**HP 41C, HP 42S, TI-60: Arithmetic-Geometric Mean**

**Arithmetic-Geometric Mean**

The program AGM calculates the arithmetic-geometric mean of two positive integers x and y. As the graphic above suggests, an iterative process is used to find the AGM, computing both the arithmetic mean and geometric mean until the two means converge.

a0 = x

g0 = y

Repeat:

Arithmetic Mean: a1 = (a0 + g0)/2

Geometric Mean: g1 = √(a0 * g0)

Transfer new to old: a0 = a1, g0 = g1

Until |a1 - g1| < tolerance

You can set the tolerance as low as you want. The programs presented on this blog set tolerance at 10^(-10) (1E-10), to fit the calculator's display.

**HP 41C Program: AGM**

01 LBL^T AGM

02 STO 01

03 X<>Y

04 STO 02

05 X<>Y

06 LBL 00

07 RCL 02

08 RCL 01

09 ENTER

10 R↑

11 R↑

12 X<>Y

13 R↓

14 ENTER

15 R↑

16 +

17 2

18 /

19 STO 01

20 R↓

21 *

22 SQRT

23 STO 02

24 R↑

25 -

26 ABS

27 1E-10

28 X≤Y?

29 GTO 00

30 CLA

31 ^T AGM =

32 ARCL 01

33 AVIEW

34 END

**HP 42S/Swiss Micros DM42/Free42 Program AGM:**

00 {53-Byte Prgm}

01 LBL "AGM"

02 STO 01

03 X<>Y

04 STO 02

05 X<>Y

06 LBL 00

07 RCL 02

08 RCL 01

09 ENTER

10 R↑

11 R↑

12 X<>Y

13 R↓

14 ENTER

15 R↑

16 +

17 2

18 /

19 STO 01

20 R↓

21 *

22 SQRT

23 STO 02

24 R↑

25 -

26 ABS

27 1E-10

28 X≤Y?

29 GTO 00

30 CLA

31 "AGM = "

32 ARCL 01

33 AVIEW

34 END

The instructions for both the HP 41C and 42S versions are same: enter X and Y on the respective stacks and XEQ AGM.

Example (ALL/STD mode is applied):

AGM(37, 78):

37, 78, XEQ AGM returns:

Alpha: AGM = 55.5947005279

**TI-60 Program: AGM**

Instructions:

1. Store X in memory register 1 and Y in memory register 2.

2. Press [ RST ] [ R/S ], the value of |a1 - g1| is displayed.

3. Keep on press [ R/S ] to repeat the calculation until |a1 - g1| falls under 10^(-10).

4. Recall either memory register 1 or 2 to get the answer.

Registers needed: 1 - 4.

Step; Key Code; Key

00; 71; RCL

01; 01; 1

02; 85; +

03; 71; RCL

04; 02; 2

05; 95; =

06; 55; ÷

07; 02; 2

08; 95; =

09; 61; STO

10; 03; 3

11; 71; RCL

12; 01; 1

13; 86; √

14; 65; ×

15; 71; RCL

16; 02; 2

17; 86; √

18; 95; =

19; 61; STO

20; 04; 4

21; 71; RCL

22; 03; 3

23; 61; STO

24; 01; 1

25; 75; -

26; 71; RCL

27; 04; 4

28; 61; STO

29; 02; 2

30; 95; =

31; 87; |X|

32; 13; R/S

Example:

AGM(37, 78)

37 STO 1

78 STO 2

RST R/S

3.778495926, R/S

0.032100702, R/S

0.000002317, R/S

2 -11 (stop)

RCL 1 (or RCL 2): 55.59470053

Source:

"Arithmetic-geometric mean" Wikipedia. https://en.wikipedia.org/wiki/Arithmetic–geometric_mean Last Edited June 12, 2020. Accessed June 12, 2020.

Onward to August...

Eddie

All original content copyright, © 2011-2020. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.