 RCL 10
129 +
130 RCL 13
131 3
132 /
133 RCL 09
134 +
135 RCL 24
136 RDN

137 XEQ IND 00
138 RCL 13
139 ST* 23
140 ST* 24
141 *
142 STO 25
143 90
144 *
145 RCL 22
146 35
147 *
148 +
149 RCL 19
150 110
151 *
152 -
153 RCL 16
154 25
155 *
156 +
157 48
158 /
159 RCL 12
160 +
161 STO 27
162 RCL 24
163 90
164 *
165 RCL 21
166 35
167 *
168 +
169 RCL 18
170 110
171 *
172 -
173 RCL 15
174 25
175 *
176 +
177 48
178 /
179 RCL 11
180 +
181 STO 26
182 RCL 23
183 90
184 *
185 RCL 20
186 35
187 *
188 +
189 RCL 17
190 110
191 *
192 -
193 RCL 14
194 25
195 *
196 +
197 48
198 /
199 RCL 10
200 +
201 RCL 13
202 1.2
203 /
204 RCL 09

205 +
206 RCL 27
207 RDN
208 XEQ IND 00
209 RCL 13
210 ST* 26
211 ST* 27
212 *
213 STO 28
214 10
215 /
216 RCL 25
217 2
218 /
219 +
220 RCL 22
221 8
222 /
223 -
224 RCL 19
225 11
226 *
227 24
228 /
229 -
230 RCL 16
231 .15
232 *
233 +
234 RCL 12
235 +
236 STO 30
237 RCL 27
238 10
239 /
240 RCL 24
241 2
242 /
243 +
244 RCL 21
245 8
246 /
247 -
248 RCL 18
249 11
250 *
251 24
252 /
253 -
254 RCL 15
255 .15
256 *
257 +
258 RCL 11
259 +
260 STO 29
261 RCL 26
262 10
263 /
264 RCL 23
265 2
266 /
267 +
268 RCL 20
269 8
270 /
271 -
272 RCL 17

273 11
274 *
275 24
276 /
277 -
278 RCL 14
279 .15
280 *
281 +
282 RCL 10
283 +
284 RCL 13
285 6
286 /
287 RCL 09
288 +
289 RCL 30
290 RDN
291 XEQ IND 00
292 RCL 13
293 ST* 29
294 ST* 30
295 *
296 STO 31
297 40
298 X^2
299 *
300 RCL 28
301 128
302 *
303 +
304 RCL 25
305 2360
306 *
307 -
308 RCL 22
309 215
310 *
311 +
312 RCL 19
313 1980
314 *
315 +
316 RCL 16
317 783
318 *
319 -
320 780
321 /
322 RCL 12
323 +
324 X<> 18
325 1980
326 *
327 RCL 30
328 40
329 X^2
330 *
331 +
332 RCL 27
333 128
334 *
335 +
336 RCL 24
337 2360
338 *
339 -
340 RCL 21

341 215
342 *
343 +
344 RCL 15
345 783
346 *
347 -
348 780
349 /
350 RCL 11
351 +
352 ENTER
353 X<> 17
354 1980
355 *
356 RCL 29
357 40
358 X^2
359 *
360 +
361 RCL 26
362 128
363 *
364 +
365 RCL 23
366 2360
367 *
368 -
369 RCL 20
370 215
371 *
372 +
373 RCL 14
374 783
375 *
376 -
377 780
378 /
379 RCL 10
380 +
381 RCL 09
382 RCL 13
383 +
384 RCL 18
385 RDN
386 XEQ IND 00
387 RCL 13
388 ST+ 09
389 ST* 17
390 ST* 18
391 *
392 STO 19
393 RCL 16
394 +
395 13
396 *
397 RCL 22
398 RCL 25
399 +
400 55
401 *
402 +
403 RCL 28
404 RCL 31
405 +
406 32
407 *
408 +
409 .5

410 %
411 ST+ 12
412 RCL 15
413 RCL 18
414 +
415 13
416 *
417 RCL 21
418 RCL 24
419 +
420 55
421 *
422 +
423 RCL 27
424 RCL 30
425 +
426 32
427 *
428 +
429 .5
430 %
431 ST+ 11
432 RCL 14
433 RCL 17
434 +
435 13
436 *
437 RCL 20
438 RCL 23
439 +
440 55
441 *
442 +
443 RCL 26
444 RCL 29
445 +
446 32
447 *
448 +
449 .5
450 %
451 ST+ 10
452 DSE 33
453 GTO 01
454 RCL 10
455 RCL 04
456 -
457 RTN
458 LBL 02
459 VIEW 07
460 RCL 07
461 XEQ 00
462 ENTER
463 ENTER
464 X<> 32
465 -
466 X#0?
467 /
468 RCL 08
469 RCL 07
470 STO 08
471 STO T
472 -
473 *
474 +
475 STO 07
476 X#Y?
477 GTO 02
478 END

