Investigating Euler s Method and Differential Equations to Approximate π. Lindsay Crowl August 2, 2001

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1 Investigating Euler s Metod and Differential Equations to Approximate π Lindsa Crowl August 2, 2001 Tis researc paper focuses on finding a more efficient and accurate wa to approximate π. Suppose tat x s (t) is te solution of an initial value problem x = F (t, x) (1) x 0 = x 0 and tat x s (1) = π. Ten we can use te tecnique called Euler s metod to approximate π. Te idea of Euler s metod is to use a series of linear approximations to step along te curve of x(t) on tangent lines determined b te derivative of x(t) at set time steps of a discrete lengt. More generall, tis metod can be used to find π if te function x(t) is equal to π at some point in time and if te initial value of te function at t = 0 is known. We want to find a differential equation of te form sown in (1), wit a special solution x s (1) = π. Te new idea explored in tis paper is to modif te differential equation in (1) so tat te special solution x s (t) remains a solution, and so tat te rest of te direction field now points more towards x s (t) tan it did originall. Tis will potentiall improve te accurac of Euler s metod using fewer steps, since all solutions point closer towards te desired value. Tis paper relies on te first order Euler formula. Using te first differencediscretization of te derivative and wic uses te derivative of a function to estimate te value of te solution at a later time. Our second order Euler formula is an extension of te first, and uses te second difference discretization of te second derivative. Our metod uses te concavit of a curve to estimate te desired value. We discuss four initial value problems wit special solutions of te form x s (1) = π. We develop a formula tat creates an attracting direction field around te special solution so tat all curves converge towards π as t approaces 1. We ave verified tis for our four examples, toug we do not et ave a rigorous proof. Since te attracting formula involves complicated functions suc as arcsine in te first derivative, te second order Euler formula becomes necessar. Tis makes a furter inspection of te concavit of 1

2 curves in te direction field elpful. Lastl, we examin possible improvements of te original attracting solution to obtain furter accurac. 2nd Order Euler Formula Euler s Approximation originates from te first order differential equation (1). Consider te definition of te slope at a point on a curve, (t + ) (t) = lim. (2) 0 Wen becomes a small set step lengt, te equalit in equation (2) becomes an approximation, called te first difference, n n+1 n, (3) wic can be rewritten to give a discrete approximation for n+1 : n+1 n + n since = F (t n, n ), (4) n+1 n + F (t n, n ), (5) ielding te 1st order Euler s formula. Te variables n and n+1 are te eigts of different points estimated to be on te curve between a certain time interval. Te second order Euler s approximation comes from taking te derivative of te derivative from equation (3) = ( ). (6) We can also form a discretized version (te second difference) of te second derivative b taking te original formula (3) and appling it to, n n+1 n, (7) and substituting in te following first differences for te first derivative terms: n+1 n+1 n (8) n n n 1. (9) 2

3 Te resulting second difference formula is n n+1 n n n 1 = n+1 2 n + n 1 2. (10) Solving for n+1, we obtain te second order Euler s formula (12) for te differential equation (1), n+1 2 n n 1 + n (11) n+1 2 n n F (t n, n ). (12) Now if te initial conditions 0, 0, and 0 are known, iterations of tis formula can be used to move along te curve to te specific point of interest, (1) = π. Using te Second Order Euler Formula to Approximate π So far, all of te initial value problems we ave found wic ave a solution x s (t) wit tat satisf x(1) = π, suc as x = f(t) x 0 = 0 (13) x = 2 arcsin(t) + 2t 1 t 2 ; (14) x = 2 arcsin(t); (15) x = 4 arctan(t); (16) ( t x = 6 arcsin, (17) 2) ave direction fields independent of x, since x depends onl on t. Euler s metod could approximate π more accuratel if te direction field ad an attracting equilibrium solution at te initial value problem were x(1) = π. Tis wa a point off te correct curve would be pused back towards te x(1) = π curve and Euler s metod would, in teor, ave less error. One wa too obtain tis attracting direction field, is te to add a function g(t) to an initial value problem. Te general form of te improved initial value problem, = x (t) + g(t, ) (18) 3

4 contains a g(t, ) modification term tat satisfies te following conditions: Wen (t) = x(t), g(t, ) = 0; Wen (t) < x(t), g(t, ) > 0; Wen (t) > x(t), g(t, ) < 0. Tese conditions sould aid in te accurac of Euler s metod since if an approximated point lies above te desired curve, te new slope formula for te next point will automaticall correct itself and point towards to te special solution x(1) = π and tat x s initial value problem is still a solution of (t). Te New Initial Value Problems Note tat we use two concave up (15), (17) and two concave down (14), (16) examples, since concavit becomes important wen te second derivative is used in te approximation formula. We give te following examples: Example (14) = 4 1 t arcsin(t) + 2t 1 t 2 (19) Example (15) = 2 1 t arcsin(t) (20) Example (16) = 4 1+t arctan(t) (21) Example (17) = 6 4 t arcsin( t 2 ) (22) were te general form of g(t, ) is x s and as an example, x = 2 arcsin(t)+ 2t 1 t 2 for te initial value problem in Example (14). Tis form of g(t, ) 4

