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| from .functions import defun, defun_wrapped | |
| def _jacobi_theta2(ctx, z, q): | |
| extra1 = 10 | |
| extra2 = 20 | |
| # the loops below break when the fixed precision quantities | |
| # a and b go to zero; | |
| # right shifting small negative numbers by wp one obtains -1, not zero, | |
| # so the condition a**2 + b**2 > MIN is used to break the loops. | |
| MIN = 2 | |
| if z == ctx.zero: | |
| if (not ctx._im(q)): | |
| wp = ctx.prec + extra1 | |
| x = ctx.to_fixed(ctx._re(q), wp) | |
| x2 = (x*x) >> wp | |
| a = b = x2 | |
| s = x2 | |
| while abs(a) > MIN: | |
| b = (b*x2) >> wp | |
| a = (a*b) >> wp | |
| s += a | |
| s = (1 << (wp+1)) + (s << 1) | |
| s = ctx.ldexp(s, -wp) | |
| else: | |
| wp = ctx.prec + extra1 | |
| xre = ctx.to_fixed(ctx._re(q), wp) | |
| xim = ctx.to_fixed(ctx._im(q), wp) | |
| x2re = (xre*xre - xim*xim) >> wp | |
| x2im = (xre*xim) >> (wp-1) | |
| are = bre = x2re | |
| aim = bim = x2im | |
| sre = (1<<wp) + are | |
| sim = aim | |
| while are**2 + aim**2 > MIN: | |
| bre, bim = (bre * x2re - bim * x2im) >> wp, \ | |
| (bre * x2im + bim * x2re) >> wp | |
| are, aim = (are * bre - aim * bim) >> wp, \ | |
| (are * bim + aim * bre) >> wp | |
| sre += are | |
| sim += aim | |
| sre = (sre << 1) | |
| sim = (sim << 1) | |
| sre = ctx.ldexp(sre, -wp) | |
| sim = ctx.ldexp(sim, -wp) | |
| s = ctx.mpc(sre, sim) | |
| else: | |
| if (not ctx._im(q)) and (not ctx._im(z)): | |
| wp = ctx.prec + extra1 | |
| x = ctx.to_fixed(ctx._re(q), wp) | |
| x2 = (x*x) >> wp | |
| a = b = x2 | |
| c1, s1 = ctx.cos_sin(ctx._re(z), prec=wp) | |
| cn = c1 = ctx.to_fixed(c1, wp) | |
| sn = s1 = ctx.to_fixed(s1, wp) | |
| c2 = (c1*c1 - s1*s1) >> wp | |
| s2 = (c1 * s1) >> (wp - 1) | |
| cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp | |
| s = c1 + ((a * cn) >> wp) | |
| while abs(a) > MIN: | |
| b = (b*x2) >> wp | |
| a = (a*b) >> wp | |
| cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp | |
| s += (a * cn) >> wp | |
| s = (s << 1) | |
| s = ctx.ldexp(s, -wp) | |
| s *= ctx.nthroot(q, 4) | |
| return s | |
| # case z real, q complex | |
| elif not ctx._im(z): | |
| wp = ctx.prec + extra2 | |
| xre = ctx.to_fixed(ctx._re(q), wp) | |
| xim = ctx.to_fixed(ctx._im(q), wp) | |
| x2re = (xre*xre - xim*xim) >> wp | |
| x2im = (xre*xim) >> (wp - 1) | |
| are = bre = x2re | |
| aim = bim = x2im | |
| c1, s1 = ctx.cos_sin(ctx._re(z), prec=wp) | |
| cn = c1 = ctx.to_fixed(c1, wp) | |
| sn = s1 = ctx.to_fixed(s1, wp) | |
| c2 = (c1*c1 - s1*s1) >> wp | |
| s2 = (c1 * s1) >> (wp - 1) | |
| cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp | |
| sre = c1 + ((are * cn) >> wp) | |
| sim = ((aim * cn) >> wp) | |
| while are**2 + aim**2 > MIN: | |
| bre, bim = (bre * x2re - bim * x2im) >> wp, \ | |
| (bre * x2im + bim * x2re) >> wp | |
| are, aim = (are * bre - aim * bim) >> wp, \ | |
| (are * bim + aim * bre) >> wp | |
| cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp | |
| sre += ((are * cn) >> wp) | |
| sim += ((aim * cn) >> wp) | |
| sre = (sre << 1) | |
| sim = (sim << 1) | |
| sre = ctx.ldexp(sre, -wp) | |
| sim = ctx.ldexp(sim, -wp) | |
| s = ctx.mpc(sre, sim) | |
| #case z complex, q real | |
| elif not ctx._im(q): | |
| wp = ctx.prec + extra2 | |
| x = ctx.to_fixed(ctx._re(q), wp) | |
| x2 = (x*x) >> wp | |
| a = b = x2 | |
| prec0 = ctx.prec | |
| ctx.prec = wp | |
| c1, s1 = ctx.cos_sin(z) | |
| ctx.prec = prec0 | |
| cnre = c1re = ctx.to_fixed(ctx._re(c1), wp) | |
| cnim = c1im = ctx.to_fixed(ctx._im(c1), wp) | |
| snre = s1re = ctx.to_fixed(ctx._re(s1), wp) | |
| snim = s1im = ctx.to_fixed(ctx._im(s1), wp) | |
| #c2 = (c1*c1 - s1*s1) >> wp | |
| c2re = (c1re*c1re - c1im*c1im - s1re*s1re + s1im*s1im) >> wp | |
| c2im = (c1re*c1im - s1re*s1im) >> (wp - 1) | |
| #s2 = (c1 * s1) >> (wp - 1) | |
| s2re = (c1re*s1re - c1im*s1im) >> (wp - 1) | |
| s2im = (c1re*s1im + c1im*s1re) >> (wp - 1) | |
| #cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp | |
| t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp | |
| t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp | |
| t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp | |
| t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp | |
| cnre = t1 | |
| cnim = t2 | |
| snre = t3 | |
| snim = t4 | |
| sre = c1re + ((a * cnre) >> wp) | |
| sim = c1im + ((a * cnim) >> wp) | |
| while abs(a) > MIN: | |
| b = (b*x2) >> wp | |
| a = (a*b) >> wp | |
| t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp | |
| t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp | |
| t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp | |
| t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp | |
| cnre = t1 | |
| cnim = t2 | |
| snre = t3 | |
| snim = t4 | |
| sre += ((a * cnre) >> wp) | |
| sim += ((a * cnim) >> wp) | |
| sre = (sre << 1) | |
| sim = (sim << 1) | |
| sre = ctx.ldexp(sre, -wp) | |
| sim = ctx.ldexp(sim, -wp) | |
| s = ctx.mpc(sre, sim) | |
| # case z and q complex | |
| else: | |
| wp = ctx.prec + extra2 | |
| xre = ctx.to_fixed(ctx._re(q), wp) | |
| xim = ctx.to_fixed(ctx._im(q), wp) | |
| x2re = (xre*xre - xim*xim) >> wp | |
| x2im = (xre*xim) >> (wp - 1) | |
| are = bre = x2re | |
| aim = bim = x2im | |
| prec0 = ctx.prec | |
| ctx.prec = wp | |
| # cos(z), sin(z) with z complex | |
| c1, s1 = ctx.cos_sin(z) | |
| ctx.prec = prec0 | |
| cnre = c1re = ctx.to_fixed(ctx._re(c1), wp) | |
| cnim = c1im = ctx.to_fixed(ctx._im(c1), wp) | |
