Approximation of π by Numerical Methods Mathematics Coursework (NM)

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1 Approximation of π by Numerical Methods Mathematics Coursework (NM) Alvin Šipraga Magdalen College School, Brackley April 1 1 Introduction There exist many ways to approximate 1 π, usually employed in computer systems. Numerical methods may be the basis for some of these approximations. In this coursework, we investigate the application of numerical methods specifically numerical integration methods to the evaluation of a particular definite integral. In this investigation we will use the function defined as follows: f(x) = x 4 x 3 + 4x 4 As mentioned, numerical integration will be employed to approximate π. The integral and its relationship is defined as follows: π = Which we shall proceed to investigate. f(x) dx = (1) x 4 x 3 dx () + 4x 4 y -4 - O x Figure 1: f(x) = 16x 16 x 4 x 3 +4x 4 The following sum was discovered by Bailey, Borwein and Plouffe (BBP) and is discussed in a paper from 1996 by them. The above integral () is derived from and is equal to it.[1, ] π = n= ( n 8n + 1 8n n ) 8n Approximate rather than calculate is used in this context, since π is irrational and any evaluation will always be an approximation never equal although the original expression may well be equal to π. (3) 1

2 1.1 Strategy SOLUTION BY SIMPSON S RULE The integral itself is appropriate because it is difficult to integrate by conventional means. The normal approach to integration by analysis is as follows: = 16 = 16 x 4 x 3 dx from () + 4x 4 x 1 x 4 x 3 + 4x 4 dx x 1 (x )(x x + ) dx At which point we can not proceed. Since there is no analytical solution within our means, our only means of solving it is by numerical methods. A brief discussion of the numerical methods we employ is also in order. 1.1 Strategy The numerical integration method we shall use is Simpson s rule. It is appropriate because it is an efficient and easy-to-apply numerical method for finding the area beneath a curve within a limit. Of interest is the error in the results obtained. Although it is not possible to integrate the expression analytically, we have the benefit of comparing the results to π, which is known to sufficiently many decimal places to never worry about it in the scope of this coursework. As such, we may calculate errors directly, and on that basis improve our solution. Simpson s rule is appropriate for the problem because it is an easy numerical method to apply, and converges relatively quickly to a desirable degree of accuracy. The Trapezium rule and Midpoint rule could also be used, but their use is superfluous because Simpson s rule directly improves on both, and so we can get better solutions using it. The rule is based on dividing the area to be integrated into equally sized strips. For an integral f(x) dx to be divided into n strips of width h, the rule is as follows: b a S n = h 3 {f + f n + 4(f 1 + f 3 + f f n 1 ) + (f + f 4 + f f n )} (4) Where f = f(a), f 1 = f(a + h), and so on such that f n = f(a + nh) = f(b). The procedure will be to apply the rule to the above integral to approximate π. Adjusting the number of strips taken (and hence width of strips) will affect the accuracy of the solution to the integral. We shall take a few of these such changes and investigate the error in their results. Given this, a more informed change to the number of strips can be made to achieve the kind of accuracy we want. Because numerical methods are rather tedious and prone to error, use of computer spreadsheet software is in order. A spreadsheet layout with formulae can be found in Appendix A. Solution by Simpson s rule Using the formula (4), we can split the region to be integrated into a set of strips for our desired degree of accuracy. The convergence of the method will be relative, so a reasonable number of strips to start with should be picked. Since we are integrating between and 1, we shall begin by using 1 strips. For reference, π to decimal places is as follows: π = (5) It is worth pointing out that given the nature of the function, were we to be able to integrate it (for example, by some more advanced partial fractions splitting) we would get an inverse trigonomic function of some kind (arctan, arcsin, etc.). However, computing π by use of trigonomic functions is paradoxical and can not give a better approximation of π without use of some further numerical method (e.g. Gregory s series, arctan x = x x3 3 + x5 5 x7 + ). This would somewhat 7 defeat the point of investigating by means of numerical integration.

