I keep neglecting to write mma2mac. (I do have a pretty good mac2mma.) Also I somewhat cavalierly Copied-as Plain text rather than Input. Nevertheless, Mma successfully read it back, even helpfully offering to strip the > comment markers as I pasted. Try this Macsyma: (c27) ELLIPTIC_E(%PI/2,1+(-1*R^2)) = 2*%PI*(1+60*R+134*R^2+60*R^3+R^4)^((-5)/4)*SUM(-1*432^N*((-1+R)^8*R*(1+R)^2*(1+60*R+134*R^2+60*R^3+R^4)^-3)^N*(-1*R*(15+68*R+90*R^2+68*R^3+15*R^4)+N*(1+(-126*R)+(-1041*R^2)+(-1764*R^3)+(-1041*R^4)+(-126*R^5)+R^6))*N!^-2*POCHHAMMER(1/12,N)*POCHHAMMER(5/12,N),N,0,INF); 2 (d27) elliptic_ec(1 - r ) = - 2 %pi inf ==== \ 1 5 n 8 n n 2 n ( > (--) (--) 432 (r - 1) r (r + 1) / 12 n 12 n ==== n = 0 6 5 4 3 2 (n (r - 126 r - 1041 r - 1764 r - 1041 r - 126 r + 1) 4 3 2 - r (15 r + 68 r + 90 r + 68 r + 15)) 2 4 3 2 3 n /(n! (r + 60 r + 134 r + 60 r + 1) )) 4 3 2 5/4 /(r + 60 r + 134 r + 60 r + 1) (c28) MATFORM(FIRST(INTOSUM(RHS(%))),N); 1 5 8 2 432 (n + --) (n + --) (r - 1) r (r + 1) 12 12 (d28) matrix([------------------------------------------, 2 4 3 2 3 (n + 1) (r + 60 r + 134 r + 60 r + 1) 5 4 3 2 15 r + 68 r + 90 r + 68 r + 15 r - 2 %pi (n - -----------------------------------------------------) 6 5 4 3 2 r - 126 r - 1041 r - 1764 r - 1041 r - 126 r + 1 2 4 3 2 (r + 6 r + 1) (r - 132 r - 250 r - 132 r + 1) 4 3 2 5/4 /(r + 60 r + 134 r + 60 r + 1) ], [0, 1]) (c29) SUBST(0.69105d0,R,LHS(D27)) = DFLOAT(PRUD(SUBST(0.69105d0,R,%),N,0,9)); (d29) 1.33925368878113d0 = [ 3.52840890972259d-78 1.33925368878113d0 ] [ ] [ 0.0d0 1.0d0 ] Note 77.5 digits! rcs>I'm having trouble parsing the EllipticE infinite series formula. Could you clarify if the 432^N is in the numerator or denominator, Numerator(!) Here, I should have used r=2/3 to illustrate the efficiency of the matrix: (c33) SUBST(2/3,R,D28); [ 1 5 59994 ] [ 64800 (n + --) (n + --) 1881502 %pi (n + ------) ] [ 12 12 940751 ] (d33) [ ----------------------- ------------------------ ] [ 2 1/4 ] [ 885012508801 (n + 1) 28803 9601 ] [ ] [ 0 1 ] --rwg rcs> and similarly for the other terms? I suspect I don't understand the grouping of terms in Mma. Rich ---- Quoting Bill Gosper <billgosper@gmail.com <http://gosper.org/webmail/src/compose.php?send_to=billgosper%40gmail.com>>: <clippage -- rcs>
This means that> EllipticE[1-r^2]==(2 \[Pi] Sum[-1/(n!)^2*432^n (((-1+r)^8 r (1+r)^2)/(1+60> r+134 r^2+60 r^3+r^4)^3)^n (-r (15+68 r+90 r^2+68 r^3+15 r^4)+n (1-126> r-1041 r^2-1764 r^3-1041 r^4-126 r^5+r^6)) Pochhammer[1/12,n]> Pochhammer[5/12,n],{n,0,\[Infinity]}])/(1+60 r+134 r^2+60 r^3+r^4)^(5/4)>> is an impressive acceleration formula for E'(r^2).>> You might be tempted to object that the k-fold speedup is cancelled by the> general term being k times more> complicated, but this is not the case. Once r is fixed, this is just a> matrix product over n of>> quadratic/quadratic linear> ( ).> 1 0>> However, it does cost a factor of 2 if you are computing megadigits of> E'(nonsquare rational).>> A correct formula for a>=b is>>> elliplen[a_, b_] := 4*a*EllipticE[Sqrt[1 - b^2/a^2]]>>>>>>> No, elliplen[a_, b_] := 4*a*EllipticE[1 - b^2/a^2]>> --rwg>>> Which is, in fact, (nonobviously) symmetrical in a and b.> --rwg