| snre = s1re = ctx.to_fixed(ctx._re(s1), wp) | |
| snim = s1im = ctx.to_fixed(ctx._im(s1), wp) | |
| c2re = (c1re*c1re - c1im*c1im - s1re*s1re + s1im*s1im) >> wp | |
| c2im = (c1re*c1im - s1re*s1im) >> (wp - 1) | |
| s2re = (c1re*s1re - c1im*s1im) >> (wp - 1) | |
| s2im = (c1re*s1im + c1im*s1re) >> (wp - 1) | |
| t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp | |
| t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp | |
| t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp | |
| t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp | |
| cnre = t1 | |
| cnim = t2 | |
| snre = t3 | |
| snim = t4 | |
| n = 1 | |
| termre = c1re | |
| termim = c1im | |
| sre = c1re + ((are * cnre - aim * cnim) >> wp) | |
| sim = c1im + ((are * cnim + aim * cnre) >> wp) | |
| n = 3 | |
| termre = ((are * cnre - aim * cnim) >> wp) | |
| termim = ((are * cnim + aim * cnre) >> wp) | |
| sre = c1re + ((are * cnre - aim * cnim) >> wp) | |
| sim = c1im + ((are * cnim + aim * cnre) >> wp) | |
| n = 5 | |
| while are**2 + aim**2 > MIN: | |
| bre, bim = (bre * x2re - bim * x2im) >> wp, \ | |
| (bre * x2im + bim * x2re) >> wp | |
| are, aim = (are * bre - aim * bim) >> wp, \ | |
| (are * bim + aim * bre) >> wp | |
| #cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp | |
| t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp | |
| t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp | |
| t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp | |
| t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp | |
| cnre = t1 | |
| cnim = t2 | |
| snre = t3 | |
| snim = t4 | |
| termre = ((are * cnre - aim * cnim) >> wp) | |
| termim = ((aim * cnre + are * cnim) >> wp) | |
| sre += ((are * cnre - aim * cnim) >> wp) | |
| sim += ((aim * cnre + are * cnim) >> wp) | |
| n += 2 | |
| sre = (sre << 1) | |
| sim = (sim << 1) | |
| sre = ctx.ldexp(sre, -wp) | |
| sim = ctx.ldexp(sim, -wp) | |
| s = ctx.mpc(sre, sim) | |
| s *= ctx.nthroot(q, 4) | |
| return s | |
| def _djacobi_theta2(ctx, z, q, nd): | |
| MIN = 2 | |
| extra1 = 10 | |
| extra2 = 20 | |
| if (not ctx._im(q)) and (not ctx._im(z)): | |
| wp = ctx.prec + extra1 | |
| x = ctx.to_fixed(ctx._re(q), wp) | |
| x2 = (x*x) >> wp | |
| a = b = x2 | |
| c1, s1 = ctx.cos_sin(ctx._re(z), prec=wp) | |
| cn = c1 = ctx.to_fixed(c1, wp) | |
| sn = s1 = ctx.to_fixed(s1, wp) | |
| c2 = (c1*c1 - s1*s1) >> wp | |
| s2 = (c1 * s1) >> (wp - 1) | |
| cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp | |
| if (nd&1): | |
| s = s1 + ((a * sn * 3**nd) >> wp) | |
| else: | |
| s = c1 + ((a * cn * 3**nd) >> wp) | |
| n = 2 | |
| while abs(a) > MIN: | |
| b = (b*x2) >> wp | |
| a = (a*b) >> wp | |
| cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp | |
| if nd&1: | |
| s += (a * sn * (2*n+1)**nd) >> wp | |
