/* ellie.c * * Incomplete elliptic integral of the second kind * * * * SYNOPSIS: * * double phi, m, y, ellie(); * * y = ellie( phi, m ); * * * * DESCRIPTION: * * Approximates the integral * * * phi * - * | | * | 2 * E(phi_\m) = | sqrt( 1 - m sin t ) dt * | * | | * - * 0 * * of amplitude phi and modulus m, using the arithmetic - * geometric mean algorithm. * * * * ACCURACY: * * Tested at random arguments with phi in [-10, 10] and m in * [0, 1]. * Relative error: * arithmetic domain # trials peak rms * IEEE -10,10 150000 3.3e-15 1.4e-16 */ /* * Cephes Math Library Release 2.0: April, 1987 * Copyright 1984, 1987, 1993 by Stephen L. Moshier * Direct inquiries to 30 Frost Street, Cambridge, MA 02140 */ /* Copyright 2014, Eric W. Moore */ /* Incomplete elliptic integral of second kind */ #include "mconf.h" extern double MACHEP; static double ellie_neg_m(double phi, double m); double ellie(double phi, double m) { double a, b, c, e, temp; double lphi, t, E, denom, npio2; int d, mod, sign; if (cephes_isnan(phi) || cephes_isnan(m)) return NPY_NAN; if (m > 1.0) return NPY_NAN; if (cephes_isinf(phi)) return phi; if (cephes_isinf(m)) return -m; if (m == 0.0) return (phi); lphi = phi; npio2 = floor(lphi / NPY_PI_2); if (fmod(fabs(npio2), 2.0) == 1.0) npio2 += 1; lphi = lphi - npio2 * NPY_PI_2; if (lphi < 0.0) { lphi = -lphi; sign = -1; } else { sign = 1; } a = 1.0 - m; E = ellpe(m); if (a == 0.0) { temp = sin(lphi); goto done; } if (a > 1.0) { temp = ellie_neg_m(lphi, m); goto done; } if (lphi < 0.135) { double m11= (((((-7.0/2816.0)*m + (5.0/1056.0))*m - (7.0/2640.0))*m + (17.0/41580.0))*m - (1.0/155925.0))*m; double m9 = ((((-5.0/1152.0)*m + (1.0/144.0))*m - (1.0/360.0))*m + (1.0/5670.0))*m; double m7 = ((-m/112.0 + (1.0/84.0))*m - (1.0/315.0))*m; double m5 = (-m/40.0 + (1.0/30))*m; double m3 = -m/6.0; double p2 = lphi * lphi; temp = ((((m11*p2 + m9)*p2 + m7)*p2 + m5)*p2 + m3)*p2*lphi + lphi; goto done; } t = tan(lphi); b = sqrt(a); /* Thanks to Brian Fitzgerald * for pointing out an instability near odd multiples of pi/2. */ if (fabs(t) > 10.0) { /* Transform the amplitude */ e = 1.0 / (b * t); /* ... but avoid multiple recursions. */ if (fabs(e) < 10.0) { e = atan(e); temp = E + m * sin(lphi) * sin(e) - ellie(e, m); goto done; } } c = sqrt(m); a = 1.0; d = 1; e = 0.0; mod = 0; while (fabs(c / a) > MACHEP) { temp = b / a; lphi = lphi + atan(t * temp) + mod * NPY_PI; denom = 1 - temp * t * t; if (fabs(denom) > 10*MACHEP) { t = t * (1.0 + temp) / denom; mod = (lphi + NPY_PI_2) / NPY_PI; } else { t = tan(lphi); mod = (int)floor((lphi - atan(t))/NPY_PI); } c = (a - b) / 2.0; temp = sqrt(a * b); a = (a + b) / 2.0; b = temp; d += d; e += c * sin(lphi); } temp = E / ellpk(1.0 - m); temp *= (atan(t) + mod * NPY_PI) / (d * a); temp += e; done: if (sign < 0) temp = -temp; temp += npio2 * E; return (temp); } /* N.B. This will evaluate its arguments multiple times. */ #define MAX3(a, b, c) (a > b ? (a > c ? a : c) : (b > c ? b : c)) /* To calculate legendre's incomplete elliptical