arithmetic-geometric mean

arithmetic-geometric mean

If

x

and

y

are non-negative

real numbers

, we can form their

arithmetic
mean

a

0

=

(

x

+

y

)

/

2

as well as their

geometric mean

g

0

=

x

⁢

y

.
This procedure can be repeated to form a sequence of

arithmetic

and
geometic means

a

n

+

1

=

(

a

n

+

g

n

)

/

2

and

g

n

+

1

=

a

n

⁢

g

n

.
By the

arithmetic-geometric means inequality

we have

a

n

≥

a

n

+

1

≥

g

n

+

1

≥

g

n

(with equality holding only when

a

n

=

g

n

),
hence these sequences converge to a number between

x

and

y

,
with the rate of convergence being superlinear.
The

arithmetic-geometric mean

M

⁢

(

x

,

y

)

of

x

and

y

is defined as this limit

M

⁢

(

x

,

y

)

=

lim

n

→

∞

⁡

a

n

,

g

n

.

The origin of the name is

obvious

from the construction.

Alternative notations

for

M

⁢

(

x

,

y

)

are

agm

⁡

(

x

,

y

)

or

AGM

⁡

(

x

,

y

)

.

The AGM lies between the arithmetic and geometric
means of

x

and

y

,

x

+

y

2

≥

M

⁢

(

x

,

y

)

≥

x

⁢

y

,

with equality holding only in case of equality

x

=

y

. The AGM is also a

homogeneous function of degree

1

, namely

M

⁢

(

α

⁢

x

,

α

⁢

y

)

=

α

⁢

M

⁢

(

x

,

y

)

for

α

>

0

. It is also

symmetric

M

⁢

(

x

,

y

)

=

M

⁢

(

y

,

x

)

.
These

properties

are obvious from the construction.

The AGM can be used to numerically evaluate

elliptic integrals

of the
first and

second kinds

. For example,

M

⁢

(

x

,

y

)

=

π

4

⁢

x

+

y

K

⁢

(

|

x

-

y

|

x

+

y

)

,

(1)

where

K

⁢

(

k

)

is the elliptic integral of the

first kind

as

function

of
the

modulus

k

.

As a numerical method, the arithmetic-geometric mean has much to recommend it.
By its nature, it automatically provides upper and lower bounds for the
answer, so one does not have to separately estimate error. To compute
the arithmetic-geometric mean to a certain accuracy, we only need to carry
out the computation until the

difference

between

a

n

and

g

n

is smaller
than the desired accuracy.

Because convergence is superlinear, only a few

iterations

are necessarry to
obtain the answer. For instance, if we compute

M

⁢

(

1

,

k

)

with

k

less than
a billion, we already obtain at least fifteen-place accuracy after eight
iterations, as the following computation of

M

⁢

(

1

,

123456789

)

shows:

n

g

n

a

n

0

1.0

123456789.0

1

11111.111060555555

61728395.0

2

828173.3227017411

30869753.055530276

3

5056234.365511624

15848963.189116009

4

8951875.352937901

10452598.777313817

5

9673177.418448625

9702237.06512586

6

9687696.345716598

9687707.241787244

7

9687701.793750389

9687701.793751922

8

9687701.793751154

9687701.793751154

The fact that relatively few iterations are necessarry to obtain a highly
accurate result also means that one does not have to worry much about the
cumulative effect of roundoff errors in the various steps of the computation.

Title

arithmetic-geometric mean

Canonical

name

ArithmeticgeometricMean

Date of creation

2013-03-22 14:23:46

Last modified on

2013-03-22 14:23:46

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

7

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 26E60

Classification

msc 33E05

Synonym

agm

Synonym

AGM

Related topic

EllipticIntegralsAndJacobiEllipticFunctions

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