An approximation to the arithmetic-geometric mean. G.J.O. Jameson, Math. Gazette 98 (2014), 85 95

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1 An pproximtion to the rithmetic-geometric men G.J.O. Jmeson, Mth. Gzette 98 (4), Given positive numbers > b, consider the itertion given by =, b = b nd n+ = ( n + b n ), b n+ = ( n b n ) /. At ech stge, the two new numbers re the rithmetic nd geometric mens of the previous two. It is esily seen tht b n < n, ( n ) is decresing, (b n ) is incresing nd n+ b n+ < ( n b n ), nd hence tht ( n ) nd (b n ) converge to common limit, which is clled the rithmetic-geometric men of nd b. We will denote it by M(, b). Furthermore, the convergence is qudrtic, hence very rpid. We illustrte this by the clcultion of M(, ): n b n n Of course, 5 nd b 5 re not ctully equl, but more deciml plces would be required to show the difference; in fct, 5 b 5 < 9. A fmous result of Guss, dting from 799, reltes M(, b) to the integrl I(, b) = π/ dx = (x + ) / (x + b ) / ( cos θ + b sin dθ. () θ) / The equivlence of the two integrls is seen by substituting x = b tn θ. Guss s theorem sttes tht I(, b) = π M(, b). () This result hs been discussed in number of Gzette rticles, e.g. [], [], [3]. The min step of the proof is to show, in the bove nottion, tht I(, b ) = I(, b). It then follows tht I(, b) = I( n, b n ) for ll n. By the first integrl in (), I( n, b n ) > dx = x + n π n, nd similrly I( n, b n ) < π/(b n ); () now follows. One wy to prove tht I(, b ) =

2 I(, b) is by the substitution x = t b/t. This ingenious substitution seems to hve been introduced by D.J. Newmn in [4]; more ccessible ccounts of it cn be seen in [] nd [5]. In [3], Nick Lord presents n interesting new method bsed on the re in polr coordintes. Depending on one s strting point, Guss s theorem cn be regrded either s the evlution of the integrl I(, b) (including quick wy to clculte it in specific cses), or s closed expression for M(, b). Clerly, M(c, cb) = cm(, b) nd I(c, cb) = I(, b) for c >, so it is enough to stte c results in terms of M(, ) nd I(, ). The topic of this rticle is the fct tht there re plesntly simple formule giving good pproximtions to I nd M, s follows: for lrge nd smll (positive) b, I(, ) log 4, I(, b) log 4 b, M(, ) F (), where F () = π log 4. The first two sttements re equivlent becuse I(, ) = I(, ), nd the sttement for M(, ) follows by Guss s theorem. The pproximtion is strikingly ccurte for modertely lrge : for exmple, F () = 6.7 to four d.p.: compre the vlue for M(, ) found bove. These pproximtions hve been known for long time: for exmple, version ppers in [6, p. 5], published in 97. A vriety of methods hve been used, with differing estimtes of the ccurcy of the pproximtion. A very effective method ws presented by Newmn in [5]. However, it is sketched rther briefly there, nd the sttement is given only in the form I(, ) = log 4 + O(/ ). Here we develop Newmn s method in more detil to estblish the following more specific estimtion (which is rther more precise thn ny version I hve seen in the existing literture): THEOREM. For ( + 4 b ) log 4 b 7 6 b, (5)

3 ( I(, ) > + ) log (6) The pirs of sttements re equivlent, becuse of the identity I(, ) = I(, ). Sttements (5) nd (6) my look rther technicl, but we include them becuse they emerge from the sme process of resoning, nd becuse, together with (3) nd (4), they show tht ( I(, ) = + ) log 4 + O(/ ), 4 nd similr sttement for b (so in fct the estimtion log 4 + O(/ ) in [5] is not quite correct). At the sme time, we hve gined the plesntly simple inequlity I(, ) > log 4. By Guss s theorem nd the inequlity /( + x) > x, pplied with x = /(4 ), sttement (4) trnsltes into the following pir of inequlities for M(, ): COROLLARY. Let F () be s bove. For, ( ) F () < M(, ) < F (). (7) 4 This grees well with the ccurcy seen bove for =. Since M(, b) = b M(/b, ) we cn derive the following bounds for M(, b) (where > b) in generl: ) π ( b < M(, b) < π log(4/b) 4 log(4/b). (8) The proofs cn be set out in terms of either or b: we shll use b. The method requires nothing more thn some elementry integrls nd the binomil nd logrithmic series. We will lso see tht weker version of Theorem cn be estblished very quickly. At this point, we mke two preliminry observtions bout I(, b). inequlity of the mens, we hve (x + ) / (x + b ) / (x + ) + (x + b ), Firstly, by the so tht I(, b) π 4 ( + ). (9) b Secondly, I(, b) equtes to stndrd elliptic integrl in the following wy. Let b = ( b ) /, so tht b + b =. Then cos θ + b sin θ = b sin θ, so tht I(, b) = π/ ( b sin dθ. θ) / 3

