Medina, Herbert A.

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Loyola Marymount University

Digital Commons@Loyola Marymount Universityand Loyola Law SchoolMathematics Faculty Works Mathematics2-1-2006A Sequence of Polynomials for ApproximatingArctangentHerbert A. MedinaLoyola Marymount University, hmedina@lmu.eduThis Article is brought to you for free and open access by the Mathematics at Digital Commons @ Loyola Marymount University and Loyola LawSchool. It has been accepted for inclusion in Mathematics Faculty Works by an authorized administrator of Digital Commons@Loyola MarymountUniversity and Loyola Law School. For more information, please contact digitalcommons@lmu.edu.Repository CitationMedina, Herbert A., "A Sequence of Polynomials for Approximating Arctangent" (2006). Mathematics Faculty Works. 95.http://digitalcommons.lmu.edu/math_fac/95Recommended CitationMedina, H. (2006). A Sequence of Polynomials for Approximating Arctangent. The American Mathematical Monthly, 113(2),156-161. doi:10.2307/27641866 NOTES Edited by William Adkins A Sequence of Polynomials for Approximating Arctangent Herbert A. Medina 1. INTRODUCTION. The Taylor series x? x" ^ (-1)* + V-+- = E arctan x ? x-1-1-= > -x ?2Jt+l 3 5 Ulk + ^ was discovered by the Scotsman James Gregory in 1671 [1, chap. 12]. The series converges uniformly to arctan x on [?1, 1]; thus, we get {Tn(x)}= \Yji-i)kx2k+xli2k+iy \k=o ) the sequence of Taylor polynomials centered at 0 that converges to arctan x on [?1, 1]. Like the Taylor polynomials for several other classical functions (e.g., cosx, sinx, and ex), this sequence of polynomials is very easy to describe and work with; but unlike those Taylor sequences with factorials in the denominators of their coefficients, it does not converge rapidly for all "important" values of x. In particular, it converges extremely slowly to arctanx when |jc| is near 1. For example, if x = 0.95, we need to use T2S, a polynomial of degree 57, to get three decimal places of accuracy for arctan(0.95); if x = 1, we need to use r50o, a polynomial of degree 1001, to get three decimal places for arctan 1. Indeed, for x in [0, 1] it is easy to show that | arctanx ? Tnix)\ > x2n~^3 2i2n + 3) Thus, as x -> 1, Tnix) cannot approximate arctan x any better than 1 1 2(2n + 3) 2(degreeT/7)+4 The same is true near ?1. It is only fair to note that {Tn} converges to arctanx reason ably fast for x near 0. In this note we present another elementary, easily-described sequence in Q[x] that approximates arctan x uniformly on [0, 1] and converges much more rapidly than the sequence {Tn}. Such an approximating sequence provides, via the identities n (1 arctan x ? ? arctan (?x) =-arctan - 2 \x 156 ? THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 113 a method of approximating arctan x for all x in R. The approximating sequence arises }meN from the family of rational functions {x4m(l ? x)4m /(I + x2)}? 2. THE SEQUENCE AND ITS RATE OF CONVERGENCE. We begin with an algebraic computation whose proof is easy via induction. Lemma 1. Ifp\(x) ? 4- 4x2 + 5x4 - 4x5 + x6 and pm(x) = x4(l - x)4pm^x(x) + (-4)m~]Pl(x) for m > 2, then x4m(l -x)4m (-4)m -? ?-=Pm(x) + ??- (meN). 1 + xz 1 + xz A calculus computation shows that x(l ? x) < 1/4 on [0, 1]. Thus, 1+jc2 - V4 on [0, 1], whence ? x Am(\ _ f\4m / i \ 4w / 1 \ 4m ( } dt<[^) x < (i) (^6 [0,1]). l+f2 " - 14 I - - \ 4 /o The result of the lemma can be rewritten as x4mil-x)4m (-l)m+l4m i + x2 =/7mW--TT^' As a result, ^_l^m+\^m /l\4m Jo i -\-1  \4/ ' Dividing by (?l)m+14m and integrating the second term on the left we obtain L x /_1\ra + l /o 4 for all x in [0, 1]. It follows that I x 5m pm (0 dt ? arctan * < ( - J (1 ) -L x (_ty??