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+--- Thread: (41C) Euler-Taylor method (updated) (/thread-14122.html)
(41C) Euler-Taylor method (updated) - Namir - 12-07-2019 04:21 PM
Euler-Taylor method
===================
To integrate f'(x,y) from x=a to x=b, give y(a), we use an extended version of the Euler method that includes the first AND second derivatives. Since we know f'(x,y) we can derive f''(x,y) and use the following formula:
y = y + h*f'(x,y) + h^2/2*f''(x,y)
Memory Map
==========
Code:
R00 = a,x
R01 = b
R02 = h
R03 = y
R04 = nsteps
R05 = usedLABELS
======
A - Main routine.
B - calculate first derivative f'(x,y).
C - calculate second derivative f''(x,y).
E - calculate exact f(x,y) (used for results comparison). If the exact solution is not known, then just have the label just return a value like 0 or insert "N/A" into register X.
Listing
=======
Code:
1 LBL "EULTLR"
2 LBL A
3 A/^B?
4 PROMPT
5 STO 01
6 RDN
7 STO 00
8 Y/^H?
9 PROMPT
10 STO 02
11 RDN
12 stO 03
13 RCL 01
14 RCL 00
15 -
16 RCL 02
17 /
18 0.5
19 +
20 INT
21 0.001
22 +
23 STO 04 # calclate and store nsteps
24 LBL 00
25 XEQ B # calculate f'(x,y)
26 RCL 02
27 *
28 STO 05
29 XEQ C # calculate f''(x,y)
30 RCL 02
31 STO+00
32 X^2
33 *
34 2
35 /
36 STO+ 05
37 RCL 05
38 STO+ 03
39 VIEW 03
40 DSE 04
41 GTO 00
42 CLD
43 RCL 01
44 XEQ E
45 RCL 03 # recall calculated y at x=b
46 RTN
47 LBL B # calculate f'(x,y) = y * x
48 RCL 03
49 RCL 00
50 *
51 RTN
52 LBL C # calculate f''(x,y) = x*dy/dx + y
53 XEQ B
54 RCL 00
55 *
56 RCL 03
57 +
58 RTN
59 LBL E # exact f(x,y) = g(x) = exp(0.5*x^2)
60 X^2
61 2
62 /
63 EXP
64 RTNUsage
=====
1. Run the program using XEQ "ELRTLR" or pressing key [A].
2. Program prompts "A/^B?". Key in the value for a, press [ENTER], key in the value for b, and then press [R/S] key.
3. Program prompts "Y/^H?". Key in the value for y(a), press [ENTER], key in the value for increment h, and then press [R/S] key.
4) Program displays intermediate values of the integrated y. The program stops with the calculated value of y at x=b.
5) Press [X<>Y] to view the exact value of y at x=b.
You calculate the value for y at any value of x by entering the value of x and the pressing the [E] key.
RE: (41C) Euler-Taylor method - Namir - 12-15-2019 01:33 AM
The next version of the Euler-Taylor program uses the approximation for the second derivative that Albert Chan came up with (click here). Thus we can eliminate LBL C and its code. The program interacts with the user like in the first version:
Code:
01 LBL "EULTLR"
02 LBL A
03 "A/^B?"
04 PROMPT
05 STO 01
06 RDN
07 STO 00
08 "Y/^H?"
09 PROMPT
10 STO 02
11 RDN
12 StO 03
13 RCL 01
14 RCL 00
15 -
16 RCL 02
17 /
18 0.5
19 +
20 INT
21 0.001
22 +
23 STO 04 # calclate and store nsteps
24 LBL 00
25 XEQ B # calculate f'(x,y)
26 STO 05 # Calculate D1=f'(x,y)
27 RCL 02
28 STO+ 00 # x = x +h
29 *
30 STO+ 03 # y = y + h*D1
31 XEQ B # calculate f'(x,y)
32 RCL 05
33 -
34 RCL 02
35 *
36 2
37 /
38 STO+ 03 # y = y + h/2*(f'(x,y)-D1)
39 VIEW 03
40 DSE 04
41 GTO 00
42 CLD
43 RCL 01
44 XEQ E
45 RCL 03 # recall calculated y at x=b
46 RTN
47 LBL B # calculate f'(x,y) = y * x
48 RCL 03
49 RCL 00
50 *
51 RTN
52 LBL E # exact f(x,y) = g(x) = exp(0.5*x^2)
53 X^2
54 2
55 /
56 EXP
57 RTNRE: (41C) Euler-Taylor method - Namir - 12-18-2019 08:12 PM
A third version of the program uses approximations for the second and third function derivatives, yielding better accuracy.
Memory Map
==========
Code:
R00 = a,x
R01 = b
R02 = h
R03 = y
R04 = nsteps
R05 = D1
R06 = D2
R07 = D3LABELS
======
Code:
A - Main routine.
B - calculate first derivative f'(x,y).
E - calculate exact f(x,y) (used for results comparison). If the exact solution is not known, then just have the label just return a value like 0 or insert "N/A" into register X.Listing
=======
Code:
1 LBL "EULTL3"
2 LBL A
3 A/^B?
4 PROMPT
5 STO 01
6 RDN
7 STO 00
8 Y/^H?
9 PROMPT
10 STO 02
11 RDN
12 StO 03
13 RCL 01
14 RCL 00
15 -
16 RCL 02
17 /
18 0.5
19 +
20 INT
21 0.001
22 +
23 STO 04 # calclate
24 SF 00 # set flag
25 LBL 00
26 XEQ B # calculate
27 STO 05 # Calculate
28 RCL 02
29 STO+ 00 # x = x +h
30 *
31 STO+ 03 # y = y + h
32 XEQ B # calculate
33 RCL 05
34 -
35 RCL 02
36 *
37 2
38 /
39 STO 06
40 STO+ 03 # y = y + D
41 FS?C 00
42 GTO 01
43 RCL 06
44 RCL 07
45 -
46 3
47 /
48 STO+ 03 # y =y + (
49 LBL 01
50 VIEW 03
51 RCL 06
52 STO 07 # D3 = D2
53 DSE 04
54 GTO 00
55 CLD
56 RCL 01
57 XEQ E
58 RCL 03 # recall ca
59 RTN
60 LBL B # calculate
61 RCL 03
62 RCL 00
63 *
64 RTN
65 LBL E # exact f(x,
66 X^2
67 2
68 /
69 EXP
70 RTN