4 In the lnguge of elliptic integrls, this is complete elliptic integrl of the first kind, denoted by K(b ). However, we do not ssume ny prior knowledge of elliptic intgrls. We now embrk on the proof. Tking =, write H(x) =. () (x + ) / (x + b ) / The strting point is the observtion tht the substitution x = b/y gives so tht b / H(x) dx = b / H(y) dy, b / I(, b) = H(x) dx. () Now ( + y) / > y for < y <, s is esily seen by multiplying out ( + y)( y), so for < x <, ( x ) (x + b ) < H(x) <. () / (x + b ) / Now observe tht the substitution x = by gives b / /b / dx = (x + b ) / (y + ) dy = / sinh b. / LEMMA. We hve log b < / sinh < log b/ b + b. (3) / 4 Proof. Recll tht sinh x = log[x + (x + ) / ], which is clerly greter thn log x. Also, since ( + b) / < + b, ( ) / b + / b + = b [ + ( + / b)/ ] < b ( + b) = / b ( + b). / 4 The right-hnd inequlity in (3) now follows from the fct tht log( + x) x. This is lredy enough to prove the following interim version of Theorem : THEOREM A. For < b nd, log 4 b b < I(, b) < log 4 b + b, (4) log 4 < I(, ) < log 4 +. (5) 4

5 where Proof. The two sttements re equivlent, since I(, ) = I(, ). By () nd (), Since (x + b ) / > x, we hve sinh b / R (b) < I(, b) < sinh b /, R (b) = b / x dx. (6) (x + b ) / R (b) < b / x b / x dx = x dx = b. Both inequlities in (4) now follow from (3), since log b / = log 4 b. Theorem A is sufficient to show tht M(, ) F () s. To see this, write I(, ) = log 4 + r(). Then I(, ) which tends to s, since r(). log 4 = r() log 4[log 4 + r()], Some reders my be content with Theorem A, but for those with the ppetite for it, we now refine the method to estblish the full strength of Theorem. All we need to do is improve our estimtions by the insertion of further terms. The upper estimte in () only used ( + y) / <. Insted, we now use the stronger bound ( + y) / < y y for < y <. These re the first three terms of the binomil expnsion, nd the stted inequlity holds becuse the terms of the expnsion lternte in sign nd decrese in mgnitude. So we hve insted of (): ( x ) (x + b ) < H(x) < ( / x x4 ). (7) (x + b ) / We lso need further degree of ccurcy in the estimtes for sinh b / LEMMA. For < b, we hve log b + b 3 / 4 3 b < sinh < log b/ b + b / 4 Proof. Agin by the binomil series, we hve 6 b + nd R (b). 3 b3. (8) + b 8 b < ( + b) / < + b 8 b + 6 b3. (9) 5

6 So, s in Lemm, ( ) / b + / b + b ( + b / 8 b ) = ( + B), b/ where B = 4 b 6 b (so B < 4 b). By the log series, log( + x) > x x for < x <, so log( + B) > B B > 4 b 6 b 3 b = 4 b 3 3 b. LEMMA 3. For < b, we hve: R (b) b + 4 b b log b / 3 6 b3, () R (b) < b 5 b. () Proof: We cn evlute R (b) explicitly by the substitution x = b sinh t in (6): R (b) = c(b) c(b) b sinh t dt = b (cosh t ) dt = 4 b sinh c(b) b c(b), where c(b) = sinh (/b / ). Now so sinh c(b) = sinh c(b) cosh c(b) = ( ) / b / b + = b ( + b)/, R (b) = b( + b)/ b c(b). Sttement () now follows from ( + b) / < + b nd c(b) > log. Sttement () b / follows from the left-hnd inequlity in (9) nd the right-hnd one in (3). Unfortuntely, () does not quite follow from (). We prove it directly from the integrl, s follows. Write x /(x + b ) / = g(x). Since g(x) < x, we hve b / g(x) dx < b (b b ). For < x < b, we hve by the binomil expnsion gin ) x x g(x) = ( x b( + x /b ) / b b + 3x4, 8b 4 nd hence b g(x) dx < ( )b < 3 b. 6

