UCLA Department of Statistics Papers

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1 UCLA Department of Statistics Papers Title Rate of Convergence of the Arithmetic-Geometric Mean Process Permalink Author de Leeuw, Jan Publication Date escholarship.org Powered by the California Digital Library University of California

2 RATE OF CONVERGENCE OF THE ARITHMETIC-GEOMETRIC MEAN PROCESS JAN DE LEEUW Abstract. This (didactic) note gives a simple counter-example to the notion that Picard iterations converge super-linearly if and only if the sup-norm of the Jacobian at the solution is equal to zero and sub-linearly if and only if it is equal to one.. Introduction Suppose S is an open subset of R n and Γ : S S is a differentiable map. Assume the Picard iterations x (k+) = Γ (x (k) ) starting from some x (0) S converge to x S. We can derive information about the rate of convergence from the sup-norm (the eigenvalue of maximum modulus) of the derivative DΓ (x). If DΓ (x) = λ < we have linear convergence with rate λ, and if DΓ (x) = 0 we have super-linear convergence [Ortega and Rheinboldt, 970, Chapter 0]. DΓ (x) = often indicates sub-linear convergence. Our elementary example below, however, has DΓ (x) = and quadratic convergence.. The Arithmetic-Geometric Mean Suppose a and b are two positive numbers. Their arithmetic mean is defined as AM(a, b) = (a + b) and their geometric mean as GM(a, b) = ab. Result. AM(a, b) GM(a, b) with equality if and only if a = b. Proof. 0 ( a b) = (AM(a, b) GM(a, b)). Date: October 3, 008 8h 4min Typeset in Lucida Bright.

3 JAN DE LEEUW From now on suppose, without loss of generality, that a > b. Let a 0 = a and b 0 = b and define the sequences (a) (b) a n = AM(a n, b n ), b n = GM(a n, b n ). Result. a n > b n Proof. From Result. Result 3. {a n } is a decreasing sequence, which is bounded below, and thus converges to some a. {b n } is an increasing sequence, which is bounded above, and thus converges to some b. Proof. a n < max(a n, b n ) = a n and b n > min(a n, b n ) = b n. Moreover a n > b n > b and b n < a n < a. Result 4. a = b. Proof. Take limits on both sides of (). This gives a = AM(a, b ), b = GM(a, b ). Both equations imply a = b. The common limit a = b is called the arithmetic-geometric mean of a and b, written as AGM(a, b). The arithmetic-geometric mean was studied by Legendre and Gauss, and it has fascinating applications in many areas of mathematics and numerical analysis. There are excellent reviews of these applications in Carlson [97], Cox [984], and Almqvist and Berndt [988]. Result 5. b < GM(a, b) < AGM(a, b) < AM(a, b) < a Proof. AGM(a, b) = a < a = AM(a, b) < a 0 = a and AGM(a, b) = b > b = GM(a, b) > b 0 = b.

4 ARITHMETIC-GEOMETRIC MEAN 3 For another proof of the convergence to a common limit we define the sequence = a n b n. It should be noted that is a reasonable way to measure distance to the solution, since a n AGM(a, b) + b n AGM(a, b) = a n AGM(a, b) + AGM(a, b) b n =. Result 6. { } is a decreasing sequence bounded below by zero, and thus converges to some δ 0. Proof. Since a n < a n and b n > b n we have = a n b n < a n b n =. Moreover > 0 for all n. Result 7. δ = 0 Proof. (a) (b) = AM(a n, b n ) GM(a n, b n ) = ( a n b n ), = a n b n = ( a n b n )( a n + b n ), and thus < It follows that 0 < < ( )n δ 0 and thus lim n = 0. The proof shows that convergence of { } is faster than that of a geometric sequence with radius. But we can be more precise. Result 8. Convergence of the sequence { } to zero is superlinear, i.e. Proof. From Equations () lim n = 0. = an b n an + 0. b n In fact, we can be even more precise. Result 9. Convergence of the sequence { } to zero is quadratic. lim n δ n = 8 AGM(a, b).

5 4 JAN DE LEEUW Proof. From Equations () δ n = ( a n + b n ) 8 AGM(a, b). In a sense, the sequences {a n } and {b n } converge equally fast. Result 0. (a n a n ) (b n b n ), i.e. a n a n lim =. n b n b n Proof. and thus a n a n = ( a n b n b n = b n ( a n b n )( a n + b n ). a n a n = an + b n. b n b n bn b n ), 3. Counterexample Equation () defines a mapping Γ : R R. The derivative of this mapping is and thus DΓ (a, b) = b a a b, DΓ (AGM(a, b), AGM(a, b)) = which has eigenvalues one and zero. The fact that DΓ (AGM(a, b), AGM(a, b)) = seems to suggest sublinear convergence, while in fact we know convergence is quadratic. If γ n is the two-element vector with elements a n a n and b n b n, normalized to length one, then Result 0 shows that γ n converges to a vector with elements and +. This eigenvector corresponds with the smallest eigenvalue of the Jacobian at the solution, and that smallest eigenvalue is equal to zero.,

6 ARITHMETIC-GEOMETRIC MEAN 5 References G. Almqvist and B. Berndt. Gauss, Landen, Ramanujan, the Arithmetic- Geometric Mean, Ellipses, π, and the Ladies Diary. The American Mathematical Monthly, 95: , 988. B.G. Carlson. Algorithms Involving Arithmetic and Geometric Means. The American Mathematical Monthly, 78: , 97. D.A. Cox. The Arithmetic-Geometric Mean of Gauss. L Enseignement Mathématique, 30:75 330, 984. J. M. Ortega and W. C. Rheinboldt. Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York, N.Y., 970.

7 6 JAN DE LEEUW Appendix A. Code agm<-function(a,b,eps=e-8,itmax=000,verbose=true) { 3 xold<-max(a,b); yold<-min(a,b); dold<-xold-yold; itel<- 4 repeat { 5 xnew<-(xold+yold)/; ynew<-sqrt(xold*yold) 6 dnew<-xnew-ynew; rat<-dnew/dold; rat<-dnew/(dold^) 7 if (verbose) cat( 8 "Iteration: ",formatc(itel,width=3, format="d"), 9 "old: ",formatc(c(xold,yold,dold),digits=8, 0 width=,format="f"), "old: ",formatc(c(xnew,ynew,dnew),digits=8, width=,format="f"), 3 "rat: ",formatc(c(rat,rat),digits=8, 4 width=,format="f"), 5 "\n") 6 if ((dnew < eps) (itel == itmax)) 7 return(c(xnew,ynew)) 8 xold<-xnew; yold<-ynew; dold<-dnew; itel<-itel+ 9 } 0 } Department of Statistics, University of California, Los Angeles, CA address, Jan de Leeuw: deleeuw@stat.ucla.edu URL, Jan de Leeuw: