Summary: The best (five extrema, equal ripple) quadratic approximation to cos(π x/2) on [-1,1] is -x^2 + (1/2)*(1 + x0^2 + Sqrt[Pi^2 - 16*x0^2]/Pi) where (4*x0)/Pi == Sin[(Pi*x0)/2] i.e., x0 = -(2nd) and 4th extrema ~ .6995333146440 . (which RIES inverts in .567 sec.) (Boy do we need the Kepler's eqn inverter fn.) Ripple amplitude = 1/2 - x0^2/2 - Sqrt[Pi^2 - 16*x0^2]/(2*Pi) ~ 0.0280048 So the coefficient(s), amplitude, and turning point are all expressible in terms of this single constant x0. http://pi.lacim.uqam.ca/eng/ gives no match. http://isc.carma.newcastle.edu.au/ and oldweb.cecm.sfu.ca/projects/ISC/ISCmain.html seem down. --rwg ---------------------------- Date: Sat, 25 Mar 2000 21:24:36 -0800 (PST) From: rwg@best.com Message-Id: <200003260524.VAA21784@shell12.ba.best.com> To: math-fun@optima.CS.Arizona.EDU A&S 9.1.44 with z=r, cos(theta) = t is inf ==== \ k cos(r t) = 2 > (- 1) J (r) T (t) + J (r), / 2 k 2 k 0 ==== k = 1 whose truncations give optimal (in the Tchebychev sense) approximations to cos(r t) in [-1,1]. Thus, taking only terms 0,1, with r=pi/2, gives %pi t %pi 2 %pi %pi cos(-----) ~ - 4 J (---) t + 2 J (---) + J (---) 2 2 2 2 2 0 2 2 ~ 0.9714045 - 0.99880666 t as the parabola "best" approximating the central arch of cos(pi t/2). The error ripple heights are ~ [0.028596, -0.028000, 0.027402] at 0, +-t0, and +-1, where t0 satisfies %pi t0 %pi sin(------) 2 t0 = ----------------- ~ .701432 . %pi 16 bessel_j (---) 2 2 While T_n is an equal-ripple approximation to 0 (amplitude 2^-n, when scaled to be monic), the cumulative effect of approximating all the higher order (>2) terms is no longer equal ripple, but is pretty close. In so simple a case, we can actually write the equal-ripple (Remez) coefficients, a and b, in %pi x 2 cos(-----) ~ a x + b, 2 to wit, 2 2 %pi x0 2 %pi x0 (%pi cos (------) - 8) sin (------) 4 4 a = - 1, b = mu + 1, mu = ------------------------------------, 8 where mu is the ripple amplitude and x0 is the turning point, satisfying %pi x0 %pi sin(------) 2 x0 = --------------- ~ 0.6995333, 4 nearly the same equation as for t0, the Tchebychev turning point. From this, or from the approximate equality of the lead coefficients, i.e., -1 ~ -4 J2(pi/2), we can expect J2(pi/2) to be nearly 1/4. In fact it's ~ .2497 . This x0 then gives mu ~ -0.028005 (necessarily edging out the Tcheby worst case of .0286 at the origin.) So the Remez approximation is %pi x 2 cos(-----) ~ 0.9719952 - x 2 (equal-ripple in [-1,1].)