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1 Exerise 7 Suppose that t 0 0andthat os t sin t At sin t os t Compute Bt t As ds,andshowthata and B ommute 0 Exerise 8 Suppose A is the oeffiient matrix of the ompanion equation Y AY assoiated with the nth order differential equation That is, + p 1 t 1 + p 2 t p ty + p n ty A p n p p n 2 p 1 Compute Bt t As ds,andshowthata and B ommute if and only if all the 0 oeffiient funtions p i t, i 1, 2,,n, are onstants Referenes 1 J J Duistermaat and J A C Kolk, Lie Groups, Springer, M Hirsh and S Smale, Differential Equations, Dynamial Systems, and Linear Algebra, Aademi Press, J Hubbard and B West, Differential Equations, SpringerVerlag, J H Liu, A remark on the hain rule for exponential matrix funtions, College Math J, Extending Theon s Ladder to Any Square Root Shaun Giberson and Thomas J Osler Rowan University, Glassboro, NJ Introdution Little is known of the life of Theon of Smyrna ira 140 AD At this time in the history of mathematis, there was a tendeny to deemphasize demonstrative and dedutive methods in favor of pratial mathematis An exellent example of this is known as Theon s ladder, whih desribes a remarkably simple way to alulate rational approximations to 2 See [2], [3], and [5] THE MATHEMATICAL ASSOCIATION OF AMERICA
2 Eah rung of the ladder ontains two numbers Call the left number on the nth rung and the right number We will use the notation [, ] to denote the nth rung of Theon s ladder We see that + 1 and that + So the next rung of the ladder is [70, 99] beause , and The ratios of the two numbers on eah rung give us suessively better approximations to 2: 1 1 1/ / / / / / / Notie that the numbers are alternately above and below The onvergene of / to 2 is slow From the above alulations, it appears that we gain an extra deimal digit in 2 after alulating another one or two rungs of the ladder We will investigate more features of this ladder We will show how to modify it to alulate the square root of anumber, we will look at several reursion relations, and we will show how to inrease the speed of the onvergene First extension: Finding any square root Suppose we wish to find rational approximations to using the basi idea of Theon s ladder We assume always that 1 < The reursion relations that ahieve this end are, for n > 1, and + 1, Notie that 1 is the same for any root Throughout this paper, we assume that the first rung of the ladder is [1, 1] From the above relations, it follows that 1 and 1 Asanexample,tofind 3weuse + 1 and + 2 to obtain the ladder 1 1 1/ / / / / / / We now show why the reursion relations 1 and 2 always lead to rational approximations of Our examination in this setion is simple, but not rigorous, sine we are required to assume that lim n / exists Later, independent of this setion, we will prove that this limit exists Dividing 2 by 1 gives us VOL 35, NO 3, MAY 2004 THE COLLEGE MATHEMATICS JOURNAL 223
3 Dividing the numerator and the denominator on the right hand side by,weget + y Replaing on the right hand side by 2, we get 1 + y + 1 y Assuming that the limit exists, we let r lim n / Thenwehave r + r r, whih redues to r 2 Thus we see that the ladder gives rational approximations to Connetion to n and 1 n We now show that the rungs of our ladder [, ] an be generated by powers of simple binomials + n and 1 n 3 We will prove 3 by indution Notie that 3 is true when n 1 Assume 3 is true for n NThen N+1 N yn + x N x N + y N + x N + y N But from the reursion relation 1, x N + y N x N+1 Wenowhave N+1 xn + y N + x N+1 4 Also x N + y N x N + x N + y N x N + x N+1 y N+1,wherewe have used 1 and 2 Now 4 beomes N+1 yn+ x N+1, 5 and our indution proof for n is finished In the same way we an show that 1 n More reursion relations In addition to the reursion relations 1 and 2, we an easily derive others For example, eliminating from 2 using 1, we get Relations 1 and 6 allow us to alulate the next rung using only the two numbers on the present rung If we reindex 2 to read and use it to eliminate 1 from 1, we get THE MATHEMATICAL ASSOCIATION OF AMERICA
4 From 6, we have +1 8 Multiply 1 by and get + 1 Now use 8 to remove and to get Relations 7 and 9 form an interesting pair They allow us to alulate the x s without referene to the y s and vie versa A rigorous proof that the ladder onverges to Earlier, in setion 2, we gave a simple nonrigorous demonstration that our extended ladder onverges to Now we are able to give a rigorous demonstration Dividing 3 by and realling that 1,wehave n x n From 7 we know that > 2 So it follows that > 2 4,and > 3 6 Continuing in this way, we see that x 2n > n and x 2n+1 > n It now follows that for even rungs y 2n 2n 2n x < 2n x 2n n n, and for odd rungs y 2n+1 2n+1 x < 2n+1 x 2n+1 2n+1 1 n n + 1 Thus it follows that lim Seond extension: Leaping over rungs We saw earlier that the onvergene of / to was slow We now show how to radially aelerate this speed by jumping over many rungs of the ladder Using 3, we have Now we see that y 2n + x 2n 2n 1 + n 2 yn + 2 x 2 n + y2 n + 2xn x 2n 2 and y 2n x 2 n + y2 n 10 Using 10, we ould start on the 10th rung [x 10, y 10 ] and jump immediately to the 20th rung [x 20, y 20 ] then to [x 40, y 40 ],then[x 80, y 80 ],In only 4 steps we jump from rung 10 to rung 80! As a numerial example, onsider the ladder given in setion 1 with 2, where our last entry was x 7 169, y 7 239, and y 7 /x Using 10, VOL 35, NO 3, MAY 2004 THE COLLEGE MATHEMATICS JOURNAL 225
5 we alulate x and y Now our alulator gives y 14 /x and We have jumped from 4 aurate deimal plaes to 9 in one step From 10, we get y 2n y2 n + x2 n 1 yn + x n x 2n 2 2 Let z k y 2 k n, x 2 k n and obtain z k z k + zk This is the iteration obtained when using Newton s method [1, p68]for finding the roots of z 2 0 It is well known that this iteration onverges quadratially Roughly speaking, we double the number of aurate deimal digits of with eah iteration Referenes 1 K Atkinson, Elementary Numerial Analysis, 2nd ed, Wiley, New York, Sir Thomas Heath, A History of Greek Mathematis, Vol 1, Clarendon Press, pp J R Newman, The World of Mathematis, Vol 1, Simon and Shuster, 1956, pp T J Osler, Extending the Babylonian algorithm, Mathematis and Computer Eduation, Vol 33, No 2, 1999, pp G C Vedova, Notes on Theon of Smyrna, Amer Math Monthly, pp To all in the statistiian after the experiment is done may be no more than asking him to perform a postmortem examination: he may be able to say what the experiment died of R A Fisher Indian Statistial Congress, Sankhya, a THE MATHEMATICAL ASSOCIATION OF AMERICA
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