3 .1 1 strips SOLUTION BY SIMPSON S RULE.1 1 strips In this case, we put n = 5, hence the number of strips is n = 1 and the width, h =.1. Below is the table of values for the points on the function that we will use in (4). x f(x) They are then substituted into the Simpson s rule formula (4): S 5 =.1 { ( ) + ( )} = (6) Which is correct to 3 significant figures. This is not particularly impressive, but Simpson s rule begins to converge very quickly. We shall double the number of strips (doubling n) to and then do some error analysis on the two results such that we can make an informed improvement on our solutions.. strips We put n = 1 = h =.5 and number of strips =. Our values of f(x) are as follows: x f(x) x f(x) Into (4): S 1 =.5 { ( ) + ( )} = (7) 3

4 3 ERROR ANALYSIS Which is correct to 5 significant figures. An improvement, but we can do better. We shall analyse the errors in our results and use the properties of Simpson s rule to find a better approximation. The spreadsheet software has a limit of 15 significant figures, so it seems reasonable 3 to try and achieve this by numerical methods. 3 Error analysis From (5), (6) and (7), we have π = S 5 = S 1 = Which, naturally, we shall refer to as π, S 5 and S 1 henceforth. Our desired accuracy is 15 significant figures, or thereabouts. Anything above 6 will be sufficient to demonstrate the merits or issues with numerical integration methods for approximation of π. We must do a little analysis given the error in the approximations we have computed. We know that for Simpson s rule, the absolute error is proportional to h 4 that is, fourth order. Using this property, it is possible to find without trial and error an n value for S n where the accuracy is roughly 4 15 significant figures. Consider the absolute errors (which we henceforth denote by x) in our approximations: S 5 S 1 S n It follows that S 1 (1/16) S 5, since h changes by a factor of 1/, and (1/) 4 = 1/16. To verify: (1/16) = , so indeed the property holds to a tolerable degree. Our aim is 15 significant figures, so an absolute error of approximately Consider the following: }.45 =.5 ± So generally we can say for n significant figures, we have an error of ±5 1 n. Obviously.55.5 to 4 significant figures, but the absolute error in any case will be for an approximation to 4 significant figures. By what we know about the rule, given a desired error we can find by what power let it be m of 1/16 we multiply S 5 by to get Given that the method is fourth order, it means that our old h would have to be multiplied by (1/) m to get the new h (h = h old (1/) m ), and it follows that h = 1/n = n = 1/h, so we can find the new n of S n such that S n = = ( ) m = S 5 16 ( ) m 1 = ( ) m = 16 ( ) = m = log 1/ = Unnecessary though it may seem, the scope of our precision pails in significance to modern computations of π. The current world record is just under.7 trillion digits (,699,999,99, exactly) by Fabrice Bellard in 9, so 15 does indeed seem reasonable given the software constraints and scope of this essay.[3] 4 We discuss this tentatively because the property of absolute error h 4 is approximate, and as such the language with which we discuss its application must also be so. 4

5 3 ERROR ANALYSIS and h = h old ( ) 1 m ( 1 = h = h old ( 1 =.1 =.9 ) m ) and n = 1 h = n = 1 h 1 =.9 = So for an absolute error of , we need (1/16) multiplied by the error of S 5 = the h of S 5 must be multiplied by a factor of (1/) to yield this = the new n of S n is 1/h = 1/(.1 (1/) ) 18. This is an awful lot, so perhaps we can see what other errors will approximately be. The following table shows the respective approximations of n for the desired number of significant figures. Remember that n significant figures = S n = 5 1 n, and so the method used above follows for each number of significant figures. s.f n For us to give an approximation to a reasonable degree of accuracy, we must also consider the subsequent Simpson rule approximations, so as to confirm the convergence (and hence correctness). Thus, we shall aim for 7 significant figures. So for our purposes, we shall consider S, S 4 and S 8 since our n values are approximations, we round the n values generously to the nearest 1 (8 rather than 7 is taken for the sake of simplicity in calculation and for continuity in doubling the n values each time) for significant figures 7, 8 and 9 (approximately). Subsequent calculations ultimately give us the following: π = S 5 = S 1 = S = S 4 = S 8 = The increase in accuracy seems to be reasonably consistent, with roughly one more correct significant figure every time n is doubled, which is nice. S is correct to only 6 significant figures, due to the process of rounding and approximate nature of the calculations likewise only 7 significant figures for S 4. S 8 is however correct to 9. We will hence use S 4 as our final approximation, since we stated 7 significant figures as a target. We can thus conclude with absolute confidence that the integral () is equal to π to 7 significant figures. Formally: x 4 x 3 dx = = π (to 7 significant figures) (8) + 4x 4 We can be sure that, to this many significant figures, the solution is correct. Further calculation of S 8 shows convergence on a value that is the same to 7 significant figures, and the function is continuous in the integral s interval, so the numerical method will not have broken down to give us incorrect convergence. However, we can not infer that the integral () is exactly equal to π, because π is irrational. The relationship can only be proven by analysis, since numerical methods will always give only a numerical approximation and π can not be expressed as a fraction. 5

