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GENERATING ACCURATE VALUES FOR ARCTAN

One of the more important trigonometric functions is arctan(x) defined by- 012 0 2 )12()1(

1)arctan(

nnnx t nx tdtx Using the double angle formula for tangent one also has that- )1arctan()arctan()arctan(xyyxyx

From this equality follow the two identities-

)12arctan(21)arctan( 2 xxx and-

2_)arctan()1arctan()arctan(

xx

A plot of arctan(x) in 0 From the graph and from the above series one sees that arctan(x) is an odd function with arctan(x)= -arctan(-x) and it has values ranging from y(- )= -ʌ/2 through y(0)=0 to y()=ʌ/2. One also sees that accurate values of arctan(x) need only be found in the range 0There are certain points within 1

x 0 for which arctan(x) is known exactly without needing to go to a numerical approximation. These include- S Further exact values may be generated by the formula- )11arctan()arctan(21 2 xxx or by combining some of the above terms. For example, we find the single term formula for Pi )32126arctan[(24 To evaluate arctan(x) for values of x for which no convenient exact value exists but which lies close to a known value arctan(x 0 ), one can make use of the identity- )1arctan()arctan()arctan( 00 xxxx H where =x-x 0 < 02 0 0 xxxxb HH and now concentrate on how to find arctan(b). If b is extremely small then the first few terms of the standard arctan series will yield excellent results. However for larger but still small values of b such a series approximation becomes impractical when accuracies of the order of twenty digits or higher are required. In the latter case we resort to a new approach introduced by us several years ago. The method is based upon the use of even Legendre polynomials appearing in the integral- )arctan(),(),(1)(),( 1 0 222
bbnNbnMdt b ttPbnI tn The even Legendre polynomials are oscillatory functions wit zeroes in 02468357246352432 bbbbbbbbbRandbbbbbbbRbbbbbRbbbR These ratios yield continuously improving approximations for arctan(b) as n gets large and b remains small. The quotients are somewhat reminiscent of Pade approximates but are much easier to derive than the latter for the same order of accuracy. You will note that as b approaches zero all values of R(n,b) go as b. Also one has arctan(x)=arctan(x 0 )+R(,b). To demonstrate the accuracy of these approximations consider the case of finding the value of arctan(0.4) to more than 20 places. Here we have - )22941(1

This yields the approximation-

)229411,3(8)4.0arctan( |R = 0.3805063771123648863035879168590118605246 Comparing this to the 40 place computer evaluation - shows that the R(3,b) approximation already yields a 28 digit accurate result. Going to R(4,b) yields the even better approximation- which is accurate to 36 places. Further improvements are possible by going to still larger n. however, the quotients R(n,b) become progressively larger making an evaluation more difficult. There are numerous arctan formulas available for evaluating Pi. Among the earliest and simplest is the Leibnitz Formula- )31arctan()21[arctan(4] Although the series for these two arctan terms converge slowly, one can speed things up by expanding about the exact values arctan(1/sqrt(3)) and arctan(sqrt(2)-1). This produces the equivalent form- )]2571arctan()3581[arctan(24 whose series converge much more rapidly. Using our R(n,b) approximation we can estimate the value of Pi to be- )2571,()5581,([24nRnR

It yields the eighteen digit accurate result-

=3.14159265358979323 whe is taken as 3. Better results could be gotten by choosing n larger or evaluating things when b 1 and b 2 are made smaller. They had values of 1/19.180 and 1/14.071 in this calculation. Finally we look at finding the value of Pi using the even simpler Gregory formula- ...91quotesdbs_dbs2.pdfusesText_2