5 satisfies te tree conditions needed to form an attracting direction field, since g(t, ) is te difference between te original initial value problem and te new initial value problem. Tis turns te special solution (1) = π into an uncanged attracting solution witin te altered direction field. W First Order Euler is Not Helpful Te problem wit te new initial value problems is tat all of teir first derivatives contain some tpe of inverse trig function, wic makes an iterations containing tese complicated functions to solve for π pointless since, for example, arcsin(1) = π so te solution can easil be calculated in one step. However, a clean form void of inverse trig functions exists for te second derivative for all te examples in tis paper, and using te second order Euler formula (12), te second derivative can be used to solve explicitl for te next step, n+1, witout te use of complicated functions. Take a look at te formula for Example (14) = 8t 1 t t 2. (23) Te new equation no longer contains a complicated function after taking te second derivative. Te general formula for te second derivative of an attracting direction field around an special solution, x s, = x + x (24) can be used as long as first order Euler approximations for and ( ) are inserted into (24), resulting wit n+1 2 n + n 1 2 = x + x n+1 n. (25) Tis equation can now be rewritten to give a discrete approximation for n+1, n+1 = n(2 + ) n (x x ). (26) 1 + Wit iterations of (26), (1) = π can be approximated along te special solution curve to obtain a more accurate estimate for π once te initial values for t, n, and n 1 are known. We cose tese values for simplicit to 5

6 approximate π, ; (27) 1 = 0 ; (28) t 0 = 0. (29) Te initial value for 1 must be approximated using te first order Euler metod using te curve s initial slope to obtain a value for te n+1 term. Te slope is unique for eac special solution for x s (t). Te reasoning beind tis is tat an initial point as well as an initial slope are necessar for te existence and uniqueness of a second order differential equation. Examining te Second Order Formula and Concavit Te idea beind an attracting direction field is to improve te original Euler metod. However, since te attracting direction field equation = x + x s (30) alwas contains an inverse trig function witin te x s term, te second order formula is necessar. From te second order Euler formula comes te equation n+1 = 2 n n 1 + n (31) = n + ( n n 1 ) + n. (32) Tis is a formula tat steps along a curve x s b placing a new point n+1, a step in time aead and finds te new eigt b taking te eigt of te last point n, adding te slope from te previous two points n n 1, and ten subtracting a factor dependent on te degree of concavit of te curve n 2, not on te slope n as in te original Euler metod. Examining te Attracting Solution Te attracting direction field is given b te differential equation = x (t) + g(t, ) (33) 6

7 were g(t, ) = x s (t) and x s (t) is te special solution wit x s (1) = π. Tis equation can be manipulated in order to find a simple correlation between and its concavit, = x + x s (34) = x + x = x + (x x ) x s + x = x s. (35) So as increases, te concavit, increases b te same magnitude. B taking a closer look at te original second order Euler formula (32), n+1 = n + ( n n 1 ) + 2 n, (36) we see tat te correction term 2 n directl depends on te curve s concavit. Te value of canges te curve from from concave up to concave down (and vice versa) onl wen it passes troug = 0. At tis point, te equation (36) simplifies into = x s x. (37) For eac x s te range were and x s are bot concave up or concave dowd is onl dependent on time. Terefore, tere is a range of were accurate approximations for eac special solution can exist in te direction field since te correction term must direct te next point towards te special solution, or te next point n+1 would diverge and become a more inaccurate approximation. If te special solution is concave up as in Examples (15) and (17), (t) crosses over into concave down curves if te estimate for n is too low. If te special solution is concave down as in Examples (14) and (16), (t) crosses over into concave up curves if n is too ig. However, all of tese ranges of congergence are witin te bounds of te initial first Euler step, 1 = 0 (at least for all four function examined in tis paper). Wen = 0, one of te bounds of convergence becomes = x x s. (38) 7

8 In order to take a closer look at te range of convergence, I investigated te examples: Example (14) Concave down range 0 < < 2 arcsin(t) + 2t 1 t 2 Example (15) Concave up range 2 arcsin(t) + Example (16) Concave down range 0 < < 4 arctan(t) Example (17) Concave up range 6 arcsin(t) + 4t (1 t 2 ) 3/2 (39) 2t < < (40) (1 t 2 ) 3/2 4t (1 + t 2 ) 2 (41) 6t < < (42) (4 t 2 ) 3/2 As long as te range is large enoug to old all estimated points, te approximation of π will not diverge and become less accurate. In all of te examples tested witin tis paper, te estimated points remain witin te convergence range of te first step for all step sizes. Improvement Beond te New IVP In order to furter improve te accurac of te attracting solution initial value problem, = x x (43), we multipl te modification term g(t, ) b some integer k > 1 so tat te new direction field will become even more attracted to te special solution. Te Improved New IVP: = x + k(x s ) (k > 1) (44) 8

9 Te new term, k increases te accurac of te second order Euler formula b causing te solutions of te IVP to converge faster and closer to te special solution. However if k is too large and te step size is also too large, te approximation becomes inaccurate once again. Tis is because te concavit convergence range comes closer to te special solution and te approximation formula is more prone to diverge, = x s x k 2. (45) B dividing x b k 2, te range of convergence gets smaller, so te approximation will be more likel to diverge if te step size is not small enoug to sta witin te range now tat te limit (36) moves closer to te special solution = x s (t). In (46) we can see tat as k approaces infinit, ten x k 2 approaces zero. Tis would cause one side of te convergence range to close into = x s, allowing no room for error on one side of te approximate curve witout divergence from te special solution. Tere seems to be a balance between accurac troug step size and accurac troug raising tis new convergence constant k. Furter Stud Find te true error bounds for te new (improved?) modified formulas Discover more functions tat satisf x s (1) = π Prove tat te second order Euler formula (wit an attracting direction field) works better tan te first order Euler formula (wit a direction field independent of ) for a small enoug step size Prove tat te approximation metod examined in tis paper is safe from divergence if it is witin te convergence region defined b te limit = x s x Analze additional steps besides te first step, 0, for te range of convergence 9