7 Together, these estimtes give R (b) sinh b / R (b) > log 4 b + b 3 6 b ( b 5 b ) > log 4 b, (with spre term 8 b ). Of course, the key fct is the cncelltion of the term b. Also, using () insted of (), we hve I(, b) > log 4 b + b 3 6 b ( b + 4 b 4 b log 4 b ) = ( + 4 b ) log 4 b 7 6 b. where Upper bound: By (7), So by (8) nd (), R (b) = I(, b) < sinh b / R (b) R (b), b / x 4 b / dx < x 3 dx = (x + b ) / 4 b. I(, b) < log 4 b + b 8 b + 6 b3 ( b + 4 b 4 b log 4 b 3 6 b3 ) b = ( + 4 b ) log 4 b 3 6 b + 4 b3. If b < 3 4, then 3 6 b 4 b3, so I(, b) < ( + 4 b ) log 4 b. For b 3 4, we reson s follows. By (9), I(, b) π 4 (+ b ). Write h(b) = (+ 4 b ) log 4 b π ( + ). We find tht h( 3 ) >. One verifies by differentition tht h(b) 4 b 4 is incresing for for 3 b. 4 Note: We mention very briefly n older, perhps better known, pproch (cf. [7, p. ]). Use the identity I(, b) = K(b ), nd note tht b when b. Define A(b ) = π/ b sin θ ( b sin dθ θ) / nd B(b ) = K(b ) A(b ). One shows by direct integrtion tht A(b ) = log[(+b )/b] nd B() = log, so tht when b, we hve A(b ) log b nd hence K(b ) log 4 b. 7

8 This method cn be developed to give n estimtion with error term of the form O(b log b ), [8, p ]. The uthor hs refined it little further, but the detils re distinctly more lborious thn the method we hve given here, nd the end result still does not chieve the ccurcy of our Theorem. Appliction to the clcultion of logrithms nd π. In my view, Theorem is interesting enough in its own right. However, it lso hs plesing pplictions to the clcultion of logrithms (ssuming π known) nd π (ssuming logrithms known), exploiting the rpid convergence of the gm itertion. Consider first the problem of clculting log x (where x > ). Choose n, in wy to be discussed below, nd let 4 = x n, so tht n log x = log 4, which is pproximted by I(, ). We clculte M = M(, ), nd hence I(, ) = π/(m). Then log x is pproximted by (/n)i(, ). How ccurte is this pproximtion? By Theorem, log 4 = I(, ) r(), where < r() < 4 log 4. Hence in which log x = r() I(, ) n n, < r() n < log x 4 = 4 log x x n. We choose n so tht this is within the desired degree of ccurcy. It might seem nomlous tht log x, which we re trying to clculte, ppers in this error estimtion, but rough estimte for it is lwys redily vilble. For illustrtion, we pply this to the clcultion of log, tking n =, so tht = = 4. We find tht M = M(4, ) , so our pproximtion is π 4 M , while in fct log = to seven d.p. The discussion bove (just tking log < ) shows tht the pproximtion overestimtes log, with error no more thn 4/ 4 = / < / 6. A vrition (cf. [9, p. ]) is s follows: choose lrge (e.g. = n for suitble n). Then I(, ) log 4 nd xi(x, ) log 4x, so log x xi(x, ) I(, ). Clerly, this method requires two gm itertions. The error is r() r(x), nd we hve gin < r() < r (), where r () = log 4. Now log y/y is decresing for y > e /, so 4 r (x) < r (), hence the mgnitude of the error is no more thn r () itself; this cn be compred with the error 4 log x in the previous method. We now turn to the question of pproximting π. The simple-minded method is s 8

9 follows. Choose suitbly lrge. By Guss s theorem nd Theorem, where π = M(, )I(, ) = M(, )log 4 + s(), s() = M(, ) r() 4 < M(, )log 4. 3 So n pproximtion to π is M(, ) log 4, with error no more thn s(). Using our originl exmple with =, this gives to five d.p., with the error estimted s no more thn 8 5 (the ctul error is bout ). For better pproximtion, one would repet with lrger. However, there is wy to generte sequence of incresingly close pproximtions from single gm itertion, which we now describe (cf. [8, Proposition 3]). In the originl gm itertion, let c n = ( n b n) /. Note tht c n converges rpidly to. By Guss s theorem, π = M(, c )I(, c ). To pply Theorem, we need to equte I(, c ) to n expression of the form I(d n, e n ), where e n /d n tends to. This is chieved s follows. We hve so c n+ = ( n b n ), nd hence c n+ = n+ b n+ = 4 ( n + b n ) n b n = 4 ( n b n ), n = n+ + c n+, b n = n+ c n+ (this shows how to derive n nd b n from n+ nd b n+, reversing the gm itertion). Hence c n = n b n = 4 n+ c n+, so the rithmetic men of n+ nd c n+ is n, while the geometric men is c n. Therefore I( n, c n ) = I( n+, c n+ ), nd hence I(, c ) = I( n n, n c n ) = ( I, c ) n. n n n Since c n / n tends to, Theorem (or even Theorem A) pplies to show tht I(, c ) = lim log 4 n. n n n c n Now tke = nd b =. Then c = b, so M(, b ) = M(, c ), nd ( n ) converges to this vlue. Since π = M(, c )I(, c ), we hve estblished the following: THEOREM. With n, b n, c n defined in this wy, we hve π = lim n log 4 n. (3) n c n 9