| else: | |
| s += (a * cn * (2*n+1)**nd) >> wp | |
| n += 1 | |
| s = -(s << 1) | |
| s = ctx.ldexp(s, -wp) | |
| # case z real, q complex | |
| elif not ctx._im(z): | |
| wp = ctx.prec + extra2 | |
| xre = ctx.to_fixed(ctx._re(q), wp) | |
| xim = ctx.to_fixed(ctx._im(q), wp) | |
| x2re = (xre*xre - xim*xim) >> wp | |
| x2im = (xre*xim) >> (wp - 1) | |
| are = bre = x2re | |
| aim = bim = x2im | |
| c1, s1 = ctx.cos_sin(ctx._re(z), prec=wp) | |
| cn = c1 = ctx.to_fixed(c1, wp) | |
| sn = s1 = ctx.to_fixed(s1, wp) | |
| c2 = (c1*c1 - s1*s1) >> wp | |
| s2 = (c1 * s1) >> (wp - 1) | |
| cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp | |
| if (nd&1): | |
| sre = s1 + ((are * sn * 3**nd) >> wp) | |
| sim = ((aim * sn * 3**nd) >> wp) | |
| else: | |
| sre = c1 + ((are * cn * 3**nd) >> wp) | |
| sim = ((aim * cn * 3**nd) >> wp) | |
| n = 5 | |
| while are**2 + aim**2 > MIN: | |
| bre, bim = (bre * x2re - bim * x2im) >> wp, \ | |
| (bre * x2im + bim * x2re) >> wp | |
| are, aim = (are * bre - aim * bim) >> wp, \ | |
| (are * bim + aim * bre) >> wp | |
| cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp | |
| if (nd&1): | |
| sre += ((are * sn * n**nd) >> wp) | |
| sim += ((aim * sn * n**nd) >> wp) | |
| else: | |
| sre += ((are * cn * n**nd) >> wp) | |
| sim += ((aim * cn * n**nd) >> wp) | |
| n += 2 | |
| sre = -(sre << 1) | |
| sim = -(sim << 1) | |
| sre = ctx.ldexp(sre, -wp) | |
| sim = ctx.ldexp(sim, -wp) | |
| s = ctx.mpc(sre, sim) | |
| #case z complex, q real | |
| elif not ctx._im(q): | |
| wp = ctx.prec + extra2 | |
| x = ctx.to_fixed(ctx._re(q), wp) | |
| x2 = (x*x) >> wp | |
| a = b = x2 | |
| prec0 = ctx.prec | |
| ctx.prec = wp | |
| c1, s1 = ctx.cos_sin(z) | |
| ctx.prec = prec0 | |
| cnre = c1re = ctx.to_fixed(ctx._re(c1), wp) | |
| cnim = c1im = ctx.to_fixed(ctx._im(c1), wp) | |
| snre = s1re = ctx.to_fixed(ctx._re(s1), wp) | |
| snim = s1im = ctx.to_fixed(ctx._im(s1), wp) | |
| #c2 = (c1*c1 - s1*s1) >> wp | |
| c2re = (c1re*c1re - c1im*c1im - s1re*s1re + s1im*s1im) >> wp | |
| c2im = (c1re*c1im - s1re*s1im) >> (wp - 1) | |
| #s2 = (c1 * s1) >> (wp - 1) | |
| s2re = (c1re*s1re - c1im*s1im) >> (wp - 1) | |
| s2im = (c1re*s1im + c1im*s1re) >> (wp - 1) | |
| #cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp | |
| t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp | |
| t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp | |
| t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp | |
| t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp | |
| cnre = t1 | |
| cnim = t2 | |
| snre = t3 | |
| snim = t4 | |
| if (nd&1): | |
| sre = s1re + ((a * snre * 3**nd) >> wp) | |
| sim = s1im + ((a * snim * 3**nd) >> wp) | |
| else: | |
| sre = c1re + ((a * cnre * 3**nd) >> wp) | |
