Complete Elliptic Integral of the First Kind

for HP 42S
An implementation of
Arithmetic-Geometric Mean
$a_{n+1} &=& \frac{1}{2}(a_n + b_n)\\ b_{n+1} &=& \sqrt{a_n b_n}$

Jan. 8 2023
Takayuki HOSODA

The arithmetic-geometric mean can be used to compute - among others - logarithms, complete and incomplete elliptic integrals of the first and second kind, and Jacobi elliptic functions.

Usage

Operation

[XEQ] "cAGM"  — for complex number arguments
[XEQ] "AGM"   — for real number arguments

Input

REG y      : argument value b
REG X      : argument value a

Resources used

MEM  00    : convergence radius
MEM  01    : |a_n + √a_n b_n|

Output

REG X      : AGM(a, b)

Arithmetic-Geometric Mean - for HP 42S
Rev.1.0 (Jan. 8, 2023) (c) 2022 Takayuki HOSODA

00 { 158-Byte Prgm }
01▸LBL "cAGM"      ; for complex number arguments
02 0
03 ENTER
04 0
05 COMPLEX
06 X=Y?
07 RTN
08 R↓
09 X<>Y
10 R↑
11 X=Y?
12 RTN
13 RCL ST Z
14 RCL+ ST Z
15 X=Y?
16 RTN
17 R↓
18 R↓
19 ENTER
20 ABS
21 RCL ST Z
22 ABS
23 X>Y?
24 X<>Y
25 CLX
26 2?-12
27 ×
28 STO 00
29▸LBL 00
30 R↓
31 ENTER
32 RCL× ST Z
33 SQRT
34 ENTER
35 R↓
36 RCL+ ST Y
37 ABS
38 STO 01
39 CLX
40 RCL ST T
41 RCL- ST Y
42 ABS
43 R↓
44 +
45 0.5
46 ×
47 LASTX
48 R↑
49 RCL- 01
50 X>0?
51 GTO 02
52 X≠0?
53 GTO 03
54 CLX
55 RCL ST T
56 RCL÷ ST Y
57 X<>Y
58 R↓
59 COMPLEX
60 X≤0?
61 GTO 03
62▸LBL 02
63 R↑
64 +/-
65 R↓
66▸LBL 03
67 R↓
68 R↓
69 ENTER
70 RCL- ST Z
71 ABS
72 RCL- 00
73 X>0?
74 GTO 00
75 R↓
76 +
77 0.5
78 ×
79 RTN
80▸LBL "AGM"       ; for real number arguments
81 0
82 COMPLEX
83 X<>Y
84 0
85 COMPLEX
86 XEQ "cAGM"
87 COMPLEX
88 X=0?
89 GTO 01
90 COMPLEX
91 RTN
92▸LBL 01
93 R↓
94 RTN
95 .END.

Download : cagm.raw — (raw program file for Free42)

Calculation example of "AGM" and "cAGM"

1 ENTER 2 XEQ "AGM" → 1.45679103105
1 ENTER .25 XEQ "AGM" → 5.60757145072e-1
2 +/- ENTER 2 COMPLEX 1 ENTER 0 COMPLEX XEQ "cAGM" → 1.26420344318e-2 +1.32391495927i
3 +/- ENTER 2 COMPLEX 2 +/- ENTER 2 COMPLEX XEQ "cAGM" → -2.48482321791 +2.01232838848i
1 ENTER 2 COMPLEX 3 ENTER 4 COMPLEX +/- XEQ "cAGM" → -2.05175259566 +7.24586986859e-2i
1.5 ENTER 2 COMPLEX +/- 3 ENTER 4 COMPLEX XEQ "cAGM" → 1.95698167954 -1.45966858375e-1i

Calculation results by WolframAlpha

agm(1, 2)               ~= 1.4567910310469068691864323832650819749738639432213055907941723832...
agm(1, 0.25)            ~= 0.5607571450719006425319505482072666351272432856016524939797059253...
agm(-2 + 2 i, 1)        ~= 0.012642034431775224871144882985098306376405625767985282751139254... 
                         + 1.3239149592728229110180317045523087521571829295502353536455198... i
agm(-3 + 2 i, -2 + 2 i) ~= -2.48482321791345966540888302608455218648014162219542779589616405...
                         + 2.01232838847582894914613621798033481637378492177117070798341113... i
agm(1 + 2 i, -3 - 4 i)  ~= -2.051752595659329572693583864239458955398878779423072275607602...
                          + 0.07245869868588149813029197687616500840652284108968969453100464... i
agm(-1.5 - 2i, 3 + 4 i) ~= 1.95698167953681205998137584021754054093691534541639368664817292...
                         - 0.145966858375103771971664986222696913809981322179346508693130731... i



SEE ALSO

Arithmetic-Geometric Mean — in C99
Complete Elliptic Integral of the First Kind — in C99
Conductor-backed Coplanar Waveguide
Calculation of π by using arithmetic geometric mean


REFERENCE

Complete Elliptic Integral of the First Kind — Wolfram MathWorld
Arithmetic-Geometric Mean — Wolfram MathWorld
Louis. V King, "ON THE DIRECT NUMERICAL CALCULATION OF ELLIPTIC FUNCTIONS AND INTEGRALS", Cambridge at the university press, 1924









© 2000 Takayuki HOSODA.