+1 hm(x) = ???pm(t)dt (2) defines a sequence in Q[x] that converges uniformly on [0, 1] to arctan x. To get a bet ter sense of the convergence rate, note that pm has degree 8m ? 2, hence /zm has degree 8m ? 1. In (1) we write 45m = (45/8)8m-i+i an(j summarize our results in Theorem 1. Theorem 1. For m = 1, 2, ... define pm(t) as in Lemma 1 and hm(x) as in (2). Then / 1 \deg(/7,?)+l |/zm00 - arctan x | < (^t^J (3) for all x in [0, 1]. February 2006]  notes  157 3. EXAMPLES, OBSERVATIONS, AND A CLOSED-FORM EXPRESSION FOR hm. Evaluating , , x x3 x5 x1 5x9 x10 43xu x12 27x13 x14 x15 h2(x) ?x-1-1-1-1-1 3 5 7 48 20 176 4 208 28 240 at x = 0.95 and x = 1 we find that at both points the approximation to arctan x is within 2.28 x 10~7, better than six decimal places of accuracy with a polynomial of much smaller degree than the Taylor polynomials mentioned in the introduction. If we consider h7, a polynomial of degree 55, (3) guarantees that the approximation on [0, 1] is accurate to within 8.47 x 10~22. Thus, _ 506119433541064524255449 " 161102819285860855603200 gives twenty digits of accuracy for n. Like the Taylor polynomials, the hm are one-sided approximations. Indeed, it is not hard to see that Am(jc) ? arctan x is positive when m is odd and negative when m is even. Taylor polynomials are constructed by matching a function and its derivatives at a point. Hermite-interpolating (or osculating) polynomials are constructed by match ing a function and its derivatives at more than one point (see [2, sec. 3.3]). For example, we can construct a sequence {pN} such that p(^i0) ? arctan(n)(0) and Px\l) = arctan(n)(l) for n = 0, 1, ..., N. We note that such a Hermite-interpolating polynomial for arctan x cannot be in Q[x] because arctan 1 = 7r/4. Thus the hm are not Hermite-interpolating polynomials. Nevertheless, the next theorem shows that the hm are similar to Hermite-interpolating polynomials in that they match derivatives of arctan x at both 0 and 1. Theorem 2. Form = 1,2,... h%\0) = arctan(n)(0) and h (1) = arctan(n)(l) when ever 1 < n < Am. Moreover, if g(x) is a polynomial of degree 8m such that g(0) = arctan 0, g{n\0) = arctan(n)(0), and g(n\l) = arctan(n)(l) whenever 1 < n < 4m, then g ? hm. Proof We deal with the case x = 1 first. We appeal to (2) and Lemma 1 in noting that (-l)m+1 (-l)m+1 /x4m(l-x)4m (-4)" v t ? ("1)w+ ( ? ("1)w+ (J hmW = ?^-Pmix) = ?^? [  1+x2 1+x2 /_i\m+l v4ra \ i ( j \l-x)4"' + ?l?. (4) 2* Am 1+x2/ 1+x Using d/dx (arctanx) = 1/(1 + x2) on the second term in (4) and the product rule for differentiation, ifix)gix)){n) - ? (l)fin-k)ix)g{k)ix), k=0 \ks on the first, we find that ^?-SCi1)^^) (<'-"-)"+?? 158 ? THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 113 IfO<it <?-l,((l-x)4m)(?:)|x=1 = 0,so/?W(l) = arctanw(l)whenl < n < Am. To prove the assertion at x = 0, we can rewrite the first summand in (4) as (-Dm+1a-x)4m\Am 4m 1 + x2 and follow the same steps. If g is a polynomial with the properties stated, then g ? hm is of degree 8m, and because g(n)(0) ? h^(0) = 0 when 0 < n < 4m, its first 4m coefficients are 0. Hence g - hm = x4m+lq, where q is of degree 4m ? 1. Write 4m-l q(x) = ^2 a^x - 1)/C k=0 Inductive use of the product rule to compute (g ? hm){k)(\) demonstrates that ak ? Cka0 when 1 < k < 4m ? 1, with Ck ^ 0. Therefore its use on (g ? hm){4m) shows that (g - hm){4m)(\) = Ca0, where C ^ 0. It follows that a0 = 0 and thus that a* = 0 when I < k < 4m ? 