| sim = c1im + ((a * cnim * 3**nd) >> wp) | |
| n = 5 | |
| while abs(a) > MIN: | |
| b = (b*x2) >> wp | |
| a = (a*b) >> wp | |
| t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp | |
| t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp | |
| t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp | |
| t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp | |
| cnre = t1 | |
| cnim = t2 | |
| snre = t3 | |
| snim = t4 | |
| if (nd&1): | |
| sre += ((a * snre * n**nd) >> wp) | |
| sim += ((a * snim * n**nd) >> wp) | |
| else: | |
| sre += ((a * cnre * n**nd) >> wp) | |
| sim += ((a * cnim * n**nd) >> wp) | |
| n += 2 | |
| sre = -(sre << 1) | |
| sim = -(sim << 1) | |
| sre = ctx.ldexp(sre, -wp) | |
| sim = ctx.ldexp(sim, -wp) | |
| s = ctx.mpc(sre, sim) | |
| # case z and q complex | |
| else: | |
| wp = ctx.prec + extra2 | |
| xre = ctx.to_fixed(ctx._re(q), wp) | |
| xim = ctx.to_fixed(ctx._im(q), wp) | |
| x2re = (xre*xre - xim*xim) >> wp | |
| x2im = (xre*xim) >> (wp - 1) | |
| are = bre = x2re | |
| aim = bim = x2im | |
| prec0 = ctx.prec | |
| ctx.prec = wp | |
| # cos(2*z), sin(2*z) with z complex | |
| c1, s1 = ctx.cos_sin(z) | |
| ctx.prec = prec0 | |
| cnre = c1re = ctx.to_fixed(ctx._re(c1), wp) | |
| cnim = c1im = ctx.to_fixed(ctx._im(c1), wp) | |
| snre = s1re = ctx.to_fixed(ctx._re(s1), wp) | |
| snim = s1im = ctx.to_fixed(ctx._im(s1), wp) | |
| c2re = (c1re*c1re - c1im*c1im - s1re*s1re + s1im*s1im) >> wp | |
| c2im = (c1re*c1im - s1re*s1im) >> (wp - 1) | |
| s2re = (c1re*s1re - c1im*s1im) >> (wp - 1) | |
| s2im = (c1re*s1im + c1im*s1re) >> (wp - 1) | |
| t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp | |
| t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp | |
| t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp | |
| t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp | |
| cnre = t1 | |
| cnim = t2 | |
| snre = t3 | |
| snim = t4 | |
| if (nd&1): | |
| sre = s1re + (((are * snre - aim * snim) * 3**nd) >> wp) | |
| sim = s1im + (((are * snim + aim * snre)* 3**nd) >> wp) | |
| else: | |
| sre = c1re + (((are * cnre - aim * cnim) * 3**nd) >> wp) | |
| sim = c1im + (((are * cnim + aim * cnre)* 3**nd) >> wp) | |
| n = 5 | |
| while are**2 + aim**2 > MIN: | |
| bre, bim = (bre * x2re - bim * x2im) >> wp, \ | |
| (bre * x2im + bim * x2re) >> wp | |
| are, aim = (are * bre - aim * bim) >> wp, \ | |
| (are * bim + aim * bre) >> wp | |
| #cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp | |
| t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp | |
| t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp | |
| t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp | |
| t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp | |
| cnre = t1 | |
| cnim = t2 | |
| snre = t3 | |
| snim = t4 | |
| if (nd&1): | |
| sre += (((are * snre - aim * snim) * n**nd) >> wp) | |
| sim += (((aim * snre + are * snim) * n**nd) >> wp) | |
| else: | |
| sre += (((are * cnre - aim * cnim) * n**nd) >> wp) | |
| sim += (((aim * cnre + are * cnim) * n**nd) >> wp) | |
| n += 2 | |
| sre = -(sre << 1) | |
| sim = -(sim << 1) | |
| sre = ctx.ldexp(sre, -wp) | |
| sim = ctx.ldexp(sim, -wp) | |
| s = ctx.mpc(sre, sim) | |
| s *= ctx.nthroot(q, 4) | |
| if (nd&1): | |
| return (-1)**(nd//2) * s | |
| else: | |
| return (-1)**(1 + nd//2) * s | |
| def _jacobi_theta3(ctx, z, q): | |
| extra1 = 10 | |
| extra2 = 20 | |
| MIN = 2 | |
| if z == ctx.zero: | |
| if not ctx._im(q): | |
| wp = ctx.prec + extra1 | |
| x = ctx.to_fixed(ctx._re(q), wp) | |
| s = x | |
| a = b = x | |
| x2 = (x*x) >> wp | |
| while abs(a) > MIN: | |
| b = (b*x2) >> wp | |
| a = (a*b) >> wp | |
| s += a | |
| s = (1 << wp) + (s << 1) | |
| s = ctx.ldexp(s, -wp) | |
| return s | |
| else: | |
| wp = ctx.prec + extra1 | |
| xre = ctx.to_fixed(ctx._re(q), wp) | |
| xim = ctx.to_fixed(ctx._im(q), wp) | |
| x2re = (xre*xre - xim*xim) >> wp | |
| x2im = (xre*xim) >> (wp - 1) | |
| sre = are = bre = xre | |
| sim = aim = bim = xim | |
| while are**2 + aim**2 > MIN: | |
| bre, bim = (bre * x2re - bim * x2im) >> wp, \ | |
| (bre * x2im + bim * x2re) >> wp | |
| are, aim = (are * bre - aim * bim) >> wp, \ | |
| (are * bim + aim * bre) >> wp | |
| sre += are | |
| sim += aim | |
| sre = (1 << wp) + (sre << 1) | |
| sim = (sim << 1) | |
| sre = ctx.ldexp(sre, -wp) | |
| sim = ctx.ldexp(sim, -wp) | |
| s = ctx.mpc(sre, sim) | |
| return s | |
| else: | |
| if (not ctx._im(q)) and (not ctx._im(z)): | |
| s = 0 | |
| wp = ctx.prec + extra1 | |
| x = ctx.to_fixed(ctx._re(q), wp) | |
| a = b = x | |
| x2 = (x*x) >> wp | |
| c1, s1 = ctx.cos_sin(ctx._re(z)*2, prec=wp) | |
| c1 = ctx.to_fixed(c1, wp) | |
| s1 = ctx.to_fixed(s1, wp) | |
| cn = c1 | |
| sn = s1 | |
| s += (a * cn) >> wp | |
| while abs(a) > MIN: | |
| b = (b*x2) >> wp | |
| a = (a*b) >> wp | |
| cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp | |
| s += (a * cn) >> wp | |
| s = (1 << wp) + (s << 1) | |
| s = ctx.ldexp(s, -wp) | |
| return s | |
| # case z real, q complex | |
| elif not ctx._im(z): | |
| wp = ctx.prec + extra2 | |
| xre = ctx.to_fixed(ctx._re(q), wp) | |
| xim = ctx.to_fixed(ctx._im(q), wp) | |
| x2re = (xre*xre - xim*xim) >> wp | |
| x2im = (xre*xim) >> (wp - 1) | |
| are = bre = xre | |
| aim = bim = xim | |
| c1, s1 = ctx.cos_sin(ctx._re(z)*2, prec=wp) | |
| c1 = ctx.to_fixed(c1, wp) | |
| s1 = ctx.to_fixed(s1, wp) | |
| cn = c1 | |
| sn = s1 | |
| sre = (are * cn) >> wp | |
| sim = (aim * cn) >> wp | |