( 714 bytes / SIZE 034 )

STACK	INPUTS	OUTPUTS
Y	y₁'(a)	y'(a)
X	y₂'(a)	y'(a)

X-inputs are initial guesses

Example: y''' = y" + x y y' y(0) = 1 y(1) = 3 y"(0) = 2

01 LBL "T"
02 *
03 *
04 +
05 RTN

"T" ASTO 00

a = 0 STO 01 A = 1 STO 03 A'' = 2 STO 05
b = 1 STO 02 B = 3 STO 04

-If we choose N = 10 ( so h = 0.1 ) and 2 initial guesses: 1 & 2

10 STO 06

1 ENTER^
2 XEQ "GRDH3" >>>> y'(0) = 0.445221967

-With N = 20 STO 06, the same initial guesses give y'(0) = 0.445221981

Notes:

-Check that R08 = B ( at least very approximately ) ( you may add RCL 08 X<>Y after line 477 )
-If the error is too large, change the initial guesses...

-If you know y'(a) and if y"(a) is unknown, simply replace line 15 by STO 12 & line 17 by STO 11 [ or use a flag... ] and store y'(a) into R05...

-With the example above, if y'(0) = 2 , it yields y"(0) = -0.244560336 with N = 10 and y"(0) = -0.244560354 with N = 20

b) y(a) = A & y(b) = B & y(c) = C -> y'(a) = ? & y"(a) = ?

b1) 4th Order Formula

-We want to find y'(a) & y"(a) such that the solution of y''' = f(x,y,y',y") that verifies y(a) = A & y(b) = B & y(c) = C

-We know a , b , c , A , B , C and "GRDHC" computes y'(a) & y"(a) with the generalized "secant method".
-It solves a system of 2 equations in 2 unknowns ( like "SXY" listed in "Linear & Non-Linear systems for the HP41" paragraph 2°)b) )

Data Registers: • R00 = function name ( Registers R00 thru R08 are to be initialized before executing "GRDHC" )

                                      • R01 = a • R04 = A • R07 = N = number of steps between a & b R09 thru R29: temp
                                      • R02 = b        • R05 = B • R08 = N' = number of steps between b & c
• R03 = c • R06 = C

Flags: /
Subroutine:       A program which computes y''' = f(x,y,y',y") assuming x , y , y' , y" are in registers X , Y , Z , T ( respectively ) upon entry

-Lines 09-134-201 are three-byte GTOs

01 LBL "GRDHC"
02 STO 21
03 RDN
04 STO 22
05 RDN
06 STO 19
07 X<>Y
08 STO 20
09 GTO 02
10 LBL 00
11 STO 11
12 X<>Y
13 STO 12
14 RCL 02
15 STO 10
16 RCL 01
17 STO 09
18 -
19 RCL 07
20 STO 23
21 ST+ X
22 /
23 STO 13
24 XEQ 01
25 RCL 05
26 -
27 STO 24
28 RCL 03
29 RCL 02
30 -

31 RCL 08
32 STO 23
33 ST+ X
34 /
35 STO 13
36 XEQ 01
37 RCL 06
38 -
39 RCL 24
40 RTN
41 LBL 01
42 RCL 12
43 RCL 11
44 RCL 10
45 RCL 09
46 XEQ IND 00
47 RCL 13
48 ST+ 09
49 *
50 STO 16
51 RCL 12
52 +
53 STO 17
54 RCL 13
55 LASTX
56 *
57 STO 15
58 RCL 11
59 +
60 STO 18

61 LASTX
62 RCL 13
63 *
64 STO 14
65 RCL 10
66 +
67 RCL 09
68 XEQ IND 00
69 RCL 13
70 ST* 17
71 ST* 18
72 *
73 ST+ 16
74 ST+ 16
75 RCL 12
76 +
77 ENTER
78 X<> 17
79 ST+ 15
80 ST+ 15
81 RCL 11
82 +
83 ENTER
84 X<> 18
85 ST+ 14
86 ST+ 14
87 RCL 10
88 +
89 RCL 09
90 XEQ IND 00