1. The next lemma is the key to establishing formulas for the coefficients. Lemma 2. For m = 1, 2,... wnf? (l - o4m 4m_204m v^ . rw(i) 7=0 where rm is a polynomial satisfying deg(rm) < 2. T/ien the following statements hold: (i) rm(r) = (-1)?4"; (?J) ?2; = (-iy+1 E Q?)(-i)* ?^-. = (-iy+1 E (2*+i)(-1}* Proo/ Write (i-o4m = E::0(4r)(-i)^ \+t2 (l+o2 -^ /4m\ i2i 2^' / 4m \ i2*+1 ^ov-./^ - _, v2A + l/l + f where we abbreviate the terms on the left and right in the last expression by SE and 5o, respectively. Invoking the identity t2k k r?iv -i? = (-i)*+1 y(_iy+>i2o-i) + L21 i+t2 ?1 i + t (-1)* 2 February 2006]  NOTES  159 for k = 1, 2, ..., we write We collect the polynomial and nonpolynomial parts to obtain Because ?("1? )(-!)* = (-l)m4m, 2m Mm ,=o ^ the nonpolynomial part becomes (?l)m4m/(l + i2). We change the order of summa tion on the polynomial part to rewrite it as f?Hs?H t2ij-i). A similar procedure performed on S0 yields -S,(.-,'g.-..'(2;:1))-'^sG::1)'-?> Because the second summand is zero, we have established that (1 - f)4 ?/ ?,?/4m\ A ,.._., (-ir4" ? -s HsGHr"* This identity proves both parts of the lemma. Combining the results of Lemmas 1 and 2, we see that 2m Am?2 Pmit) = (-l)'M J](-iy i2?'-1} + ? fl; i4m+7, (5) 7=1 7=0 where the a-3 are as in Lemma 2. The closed-form expression for hm follows from (5) and (2). Theorem 3. For m ? 1, 2, ... 2m f_iv + l 4m-2 i__^_ x2./-l , y^ _^_ r4m+j + l 27-1 4? (?l)w+14m(4m + 7 + 1) ? W^V(-1)7 jc^i+V^ ^_x' mv Z-, 9.1-1 ?^ r_nm+i4mM - y = l ^ ,/=0 160 ? THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 113 where and The theorem makes it easy to use a computer for the computation of the hm. The author's website (http://myweb.lmu.edu/hmedina) contains Mathematica code for this computation and more details related to this note. 4. FURTHER REMARKS AND QUESTIONS. The keys to the approximating se quence {hm} are that the family of polynomials x4m(l ? x)4m(m e N) leaves an integer remainder when divided by 1 + x2 and that the members of the family are small for x in [0, 1]. There are other families of polynomials with this property, but is there an other simple one that converges more quickly than {hm} to arctan jc? Is there one with the desirable factorials in the denominator of the error bound? The results herein were stumbled upon after the author became intrigued by and curious about the fact that fl x4(l-x)4 22 i -? dx =-it ; Jo 1+*2 1 that is, 4(Ai(l) ? arctan 1) = y ? tt. Is there a simple closed-form expression for 4Am(l)? If so, it would provide a sequence in Q for approximating tt. Another prob lem, probably a very difficult one, is to find an easily describable sequence {gn} such that 4g?(l) is always a convergent in the continued fraction expansion of jr. REFERENCES 1. P. Beckmann, A History of Pi, St. Martin's Press, New York, 1976. 2. R. L. Burden and J. D. Faires, Numerical Analysis, 7th ed., Brooks/Cole, Pacific Grove, CA, 2001. Department of Mathematics, Loyola Marymount University, Los Angeles, CA 90045 hmedina @ Imu. edu A Cantor-Bernstein Theorem for Paths in Graphs Reinhard Diestel and Carsten Thomassen The Cantor-Bernstein theorem says that if for two infinite sets A and B there are injec tive functions / : A -> B and g : B -> A, then there is a bijection A <-> B. Perhaps its February 2006]  NOTES  161

A Sequence of Polynomials for Approximating Arctangent

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A Sequence of Polynomials for Approximating Arctangent

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