The AGM is defined as the limit of the interdependent
sequences and :
These two sequences
converge to the same number, the arithmeticâgeometric mean of x and y; it is denoted by M(x, y), or sometimes by agm(x, y) or AGM(x, y).
The arithmeticâgeometric mean can be extended to
complex numbers and when the
branches of the square root are allowed to be taken inconsistently, it is, in general, a
multivalued function.[1]
Example
To find the arithmeticâgeometric mean of a0 = 24 and g0 = 6, iterate as follows:
The first five iterations give the following values:
n
an
gn
0
24
6
1
15
12
2
13.5
13.416 407 864 998 738 178 455 042...
3
13.458 203 932 499 369 089 227 521...
13.458 139 030 990 984 877 207 090...
4
13.458 171 481 745 176 983 217 305...
13.458 171 481 706 053 858 316 334...
5
13.458 171 481 725 615 420 766 820...
13.458 171 481 725 615 420 766 806...
The number of digits in which an and gn agree (underlined) approximately doubles with each iteration. The arithmeticâgeometric mean of 24 and 6 is the common limit of these two sequences, which is approximately 13.4581714817256154207668131569743992430538388544.[2]
History
The first algorithm based on this sequence pair appeared in the works of
Lagrange. Its properties were further analyzed by
Gauss.[1]
Properties
The geometric mean of two positive numbers is
never greater than the arithmetic mean.[3] So (gn) is an increasing sequence, (an) is a decreasing sequence, and gn ⤠M(x, y) ⤠an. These are strict inequalities if x â y.
M(x, y) is thus a number between the geometric and arithmetic mean of x and y; it is also between x and y.
If r ⼠0, then M(rx,ry) = r M(x,y).
There is an integral-form expression for M(x,y):[4]
Since the arithmeticâgeometric process converges so quickly, it provides an efficient way to compute elliptic integrals, which are used, for example, in
elliptic filter design.[5]
that is, the sequence gn is nondecreasing and bounded above by the larger of x and y. By the
monotone convergence theorem, the sequence is convergent, so there exists a g such that:
That is to say that this
quarter period may be efficiently computed through the AGM,
Other applications
Using this property of the AGM along with the ascending transformations of
John Landen,[16]Richard P. Brent[17] suggested the first AGM algorithms for the fast evaluation of elementary
transcendental functions (ex, cos x, sin x). Subsequently, many authors went on to study the use of the AGM algorithms.[18]
^Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience.
ISBN0-471-83138-7. pages 35, 40
^G. V. Choodnovsky: Algebraic independence of constants connected with the functions of analysis, Notices of the AMS 22, 1975, p. A-486
^G. V. Chudnovsky: Contributions to The Theory of Transcendental Numbers, American Mathematical Society, 1984, p. 6
^Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience.
ISBN0-471-83138-7. p. 45
^Newman, D. J. (1985). "A simplified version of the fast algorithms of Brent and Salamin". Mathematics of Computation. 44 (169): 207â210.
doi:
10.2307/2007804.
JSTOR2007804.
The AGM is defined as the limit of the interdependent
sequences and :
These two sequences
converge to the same number, the arithmeticâgeometric mean of x and y; it is denoted by M(x, y), or sometimes by agm(x, y) or AGM(x, y).
The arithmeticâgeometric mean can be extended to
complex numbers and when the
branches of the square root are allowed to be taken inconsistently, it is, in general, a
multivalued function.[1]
Example
To find the arithmeticâgeometric mean of a0 = 24 and g0 = 6, iterate as follows:
The first five iterations give the following values:
n
an
gn
0
24
6
1
15
12
2
13.5
13.416 407 864 998 738 178 455 042...
3
13.458 203 932 499 369 089 227 521...
13.458 139 030 990 984 877 207 090...
4
13.458 171 481 745 176 983 217 305...
13.458 171 481 706 053 858 316 334...
5
13.458 171 481 725 615 420 766 820...
13.458 171 481 725 615 420 766 806...
The number of digits in which an and gn agree (underlined) approximately doubles with each iteration. The arithmeticâgeometric mean of 24 and 6 is the common limit of these two sequences, which is approximately 13.4581714817256154207668131569743992430538388544.[2]
History
The first algorithm based on this sequence pair appeared in the works of
Lagrange. Its properties were further analyzed by
Gauss.[1]
Properties
The geometric mean of two positive numbers is
never greater than the arithmetic mean.[3] So (gn) is an increasing sequence, (an) is a decreasing sequence, and gn ⤠M(x, y) ⤠an. These are strict inequalities if x â y.
M(x, y) is thus a number between the geometric and arithmetic mean of x and y; it is also between x and y.
If r ⼠0, then M(rx,ry) = r M(x,y).
There is an integral-form expression for M(x,y):[4]
Since the arithmeticâgeometric process converges so quickly, it provides an efficient way to compute elliptic integrals, which are used, for example, in
elliptic filter design.[5]
that is, the sequence gn is nondecreasing and bounded above by the larger of x and y. By the
monotone convergence theorem, the sequence is convergent, so there exists a g such that:
That is to say that this
quarter period may be efficiently computed through the AGM,
Other applications
Using this property of the AGM along with the ascending transformations of
John Landen,[16]Richard P. Brent[17] suggested the first AGM algorithms for the fast evaluation of elementary
transcendental functions (ex, cos x, sin x). Subsequently, many authors went on to study the use of the AGM algorithms.[18]
^Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience.
ISBN0-471-83138-7. pages 35, 40
^G. V. Choodnovsky: Algebraic independence of constants connected with the functions of analysis, Notices of the AMS 22, 1975, p. A-486
^G. V. Chudnovsky: Contributions to The Theory of Transcendental Numbers, American Mathematical Society, 1984, p. 6
^Borwein, Jonathan M.; Borwein, Peter B. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (First ed.). Wiley-Interscience.
ISBN0-471-83138-7. p. 45
^Newman, D. J. (1985). "A simplified version of the fast algorithms of Brent and Salamin". Mathematics of Computation. 44 (169): 207â210.
doi:
10.2307/2007804.
JSTOR2007804.