91 RCL 13
92 ST+ 09
93 ST+ X
94 ST* 17
95 ST* 18
96 *
97 ST+ 16
98 RCL 12
99 +
100 ENTER
101 X<> 17
102 ST+ 15
103 RCL 11
104 +
105 ENTER
106 X<> 18
107 ST+ 14
108 RCL 10
109 +
110 RCL 09
111 XEQ IND 00
112 RCL 13
113 ST* 17
114 ST* 18
115 *
116 RCL 16
117 +
118 3
119 /
120 ST+ 12

121 RCL 15
122 RCL 17
123 +
124 3
125 /
126 ST+ 11
127 RCL 14
128 RCL 18
129 +
130 3
131 /
132 ST+ 10
133 DSE 23
134 GTO 01
135 RCL 10
136 RTN
137 LBL 02
138 VIEW 19
139 RCL 20
140 RCL 21
141 XEQ 00
142 STO 27
143 X<>Y
144 STO 26
145 RCL 22
146 RCL 19
147 XEQ 00
148 STO 29
149 X<>Y
150 STO 28

151 RCL 20
152 RCL 19
153 XEQ 00
154 STO 25
155 ST- 27
156 ST- 29
157 X<>Y
158 STO 24
159 ST- 26
160 ST- 28
161 RCL 26
162 RCL 29
163 *
164 RCL 27
165 RCL 28
166 *
167 -
168 STO 23
169 X=0?
170 GTO 03
171 RCL 25
172 RCL 28
173 *
174 RCL 24
175 RCL 29
176 *
177 -
178 X<>Y
179 /
180 RCL 21

181 RCL 19
182 STO 21
183 -
184 *
185 ST+ 19
186 RCL 24
187 RCL 27
188 *
189 RCL 25
190 RCL 26
191 *
192 -
193 RCL 23
194 /
195 RCL 22
196 RCL 20
197 STO 22
198 -
199 *
200 ST+ 20
201 GTO 02
202 LBL 03
203 RCL 24
204 ABS
205 RCL 25
206 ABS
207 +
208 RCL 20
209 RCL 19
210 END

( 326 bytes / SIZE 030 )

STACK	INPUTS	OUTPUTS
T	y₁"(a)	/
Z	y₁'(a)	\| error \|
Y	y₂"(a)	y"(a)
X	y₂'(a)	y'(a)

Inputs are initial guesses
Z-output = | y(b) - B | + | y(c) - C | must be close to zero.
Otherwise, change the initial guesses

Example: y''' = y" + x y y' y(0) = 1 y(1) = 3 y(2) = 4

01 LBL "T"
02 *
03 *
04 +
05 RTN

"T" ASTO 00

a = 0 STO 01 A = 1 STO 04
b = 1 STO 02 B = 3 STO 05
c = 2 STO 03 C = 4 STO 06

-If you choose N = N' = 10 ( so stepsize = h = 0.1 ) and 2 initial guesses: (1;1) & (2;2)

10 STO 07 STO 08

1 ENTER^ ENTER^
2 ENTER^ XEQ "GRDHC" >>>> y'(0) = 2.906612347 ---Execution time = 48m39s--- (!)
RDN y"(0) = -1.561167630
RDN 6 E-9

-With N = N' = 100 STO 07 STO 08, the same initial guesses give ( with V41 in turbo mode ) y'(0) = 2.906622627 & y"(0) = -1.561183524

Notes:

-With free42 and N = N' = 1000 it yields: y'(0) = 2.90662263100... & y"(0) = -1.56118352608...

-It's better to choose a < b < c or c < b < a
-In most cases, N & N' should be chosen such that (b-a) / N ~ (c-b) / N'
-Thus, the stepsizes will be approximately equal...

b2) 6th Order Formula

Data Registers: • R00 = function name ( Registers R00 thru R08 are to be initialized before executing "GRDHC" )

                                      • R01 = a • R04 = A • R07 = N = number of steps between a & b R09 thru R44: temp
                                      • R02 = b        • R05 = B • R08 = N' = number of steps between b & c
• R03 = c • R06 = C

Flags: /
Subroutine:       A program which computes y''' = f(x,y,y',y") assuming x , y , y' , y" are in registers X , Y , Z , T ( respectively ) upon entry

