(41C) Euler-Taylor method (updated)
12-07-2019, 04:21 PM (This post was last modified: 12-19-2019 02:00 PM by Namir.)
Post: #1 Namir Senior Member Posts: 907 Joined: Dec 2013
(41C) Euler-Taylor method (updated)
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.
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 Messages In This Thread (41C) Euler-Taylor method (updated) - Namir - 12-07-2019 04:21 PM RE: (41C) Euler-Taylor method - Namir - 12-15-2019, 01:33 AM RE: (41C) Euler-Taylor method - Namir - 12-18-2019, 08:12 PM

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