Ramanujan's Arithmetic Geometric Mean Continued Fractions and Dynamics Dalhousie Colloquium

78 Pages·2003·0.85 MB·English

by  Borwein J.M.

Ramanujan’s Arithmetic-Geometric Mean
Continued Fractions and Dynamics
Jonathan M. Borwein, FRSC
Prepared for
Dalhousie Colloquium
Spring 2003
Research Chair, Faculty of Computer Science
Dalhousie University, Halifax, NS Canada
www.cecm.sfu.ca/~jborwein/talks.html
www.cs.dal.ca/~jborwein
Revised: February 4, 2003
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Srinivasa Ramanujan (1887–1920)
• G. N. Watson (1886–1965), on reading Ra-
manujan’s work, describes:
a thrill which is indistinguishable from
the thrill I feel when I enter the Sagres-
tia Nuovo of the Capella Medici and
see before me the austere beauty of the
four statues representing ‘Day,’ ‘Night,’
‘Evening,’ and ‘Dawn’ which Michelan-
gelo has set over the tomb of Guiliano
de’Medici and Lorenzo de’Medici.
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1. Abstract
The Ramanujan AGM continued fraction
a
R (a, b) =
η
2
b
η +
2
4a
η +
2
9b
η +
η + ...
enjoys attractive algebraic properties such as a
striking arithmetic-geometric mean relation
& elegant links with elliptic-function theory.
• The fraction presents a computational
challenge, which we could not resist.
• Joint work with Richard Crandall and vari-
ously D. Borwein, G. Fee, R. Luke and R.
Mayer.
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♣ Much of this work is to appear in Experi-
mental Mathematics [CoLab Preprints 27,
29 and 253.]
1. In Part I: we show how to rapidly eval-
uate R for any positive reals a, b, η. The
problematic case being a ≈ b—then subtle
transformations allow rapid evaluation.
• On route we find, e.g., that for rational
a = b, R is an L-series with a ’closed-
η
form.’
• We ultimately exhibit an algorithm yield-
∗
ing D digits of R in O(D) iterations.
2. Finally, in Part II of this talk, we address
the harder theoretical and computational
dilemmas arising when parameters are al-
lowed to be complex.
∗The big-O constant is independent of the positive-real
triple a,b,η.
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2. Preliminaries
PART I. Entry 12 of Chapter 18 of Ramanu-
jan’s Second Notebook [BeIII] gives the beau-
tiful:
a
R (a, b) = (1.1)
η
2
b
η +
2
4a
η +
2
9b
η +
η + ...
which we interpret—in most of the present
treatment—for real, positive a, b, η > 0.
Remarkably, for such parameters, R satisfies
an AGM relation
(cid:181) (cid:182)
√
a + b R (a, b) + R (b, a)
η η
R , ab = (1.2)
η
2 2
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1. (1.2) is one of many relations we develop
for computation of R .
η
2. “The hard cases occur when b is near
to a,” including the case a = b.
3. We eventually exhibit an algorithm uniformly
of geometric/linear convergence across the
positive quadrant a, b > 0.
4. Along the way, we find attractive identities,
such as that for R (r, r), with r rational.
η
5. Finally, we consider complex a, b—obtaining
theorems and conjectures on the domain of
validity for the AGM relation (1.2).
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Research started in earnest
when we noted R (1, 1)
1
‘seemed close to’ log 2.
Such is the value of exper-
iment: one can be led into
deep waters.
As discussed in Chapter One
of Experimentation in Math-
ematics.
• A useful simplification is
R (a, b) = R (a/η, b/η),
η 1
as can be seen by ‘cancellation’ of the η
elements down the fraction.
Such manipulations are valid because the
continued converges.
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To prove convergence we put a/R in RCF
1
(reduced continued fraction) form:
a
R (a, b) = (2.1)
1
[A ; A , A , A , . . . ]
0 1 2 3
a
:=
1
A +
0
1
A +
1
1
A +
2
A3 + ...
where the A are all positive real.
i
a
It is here that both the
profundity and limitations
of Ramanujan’s knowledge
stand out most sharply.
(G.H. Hardy)
a
Ramanujan’s work on elliptic
and modular functions.
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• Inspection of Ramanujan’s pattern in R
yields the RCF elements explicitly and gives
the asymptotics of A :
n
For even n
2 n n
n! b 2 b
−n
A = 4 ∼ .
n
(n/2)!4 an π n an
For odd n
4 n−1 n
((n − 1)/2!) a π a
n−1
A = 4 ∼ .
n
n!2 bn+1 2 ab n bn
• This representation leads immediately to:
Theorem 2.1: For any positive real pair a, b
the fraction R (a, b) converges.
1
(cid:80)
Proof: An RCF converges iff A diverges.
i
(This is the Seidel–Stern theorem [Kh,LW].)
In our case, such divergence is evident for every
choice of real a, b > 0. (cid:176)c
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(cid:80)
• Note for a = b, divergence of A is only
i
logarithmic —a true indication of slow con-
vergence (we wax more quantitatively later).
• Indeed, our interest in computational as-
pects started with asking how, for positive
a, to (rapidly) evaluate
R(a) := R (a, a)
1
and thence to prove suspected identities.
• We shall later encounter a different con-
tinued fraction for R(a), as well as other
computationally efficient constructs.
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