-Lines 09-468-535 are three-byte GTOs

01 LBL "GRDHC"
02 STO 36
03 RDN
04 STO 37
05 RDN
06 STO 34
07 X<>Y
08 STO 35
09 GTO 02
10 LBL 00
11 STO 11
12 X<>Y
13 STO 12
14 RCL 02
15 STO 10
16 RCL 01
17 STO 09
18 -
19 RCL 07
20 STO 33
21 /
22 STO 13
23 XEQ 01
24 RCL 05
25 -
26 STO 32
27 RCL 03
28 RCL 02
29 -
30 RCL 08
31 STO 33
32 /
33 STO 13
34 XEQ 01
35 RCL 06
36 -
37 RCL 32
38 RTN
39 LBL 01
40 RCL 12
41 STO 15
42 RCL 11
43 STO 14
44 RCL 10
45 RCL 09
46 XEQ IND 00
47 RCL 13
48 ST* 14
49 ST* 15
50 *
51 STO 16
52 3
53 /
54 RCL 12
55 +
56 STO 18
57 RCL 15
58 3
59 /
60 RCL 11
61 +
62 STO 17
63 RCL 14
64 3
65 /
66 RCL 10
67 +
68 RCL 13
69 3
70 /
71 RCL 09
72 +
73 RCL 18
74 RDN
75 XEQ IND 00
76 RCL 13
77 ST* 17
78 ST* 18

79 *
80 STO 19
81 1.5
82 /
83 RCL 12
84 +
85 STO 21
86 RCL 18
87 1.5
88 /
89 RCL 11
90 +
91 STO 20
92 RCL 17
93 1.5
94 /
95 RCL 10
96 +
97 RCL 13
98 1.5
99 /
100 RCL 09
101 +
102 RCL 21
103 RDN
104 XEQ IND 00
105 RCL 13
106 ST* 20
107 ST* 21
108 *
109 STO 22
110 CHS
111 RCL 19
112 4
113 *
114 +
115 RCL 16
116 +
117 12
118 /
119 RCL 12
120 +
121 STO 24
122 RCL 18
123 4
124 *
125 RCL 21
126 -
127 RCL 15
128 +
129 12
130 /
131 RCL 11
132 +
133 STO 23
134 RCL 17
135 4
136 *
137 RCL 20
138 -
139 RCL 14
140 +
141 12
142 /
143 RCL 10
144 +
145 RCL 13
146 3
147 /
148 RCL 09
149 +
150 RCL 24
151 RDN
152 XEQ IND 00
153 RCL 13
154 ST* 23
155 ST* 24
156 *

157 STO 25
158 90
159 *
160 RCL 22
161 35
162 *
163 +
164 RCL 19
165 110
166 *
167 -
168 RCL 16
169 25
170 *
171 +
172 48
173 /
174 RCL 12
175 +
176 STO 27
177 RCL 24
178 90
179 *
180 RCL 21
181 35
182 *
183 +
184 RCL 18
185 110
186 *
187 -
188 RCL 15
189 25
190 *
191 +
192 48
193 /
194 RCL 11
195 +
196 STO 26
197 RCL 23
198 90
199 *
200 RCL 20
201 35
202 *
203 +
204 RCL 17
205 110
206 *
207 -
208 RCL 14
209 25
210 *
211 +
212 48
213 /
214 RCL 10
215 +
216 RCL 13
217 1.2
218 /
219 RCL 09
220 +
221 RCL 27
222 RDN
223 XEQ IND 00
224 RCL 13
225 ST* 26
226 ST* 27
227 *
228 STO 28
229 10
230 /
231 RCL 25
232 2
233 /
234 +

235 RCL 22
236 8
237 /
238 -
239 RCL 19
240 11
241 *
242 24
243 /
244 -
245 RCL 16
246 .15
247 *
248 +
249 RCL 12
250 +
251 STO 30
252 RCL 27
253 10
254 /
255 RCL 24
256 2
257 /
258 +
259 RCL 21
260 8
261 /
262 -
263 RCL 18
264 11
265 *
266 24
267 /
268 -
269 RCL 15
270 .15
271 *
272 +
273 RCL 11
274 +
275 STO 29
276 RCL 26
277 10
278 /
279 RCL 23
280 2
281 /
282 +
283 RCL 20
284 8
285 /
286 -
287 RCL 17
288 11
289 *
290 24
291 /
292 -
293 RCL 14
294 .15
295 *
296 +
297 RCL 10
298 +
299 RCL 13
300 6
301 /
302 RCL 09
303 +
304 RCL 30
305 RDN
306 XEQ IND 00
307 RCL 13
308 ST* 29
309 ST* 30
310 *
311 STO 31
312 40

