Elliptic Curves over C - Elliptic Curves: Number Theory and Cryptography - page 289

Therefore, since a n− 1 ≥ b n− 1 , it follows immediately from (9.20) that

b n ≥ b n− 1 b n− 1 = b n− 1 .

1
2 ( a n− 1 + a n− 1 )= a n− 1

a n ≤ and

Also,

2 √ a n− 1 − b n− 1 2

1

a n − b n =

√ a n− 1 − b n− 1 √ a n− 1 + b n− 1

1
2

≤

1
2 ( a n− 1 −

= b n− 1 ) .

Therefore, a n − b n ≤ (1 / 2) n ( a − b ), so a n − b n → 0. Since the a n 's are
a decreasing sequence bounded below by the increasing sequence of b n 's, it
follows immediately that the two seque nces conv erge to the same limit, so

M ( a, b ) exists. If b n− 1 ≥ 1, then √ a n− 1 + b n− 1 ≥ 2, so

16 √ a n− 1 − b n− 1 2

a n − b n
8

1

=

b n− 1 2 √ a n − 1 + b n − 1 2
4 16 √ a n− 1 −

1

≤

= a n − 1 −

2

b n− 1

.

8

Inequality 9.22 follows easily by induction.

The limit M ( a, b ) is called the arithmetic-geometric mean of a and b .
Since

M ( ca, cb )= cM ( a, b ) ,

we can always rescale a and b to make b 1. Also, since M ( b, a )= M ( a, b )

(because a 1 and b 1 are symmetric in a, b ), we may always arrange that a

≥

≥ b .

By Inequality (9.21), a n − b n < 1 for su ciently large n .Thenumbers a n + m

and b n + m give approximations to M ( a, b ). Inequality (9.22) predicts that
the number of decimal places of accuracy doubles with each iteration. This
phenomenon occurs in the above example.
The reasons we are interested in the arithmetic-geometric mean are the
following two propositions.

PROPOSITION 9.24
Let a, b be positive realnum bers. D efine
I ( a, b )= π/ 2
0

dθ
a 2 cos 2 θ + b 2 sin 2 θ

.