(41C) Euler-Taylor method (updated)

[Namir]

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 = used

LABELS
======

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  RTN

Usage
=====

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.