(12C) Secant Method solution of f(x)=0

[Gamo]

This Secant Method used only 29 line in the program but come with a cost of more steps to setup.

Setup Procedure:
1. Store Two Guesses at R0 and R2
2. Store f(x) end point at R1
3. Store Tolerant at R4
4. Store Interval at R3
5. GTO 03
6. R/S

Example: X^X=Y
Problem: X^X=1000
f(x) Program:
GTO 28
P/R
LN
LSTx
x
EEX
3
LN
-
STO1
GTO 03
P/R

Setup:
1. 3 STO 0, 5 STO 2
2. 1 STO 1
3. EEX CHS 6 STO 4
4. 1 CHS STO 3
5. GTO 03
6. R/S

Answer: 4.555535705

Program:

Code:


01 RCL 0    // Result
02 R/S       // ........and stop
03 RCL 2    // Secant Method start here 03 to 18
04 RCL 0
05 -
06 LSTx
07 STO 2
08 CLx
09 RCL 3
10 RCL 1
11 -
12 LSTx
13 STO 3
14 Rv
15 /
16 RCL 1
17 x
18 STO-0
19 RCL 0       //Stop Condition: relative error of the root
20 /
21 2             //ABS function
22 y^x
23 SQRT
24 RCL 4
25 X<>Y
26 X<=Y
27 GTO 01    //Root is found
28 RCL 0
29 .............Start f(x) here
.
.
.
STO 1
GTO 03........Iteration Loop and End f(x)

Gamo

[Dieter]

(03-31-2018 12:43 PM)Gamo Wrote:  Setup Procedure:
1. Store Two Guesses at R0 and R2
2. Store f(x) end point at R1
3. Store Tolerant at R4
4. Store Interval at R3
5. GTO 03
6. R/S

What is "f(x) end point" in step 2 ? Why is it 1 in the example?
And what is the "interval" in step 4 ? Why is it –1 in the example?
I get an Error 0 if I run the program with the example data.

Looking at the program, I think the values in R1 (step 2) and R3 (step 4) should be f(x1) and f(x2).

(03-31-2018 12:43 PM)Gamo Wrote:  Example: X^X=Y
Problem: X^X=1000
f(x) Program:
GTO 28
P/R
LN
LSTx
x
EEX
3
-
STO1
GTO 03
P/R

Here f(x) is rewritten as x*ln(x) – ln 1000 = 0
So there is a "LN" missing after the EEX 3.

And here's a tip: You can elegantly store x1 and f(x1) during the calculation.

Code:

RCL 2
RCL 0
STO 2
-
RCL 3
RCL 1
STO 3
-
/

And another tip: if you want to check if |x| is less than R4 (error tolerance) you can also divide x by R4 and check if the integer part is zero. This is both shorter and faster, three steps instead of six:

Code:

...
RCL 4
/
INT
X=0?
GTO 01

BTW checking the relative error (division by R0) may cause problems since R0 can be zero. You can avoid this if you replace this value with 1:

Code:

...
STO-0
RCL 0
x=0?
e^x
/

Maybe you can try this version:

Code:

01 STO 4
02 Rv
03 STO 3
04 Rv
05 STO 2
06 Rv
07 STO 1
08 RCL 1
09 RCL 2
10 STO 1
11 -
12 RCL 3
13 RCL 4
14 STO 3
15 -
16 /
17 RCL 4
18 x
19 STO-2
20 RCL 2
21 X=0?
22 e^x
23 /
24 EEX
25 8
26 x
27 INTG
28 X=0?
29 GTO 31
30 GTO 33
31 RCL 2
32 GTO 00
33 RCL 2
34 ENTER    // start f(x)
35 LN
36 x
37 EEX
38 3
39 LN
40 -   .
41 STO 4    // end f(x) with
42 GTO 08   // these two lines

You have to enter the two guesses x1 and x2 as well as their respective function values f(x1) and f(x2). The program expects them on the stack. You can also store them manually in R1...R4 and remove the first seven lines, which makes the program much shorter. In this case, here is the register mapping:

R1:  x1
R2:  x2
R3:  f(x1)
R4:  f(x2)

If in the example the guesses are 3 and 5 their function values are f(3)=–3.611918412 and f(5)=1.139434281. But usually rounded values will do:

3 [ENTER] 5  [ENTER]  –3,6 [ENTER] 1,1  [R/S]
=> 4,555535706

You can even round f(x1) to –4 and f(x2) to 1 and it still works. If you're lazy you may even try just the signs of f(x), here –1 and +1 (which indeed is enough to return a correct result).

An input for the relative error threshold is not required, the program quits if it drops below 1E–8 which usually means that the result is as good as it gets.

However, I think the already posted modified Regula Falsi / Illinois algorithm works better than the Secant Rule, so it should be preferred.

Dieter

[Csaba Tizedes]

A little idea, because I hate to calculate the next estimated root. (A BIG HINT: With this method you can easily implement Muller's method if your calculator include parabolic fit!!!)

For secant method here is a "skeleton", but it can be run:

Code:

0
x^,r
STO 0
RCL 8
y^,r
LSTx
SUM-
RCL 7
STO 8
RCL 0
STO 7
PSE    // removable, you can see here the actual estimation of the root
-----------
e^x    // f(x)=0 here, eg: e^x=x^2
LSTx
ENTER
×
-
RCL 0
SUM+

Usage similar as above:

1. but first clear the stat regs: CLEAR SUM,
2. then store x0 into R7
3. estimate f(x0) and store x0, f(x0) point into stat regs: f(x0) ENTER x0 SUM+
4. then store x1 into R8
5. estimate f(x1) and store x1, f(x1) point into stat regs: f(x1) ENTER x1 SUM+
6. jump to the beginning of the program memory and run the program: CLEAR PRGM R/S
7. the estimations shows on the display
8. the approximated root is in R0
9. (during testing I got Error 2 sometimes...)
   

For e^x=x^2:

1. f CLEAR SUM
2. 0 STO R7
3. 1 ENTER 0 SUM+
4. 1 STO 8
5. 1.7 ENTER 1 SUM+
6. f CLEAR PRGM
7. R/S
   

The result:

Code:

-1.4286  R/S
-0.5100  R/S
-0.6560  R/S
-0.7074  R/S
-0.7034  R/S
-0.7035  R/S
-0.7035  R/S
...

Csaba