Let c sub 1 = 1; c sub 2 = 1; c sub 3 = 5; y sub 3 = 5*x^2 - 5*x + 1 (undefined for n=1, 2). Let: y sub n = (2*x - 1)*[c sub (n - 1)]*(x^(2*n - 3) + (x - 1)^(2*n - 3)) - 4*x^2*(x - 1)^2*[y sub (n - 1)] forn>3; c sub n = (d/dx)^(2*n - 4)[y sub n]/(2*n - 4)! for n>3, or just the coefficient of the polynomial term with the highest power. Let G(p) = 4*SIGMA[k=1 to p]((-1)^(k + 1)/(2*k - 1)), or 4 times the pth partial sum of the Gregory series. Then, let G(p, n) = G(p) + SIGMA[k=1 to n]((-1)^(p + k + 1)*[c sub k]/(4^(k - 1)*p^(2*k - 1), the “perturbation function” with the coefficients calculated above. Worked out, one has: G(p, n) = G(p) +((-1)^p)*1/(1*p^1) -((-1)^p)*1/(4*p^3) +((-1)^p)*5/(16*p^5) -((-1)^p)*61/(64*p^7) +((-1)^p)*1385/(256*p^9) -((-1)^p)*50521/(1024*p^11) +((-1)^p)*2702765/(4096*p^13) -((-1)^p)*199360981/(16384*p^15) +((-1)^p)*19391512145/(65536*p^17) -((-1)^p)*2404879675441/(262144*p^19) +((-1)^p)*370371188237525/(1048576*p^21) -((-1)^p)*69348874393137901/(4194304*p^23) +((-1)^p)*15514534163557086905/(16777216*p^25) -((-1)^p)*4087072509293123892361/(67108864*p^27) +((-1)^p)*1252259641403629865468285/(268435456*p^29)…. The Gregory series for pi/4 is notable for its beautiful simplicity but notorious for its stunningly slow convergence. But here, without proof or motivation other than the observation of a pattern in G(p)’sconvergence to pi, I show G(p, n) is a “startling” acceleration of the convergence of G(p) pi for p, n >> 0. Quantifying “startling” is subjective, but review the data below, checked carefully for repeatable accuracy, and decide for yourself. For example, while G(10000) yields 4 correct digits ofpi (counting the initial “3”), G(10000,750) yields 2635 digits correct. Why it works for very large p I can somewhat explain, but this strategy works unreasonably well, even when theterms OVERLAP! The beginning of the pattern I referenced is obvious for p chosen as a fairly large power of ten. One will note that one digit is incorrect in an “island” of correct digits. The pattern unfolds as shown above, though the formula was extended by a number of very fortunate discoveries, such as the formulae for perturbation function coefficients. In the limit the perturbation function is an infinite series, and I suspect only finite truncations work in accelerating the series, and then only for certain sets of p. I used a program I wrote in the ultra-fast interpreted language Euphoria providing an extended precision mathematical environment. Large integers were represented by “sequences” of their digits. Representing large decimals as integers, later truncated, allowed extended precision division as well as multiplication, etc. The “symbolic” manipulation of polynomials was simple. For example, (x - 1)^n was calculated by forming a Pascal triangle of coefficients with signs distributed correctly. Polynomial addition and multiplication were quite compact and elementary by manipulating the coefficients. So, apart from (huge) memory requirements, calculating the perturbation function was relatively easy. Thus, empirically, pi may apparently be calculated to thousands of digits using the Gregory series for pi/4 and a reiterative algebraic manipulation…on a desktop PC. I suggest it amounts to finding everbetter series for pi starting with Gregory series partial sums. This in itself is novel and interesting. It isalso interesting that the acceleration effect appears to fade in and fade out as n increases with a peak in accuracy for G(p, n) (p constant), at least initially. I am not mathematically sophisticated enough to provide proofs of these assertions, so instead I call this project a “discovery.” Update: It has come to my attention that the perturbation coefficients are precisely the (secant) Euler numbers..:)
Posted on: Fri, 21 Jun 2013 02:50:37 +0000
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