integral of the second kind for * negative m, we use a power series in phi for small m*phi*phi, an asymptotic * series in m for large m*phi*phi* and the relation to Carlson's symmetric * integrals, R_F(x,y,z) and R_D(x,y,z). * * E(phi, m) = sin(phi) * R_F(cos(phi)^2, 1 - m * sin(phi)^2, 1.0) * - m * sin(phi)^3 * R_D(cos(phi)^2, 1 - m * sin(phi)^2, 1.0) / 3 * * = R_F(c-1, c-m, c) - m * R_D(c-1, c-m, c) / 3 * * where c = csc(phi)^2. We use the second form of this for (approximately) * phi > 1/(sqrt(DBL_MAX) ~ 1e-154, where csc(phi)^2 overflows. Elsewhere we * use the first form, accounting for the smallness of phi. * * The algorithm used is described in Carlson, B. C. Numerical computation of * real or complex elliptic integrals. (1994) https://arxiv.org/abs/math/9409227 * Most variable names reflect Carlson's usage. * * In this routine, we assume m < 0 and 0 > phi > pi/2. */ double ellie_neg_m(double phi, double m) { double x, y, z, x1, y1, z1, ret, Q; double A0f, Af, Xf, Yf, Zf, E2f, E3f, scalef; double A0d, Ad, seriesn, seriesd, Xd, Yd, Zd, E2d, E3d, E4d, E5d, scaled; int n = 0; double mpp = (m*phi)*phi; if (-mpp < 1e-6 && phi < -m) { return phi + (mpp*phi*phi/30.0 - mpp*mpp/40.0 - mpp/6.0)*phi; } if (-mpp > 1e6) { double sm = sqrt(-m); double sp = sin(phi); double cp = cos(phi); double a = -cosm1(phi); double b1 = log(4*sp*sm/(1+cp)); double b = -(0.5 + b1) / 2.0 / m; double c = (0.75 + cp/sp/sp - b1) / 16.0 / m / m; return (a + b + c) * sm; } if (phi > 1e-153 && m > -1e200) { double s = sin(phi); double csc2 = 1.0 / s / s; scalef = 1.0; scaled = m / 3.0; x = 1.0 / tan(phi) / tan(phi); y = csc2 - m; z = csc2; } else { scalef = phi; scaled = mpp * phi / 3.0; x = 1.0; y = 1 - mpp; z = 1.0; } if (x == y && x == z) { return (scalef + scaled/x)/sqrt(x); } A0f = (x + y + z) / 3.0; Af = A0f; A0d = (x + y + 3.0*z) / 5.0; Ad = A0d; x1 = x; y1 = y; z1 = z; seriesd = 0.0; seriesn = 1.0; /* Carlson gives 1/pow(3*r, 1.0/6.0) for this constant. if r == eps, * it is ~338.38. */ Q = 400.0 * MAX3(fabs(A0f-x), fabs(A0f-y), fabs(A0f-z)); while (Q > fabs(Af) && Q > fabs(Ad) && n <= 100) { double sx = sqrt(x1); double sy = sqrt(y1); double sz = sqrt(z1); double lam = sx*sy + sx*sz + sy*sz; seriesd += seriesn / (sz * (z1 + lam)); x1 = (x1 + lam) / 4.0; y1 = (y1 + lam) / 4.0; z1 = (z1 + lam) / 4.0; Af = (x1 + y1 + z1) / 3.0; Ad = (Ad + lam) / 4.0; n += 1; Q /= 4.0; seriesn /= 4.0; } Xf = (A0f - x) / Af / (1 << 2*n); Yf = (A0f - y) / Af / (1 << 2*n); Zf = -(Xf + Yf); E2f = Xf*Yf - Zf*Zf; E3f = Xf*Yf*Zf; ret = scalef * (1.0 - E2f/10.0 + E3f/14.0 + E2f*E2f/24.0 - 3.0*E2f*E3f/44.0) / sqrt(Af); Xd = (A0d - x) / Ad / (1 << 2*n); Yd = (A0d - y) / Ad / (1 << 2*n); Zd = -(Xd + Yd)/3.0; E2d = Xd*Yd - 6.0*Zd*Zd; E3d = (3*Xd*Yd - 8.0*Zd*Zd)*Zd; E4d = 3.0*(Xd*Yd - Zd*Zd)*Zd*Zd; E5d = Xd*Yd*Zd*Zd*Zd; ret -= scaled * (1.0 - 3.0*E2d/14.0 + E3d/6.0 + 9.0*E2d*E2d/88.0 - 3.0*E4d/22.0 - 9.0*E2d*E3d/52.0 + 3.0*E5d/26.0) /(1 << 2*n) / Ad / sqrt(Ad); ret -= 3.0 * scaled * seriesd; return ret; }