| while are**2 + aim**2 > MIN: | |
| bre, bim = (bre * x2re - bim * x2im) >> wp, \ | |
| (bre * x2im + bim * x2re) >> wp | |
| are, aim = (are * bre - aim * bim) >> wp, \ | |
| (are * bim + aim * bre) >> wp | |
| cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp | |
| sre += (are * cn) >> wp | |
| sim += (aim * cn) >> wp | |
| sre = (1 << wp) + (sre << 1) | |
| sim = (sim << 1) | |
| sre = ctx.ldexp(sre, -wp) | |
| sim = ctx.ldexp(sim, -wp) | |
| s = ctx.mpc(sre, sim) | |
| return s | |
| #case z complex, q real | |
| elif not ctx._im(q): | |
| wp = ctx.prec + extra2 | |
| x = ctx.to_fixed(ctx._re(q), wp) | |
| a = b = x | |
| x2 = (x*x) >> wp | |
| prec0 = ctx.prec | |
| ctx.prec = wp | |
| c1, s1 = ctx.cos_sin(2*z) | |
| ctx.prec = prec0 | |
| cnre = c1re = ctx.to_fixed(ctx._re(c1), wp) | |
| cnim = c1im = ctx.to_fixed(ctx._im(c1), wp) | |
| snre = s1re = ctx.to_fixed(ctx._re(s1), wp) | |
| snim = s1im = ctx.to_fixed(ctx._im(s1), wp) | |
| sre = (a * cnre) >> wp | |
| sim = (a * cnim) >> wp | |
| while abs(a) > MIN: | |
| b = (b*x2) >> wp | |
| a = (a*b) >> wp | |
| t1 = (cnre*c1re - cnim*c1im - snre*s1re + snim*s1im) >> wp | |
| t2 = (cnre*c1im + cnim*c1re - snre*s1im - snim*s1re) >> wp | |
| t3 = (snre*c1re - snim*c1im + cnre*s1re - cnim*s1im) >> wp | |
| t4 = (snre*c1im + snim*c1re + cnre*s1im + cnim*s1re) >> wp | |
| cnre = t1 | |
| cnim = t2 | |
| snre = t3 | |
| snim = t4 | |
| sre += (a * cnre) >> wp | |
| sim += (a * cnim) >> wp | |
| sre = (1 << wp) + (sre << 1) | |
| sim = (sim << 1) | |
| sre = ctx.ldexp(sre, -wp) | |
| sim = ctx.ldexp(sim, -wp) | |
| s = ctx.mpc(sre, sim) | |
| return s | |
| # case z and q complex | |
| else: | |
| wp = ctx.prec + extra2 | |
| xre = ctx.to_fixed(ctx._re(q), wp) | |
| xim = ctx.to_fixed(ctx._im(q), wp) | |
| x2re = (xre*xre - xim*xim) >> wp | |
| x2im = (xre*xim) >> (wp - 1) | |
| are = bre = xre | |
| aim = bim = xim | |
| prec0 = ctx.prec | |
| ctx.prec = wp | |
| # cos(2*z), sin(2*z) with z complex | |
| c1, s1 = ctx.cos_sin(2*z) | |
| ctx.prec = prec0 | |
| cnre = c1re = ctx.to_fixed(ctx._re(c1), wp) | |
| cnim = c1im = ctx.to_fixed(ctx._im(c1), wp) | |
| snre = s1re = ctx.to_fixed(ctx._re(s1), wp) | |
| snim = s1im = ctx.to_fixed(ctx._im(s1), wp) | |
| sre = (are * cnre - aim * cnim) >> wp | |
| sim = (aim * cnre + are * cnim) >> wp | |
| while are**2 + aim**2 > MIN: | |
| bre, bim = (bre * x2re - bim * x2im) >> wp, \ | |
| (bre * x2im + bim * x2re) >> wp | |
| are, aim = (are * bre - aim * bim) >> wp, \ | |
| (are * bim + aim * bre) >> wp | |
| t1 = (cnre*c1re - cnim*c1im - snre*s1re + snim*s1im) >> wp | |
| t2 = (cnre*c1im + cnim*c1re - snre*s1im - snim*s1re) >> wp | |
| t3 = (snre*c1re - snim*c1im + cnre*s1re - cnim*s1im) >> wp | |