6 A APPENDIX: SPREADSHEET SCHEMA 4 Summary We have demonstrated concisely the application of the numerical integration method Simpson s rule, along with the merits of error analysis. By analysing the errors in our numerical approximations, we were able to save the effort of trying to achieve 15 significant figures, knowing full well that the numerical method would be too cumbersome to apply in that case. Instead, we were able to correctly approximate the numbers of strips required for certain numbers of significant figures. It is obvious that, on a grander scale, analysis of this kind can save a lot of time. It was sufficiently accurate for our purposes, but generally speaking the properties of the rule should hold for most functions that would be encountered. There are obvious limitations with the method, though. The objective was to approximate π by using numerical integration methods. Of the elementary numerical integration methods, Simpson s rule is the best in terms of convergence, but despite this we must go up to around S for only 15 significant figures. For application in computer systems, Simpson s rule is inadequate in conjunction with the function (1) there exist algorithms that can be employed by computers which are astronomically better than our method. The world record of π digits calculated was done using the Chudnovsky series: 1 π = 1 n= ( 1) n (6n)!( n) (n!) 3 (3n)! (643 3 ) n+1/ (9) Which accurately gives 14 digits per term of the series.[3, ] Our method gave approximately 1 digit per term in the series (whereby we double n). 5 The same is true for application by hand numerical methods are very prone to error and require a lot of arithmetic. The only merit is that the mathematics employed is itself rudimentary. The main concern if doing it by hand is the speed of convergence, since the process takes infinitely longer time than on a computer. The properties desired in a computer algorithm are thus similar to those for pen and pencil calculations, for which our method is still not the best. A Appendix: Spreadsheet schema Numerical methods are cumbersome to apply using pen and paper, even with a calculator. As such, computer spreadsheet software has been used in the calculation of Simpson s rule approximations of π. Every spreadsheet for S 5 to S 8 is superfluous, so a general spreadsheet schema is given below, along with an example for S 1. A.1 General schema Text in monospace denotes actual spreadsheet contents. Text in italics denotes details that vary as n varies. For clarity, FUNC(X) is used in place of (16*X-16)/(X^4-*X^3+4*X-4). SUM(A:B) is for the sum of cells A to B. Note that cells such as D999 are used to imply the summation of the rest of the column, irrespective of (inevitably variable) size. A B C D E F 1 n n =SUM(B7:B8) no. strips =*B1 =SUM(D7:D999) 3 h =1/B =SUM(F7:F999) 4 S =(B3/3)*(D1+4*D+*D3) 5 6 x f(x) x f(x) x f(x) 7 =FUNC(A7) h =FUNC(C7) h =FUNC(E7) 8 1 =FUNC(A8) 3h =FUNC(C8) 4h =FUNC(E8) 9 5h =FUNC(C9) h 1 h. 5 Note however that we do not sum our series. 6

7 A. Example for S 1 REFERENCES A. Example for S 1 A B C D E F 1 n 1 4 no. strips h S x f(x) x f(x) x f(x) B Appendix: Software used This document was typeset using the L A TEX typesetting program. All graphs were plot using gnuplot. Calculations were done using the spreadsheet software Gnumeric and a calculator. References [1] Bailey, David H., A Compendium of BBP-Type Formulas for Mathematical Constants, November, [] Weisstein, Eric W., Pi Formulas, [3] Bellard, Frabrice, Computation of 7 billion decimal places of Pi using a Desktop Computer, 4th revision, February 1, 7