(41C) Method of Successive Substitutions
10-04-2020, 04:21 PM
Post: #1 Eddie W. Shore Senior Member Posts: 1,236 Joined: Dec 2013
(41C) Method of Successive Substitutions
Repeating the Calculation Again and Again

With an aid of a scientific calculator, we can solve certain problems of the form:

f(x) = x.

Examples include:

tan cos sin x = x

e^-x = x

atan √x = x

sin cos x = x

(acos x) ^ (1/3) = x

In any case with trigonometric functions, the angle mode will need to be in radians. You will also need a good guess to get to a solution and to know that at some real number x, f(x) and x intersect.

Take the equation ln(3*x) = x with initial guess x0 = 1.512

Depending on the operating of the scientific calculator the keystrokes would be:

AOS:
1.512 [ = ]
Loop: [ × ] 3 [ = ] [ ln ]

RPN:
1.512 [ENTER]
Loop: 3 [ × ] [ ln ]

ALG:
1.512 [ENTER/=]
Loop: ln( 3 * Ans) [ ENTER/= ]

Repeat the loop as many times as you like and hope you start seeing the answers converge. After repeating the loop over and over and over again, at six decimal answers, the readout will be about 1.512135.

An approximate answer to ln(3x) = x, x ≈ 1.512134552

If your calculator has a solve function, you can check the answer, but this method can be useful if your calculator does not have a solve function.

The program SUCCESS illustrates this method.

HP 41C/DM41 Program: SUCCESS
This program calls on the subroutine, FX. FX is where you enter f(x). End FX with the RTN command.

Code:
01 LBL^T SUCCESS 02 ^T F<X>=X 03 AVIEW 04 PSE 05 ^T GUESS? 06 PROMPT 07 STO 00 08 ^T PRECISION? 09 PROMPT 10 STO 02 11 0 12 STO 03 13 1 14 STO 04 15 LBL 01 16 RCL 00 17 XEQ ^FX 18 STO 01 19 RCL 00 20 - 21 ABS 22 STO 04 23 RCL 01 24 STO 00 25 1 26 ST+ 03 27 200 28 RCL 03 29 X>Y? 30 GTO 02 31 RCL 02 32 CHS 33 10↑X 34 RCL 04 35 X>Y? 36 GTO 01 37 ^T SOL= 38 ARCL 01 39 AVIEW 40 STOP 41 ^T ITER= 42 ARCL 03 43 AVIEW 44 STOP 45 ^T DIFF= 46 ARCL 04 47 AVIEW 48 STOP 49 GTO 04 50 LBL 02 51 ^T NO SOL FOUND 52 AVIEW 53 STOP  54 LBL 04 55 END

Examples for FX:

f(x) = sin cos x.

Program:
LBL ^FX
COS
SIN
RTN

f(x) = e^-x

Program:
LBL ^FX
CHS
E↑X
RTN

Be aware, some equations cannot be solved in this manner, such as x = π / sin x and x = ln(1 / x^4).

Cheung, Y.L. "Using Scientific Calculators to Demonstrate the Method of Successive Substitutions" The Mathematics Teacher. National Council of Teachers of Mathematics. January 1986, Vol. 79 No. 1 pp. 15-17 http://www.jstor.com/stable/27964746

Blog entry: https://edspi31415.blogspot.com/2020/10/...od-of.html
10-06-2020, 12:57 AM (This post was last modified: 10-06-2020 02:05 AM by Albert Chan.)
Post: #2
 Albert Chan Senior Member Posts: 1,658 Joined: Jul 2018
RE: (41C) Method of Successive Substitutions
(10-04-2020 04:21 PM)Eddie W. Shore Wrote:  Take the equation ln(3*x) = x with initial guess x0 = 1.512

Sometimes, convergence may be slow, or not at all.
We can place a weight on it.

With my Casio FX-115MS

1.512 =
ln(3 Ans ﻿
= ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → 1.512045566
= ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → 1.512075703
﻿= ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → 1.512095633

r = (95633-75703) / (75703-45566) = 19930 / 30317

Convegence is slow (we wanted small |r|)
Assume same trend continued (constant r), estimated converged to:

1.512045566 + 0.000030317/(1-r) = 1.512134548

Let's check if assumption is good. Continued on ...
w = 1/(1-r) ≈ 3 ﻿ ﻿ ﻿ ﻿ ﻿ ﻿
x = (1-w)*x + w*ln(3*x) = -2 x + 3 ln(3*x)

-2 Ans + 3 ln( 3 Ans ﻿
= ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → 1.512135175
= ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → 1.512134542
= ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → 1.512134552, converged