313 X^2
314 *
315 RCL 28
316 128
317 *
318 +
319 RCL 25
320 2360
321 *
322 -
323 RCL 22
324 215
325 *
326 +
327 RCL 19
328 1980
329 *
330 +
331 RCL 16
332 783
333 *
334 -
335 780
336 /
337 RCL 12
338 +
339 X<> 18
340 1980
341 *
342 RCL 30
343 40
344 X^2
345 *
346 +
347 RCL 27
348 128
349 *
350 +
351 RCL 24
352 2360
353 *
354 -
355 RCL 21
356 215
357 *
358 +
359 RCL 15
360 783
361 *
362 -
363 780
364 /
365 RCL 11
366 +
367 ENTER
368 X<> 17
369 1980
370 *
371 RCL 29
372 40
373 X^2
374 *
375 +
376 RCL 26
377 128
378 *
379 +
380 RCL 23
381 2360
382 *
383 -
384 RCL 20
385 215
386 *
387 +
388 RCL 14
389 783
390 *

391 -
392 780
393 /
394 RCL 10
395 +
396 RCL 09
397 RCL 13
398 +
399 RCL 18
400 RDN
401 XEQ IND 00
402 RCL 13
403 ST+ 09
404 ST* 17
405 ST* 18
406 *
407 STO 19
408 RCL 16
409 +
410 13
411 *
412 RCL 22
413 RCL 25
414 +
415 55
416 *
417 +
418 RCL 28
419 RCL 31
420 +
421 32
422 *
423 +
424 .5
425 %
426 ST+ 12
427 RCL 15
428 RCL 18
429 +
430 13
431 *
432 RCL 21
433 RCL 24
434 +
435 55
436 *
437 +
438 RCL 27
439 RCL 30
440 +
441 32
442 *
443 +
444 .5
445 %
446 ST+ 11
447 RCL 14
448 RCL 17
449 +
450 13
451 *
452 RCL 20
453 RCL 23
454 +
455 55
456 *
457 +
458 RCL 26
459 RCL 29
460 +
461 32
462 *
463 +
464 .5
465 %
466 ST+ 10
467 DSE 33

468 GTO 01
469 RCL 10
470 RTN
471 LBL 02
472 VIEW 34
473 RCL 35
474 RCL 36
475 XEQ 00
476 STO 42
477 X<>Y
478 STO 41
479 RCL 37
480 RCL 34
481 XEQ 00
482 STO 44
483 X<>Y
484 STO 43
485 RCL 35
486 RCL 34
487 XEQ 00
488 STO 40
489 ST- 42
490 ST- 44
491 X<>Y
492 STO 39
493 ST- 41
494 ST- 43
495 RCL 41
496 RCL 44
497 *
498 RCL 42
499 RCL 43
500 *
501 -
502 STO 38
503 X=0?
504 GTO 03
505 RCL 40
506 RCL 43
507 *
508 RCL 39
509 RCL 44
510 *
511 -
512 X<>Y
513 /
514 RCL 36
515 RCL 34
516 STO 36
517 -
518 *
519 ST+ 34
520 RCL 39
521 RCL 42
522 *
523 RCL 40
524 RCL 41
525 *
526 -
527 RCL 38
528 /
529 RCL 37
530 RCL 35
531 STO 37
532 -
533 *
534 ST+ 35
535 GTO 02
536 LBL 03
537 RCL 39
538 ABS
539 RCL 40
540 ABS
541 +
542 RCL 35
543 RCL 34
544 END

( 833 bytes / SIZE 045 )

STACK	INPUTS	OUTPUTS
T	y₁"(a)	/
Z	y₁'(a)	\| error \|
Y	y₂"(a)	y"(a)
X	y₂'(a)	y'(a)

Inputs are initial guesses
Z-output = | y(b) - B | + | y(c) - C | must be close to zero.
Otherwise, change the initial guesses

Example: y''' = y" + x y y' y(0) = 1 y(1) = 3 y(2) = 4

01 LBL "T"
02 *
03 *
04 +
05 RTN

"T" ASTO 00

   a = 0 STO 01    A = 1 STO 04
   b = 1 STO 02 B = 3 STO 05
c = 2 STO 03 C = 4 STO 06

-If you choose N = N' = 10 ( so stepsize = h = 0.1 ) and 2 initial guesses: (1;1) & (2;2)

10 STO 07 STO 08

1 ENTER^ ENTER^
2 ENTER^ XEQ "GRDHC" >>>> y'(0) = 2.906622560
RDN y"(0) = -1.561183430
RDN 7 E-9

-With N = N' = 20 STO 07 STO 08, the same initial guesses yield:    y'(0) = 2.906622627 & y"(0) = -1.561183523

Reference:

[1] Francis Scheid - "Numerical Analysis" - ( McGraw-Hill ) ISBN: 07-055197-9

hp41programs

Garden-Hose Method for the HP-41