10 Denote this pproximtion by p n. With enough determintion, one could derive n error estimtion, but we will just illustrte the speed of convergence by performing the first three steps. Vlues re rounded to six significnt figures. To clculte c n, we use c n = c n /(4 n ) rther thn ( n b n ), since this would require much greter ccurcy in the vlues of n nd b n. n n b n c n p n More ccurte clcultion gives the vlue p 3 = , showing tht it ctully grees with π to eight deciml plces. While this itertion does indeed converge rpidly to π, it hs the disdvntge tht logrithms must be clculted, or ssumed known. This is overcome by the Guss-Brent- Slmin lgorithm, which lso uses the gm itertion, but depends on identities involving integrls of the kind π/ sin θ ( cos θ + b sin dθ θ) / insted of Theorem. A user-friendly ccount of this lgorithm, with cre tken to use the most elementry methods vilble, is given in []. Series expressions for I(, b). Recll tht K(b) = π/ ( b sin θ) / dθ nd I(, b) = K(b ). By integrting the binomil expnsion, one finds tht K(b) = π n= d nb n for < b <, where d = nd d n =.3... (n ).4... (n). By more elborte methods, identifying nd solving second-order differentil eqution stisfied by I(, b), one cn estblish the following identity [9, p. 9]: I(, b) = K(b ) = π K(b) log 4 b d n e n b n, (4) where e n = n r= ( ). Together with the stted series for K(b), this gives r r I(, b) = log 4 b + d n (log 4 ) b e n b n. (5) n= Let us compre the informtion delivered by these series with our Theorem. Write I(, b) = log 4 + r b (b). The first term of the series in (5) is 4 b (log 4 ), so we cn stte b ( r (b) = 4 b log 4 ) ( b + O b 4 log ), (6) b n=

11 which is little more ccurte thn our estimte. For the gm, this trnsltes into M(, ) = F () F () log 4 + O(/ 3 ) s, 4 log 4 An ctul lower bound cn be derived s follows. We hve e n < log for ll n, since (e n ) is incresing with limit log. Hence log 4 b e n >, so from (5), r (b) > 4 b ( log 4 b ), (7) which is slightly stronger thn (5). However, some quite delicte clcultions re needed to recpture our upper bound in (3). Acknowledgement: I m grteful to Nick Lord for his constnt interest nd dvice while this rticle evolved through severl preliminry versions, nd to Mike Hirschhorn for some useful suggestions. References. Nick Lord, Recent clcultions of π: the Guss-Slmin lgorithm, Mth. Gz. 76 (99), Robert M. Young, On the re enclosed by the curve x 4 + y 4 =, Mth. Gz. 93 (9), Nick Lord, Evluting integrls using polr res, Mth. Gz. 96 (), D.J. Newmn, Rtionl pproximtion versus fst computer methods, in Lectures on pproximtion nd vlue distribution, pp , Sém. Mth. Sup. 79 (Presses Univ. Montrél, 98). 5. D.J. Newmn, A simplified version of the fst lgorithm of Brent nd Slmin, Mth. Comp. 44 (985), 7, reprinted in Pi: A Source Book (Springer, 999), E.T. Whittker nd G.N. Wtson, A Course of Modern Anlysis (Cmbridge Univ. Press, 97). 7. F. Bowmn, Introduction to Elliptic Functions, (Dover Publictions, 96). 8. J.M. Borwein nd P.B. Borwein, The rithmetic-geometric men nd fst computtion of elementry functions, SIAM Review 6 (984), , reprinted in Pi: A Source Book (Springer, 999), J.M. Borwein nd P.B. Borwein, Pi nd the AGM, (Wiley, 987). G.J.O. JAMESON Dept. of Mthemtics nd Sttistics, Lncster University, Lncster LA 4YF emil: g.jmeson@lncster.c.uk

Euler, Ioachimescu and the trapezium rule. G.J.O. Jameson (Math. Gazette 96 (2012), )

Euler, Ioachimescu and the trapezium rule. G.J.O. Jameson (Math. Gazette 96 (2012), ) Euler, Iochimescu nd the trpezium rule G.J.O. Jmeson (Mth. Gzette 96 (0), 36 4) The following results were estblished in recent Gzette rticle [, Theorems, 3, 4]. Given > 0 nd 0 < s