| t4 = (snre*c1im + snim*c1re + cnre*s1im + cnim*s1re) >> wp | |
| cnre = t1 | |
| cnim = t2 | |
| snre = t3 | |
| snim = t4 | |
| sre += (are * cnre - aim * cnim) >> wp | |
| sim += (aim * cnre + are * cnim) >> wp | |
| sre = (1 << wp) + (sre << 1) | |
| sim = (sim << 1) | |
| sre = ctx.ldexp(sre, -wp) | |
| sim = ctx.ldexp(sim, -wp) | |
| s = ctx.mpc(sre, sim) | |
| return s | |
| def _djacobi_theta3(ctx, z, q, nd): | |
| """nd=1,2,3 order of the derivative with respect to z""" | |
| MIN = 2 | |
| extra1 = 10 | |
| extra2 = 20 | |
| if (not ctx._im(q)) and (not ctx._im(z)): | |
| s = 0 | |
| wp = ctx.prec + extra1 | |
| x = ctx.to_fixed(ctx._re(q), wp) | |
| a = b = x | |
| x2 = (x*x) >> wp | |
| c1, s1 = ctx.cos_sin(ctx._re(z)*2, prec=wp) | |
| c1 = ctx.to_fixed(c1, wp) | |
| s1 = ctx.to_fixed(s1, wp) | |
| cn = c1 | |
| sn = s1 | |
| if (nd&1): | |
| s += (a * sn) >> wp | |
| else: | |
| s += (a * cn) >> wp | |
| n = 2 | |
| while abs(a) > MIN: | |
| b = (b*x2) >> wp | |
| a = (a*b) >> wp | |
| cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp | |
| if nd&1: | |
| s += (a * sn * n**nd) >> wp | |
| else: | |
| s += (a * cn * n**nd) >> wp | |
| n += 1 | |
| s = -(s << (nd+1)) | |
| s = ctx.ldexp(s, -wp) | |
| # case z real, q complex | |
| elif not ctx._im(z): | |
| wp = ctx.prec + extra2 | |
| xre = ctx.to_fixed(ctx._re(q), wp) | |
| xim = ctx.to_fixed(ctx._im(q), wp) | |
| x2re = (xre*xre - xim*xim) >> wp | |
| x2im = (xre*xim) >> (wp - 1) | |
| are = bre = xre | |
| aim = bim = xim | |
| c1, s1 = ctx.cos_sin(ctx._re(z)*2, prec=wp) | |
| c1 = ctx.to_fixed(c1, wp) | |
| s1 = ctx.to_fixed(s1, wp) | |
| cn = c1 | |
| sn = s1 | |
| if (nd&1): | |
| sre = (are * sn) >> wp | |
| sim = (aim * sn) >> wp | |
| else: | |
| sre = (are * cn) >> wp | |
| sim = (aim * cn) >> wp | |
| n = 2 | |
| while are**2 + aim**2 > MIN: | |
| bre, bim = (bre * x2re - bim * x2im) >> wp, \ | |
| (bre * x2im + bim * x2re) >> wp | |
| are, aim = (are * bre - aim * bim) >> wp, \ | |
| (are * bim + aim * bre) >> wp | |
| cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp | |
| if nd&1: | |
| sre += (are * sn * n**nd) >> wp | |
| sim += (aim * sn * n**nd) >> wp | |
| else: | |
| sre += (are * cn * n**nd) >> wp | |
| sim += (aim * cn * n**nd) >> wp | |
| n += 1 | |
| sre = -(sre << (nd+1)) | |
| sim = -(sim << (nd+1)) | |
| sre = ctx.ldexp(sre, -wp) | |
| sim = ctx.ldexp(sim, -wp) | |
| s = ctx.mpc(sre, sim) | |
| #case z complex, q real | |
| elif not ctx._im(q): | |
| wp = ctx.prec + extra2 | |
| x = ctx.to_fixed(ctx._re(q), wp) | |
| a = b = x | |
| x2 = (x*x) >> wp | |
| prec0 = ctx.prec | |
| ctx.prec = wp | |
| c1, s1 = ctx.cos_sin(2*z) | |
| ctx.prec = prec0 | |
| cnre = c1re = ctx.to_fixed(ctx._re(c1), wp) | |
| cnim = c1im = ctx.to_fixed(ctx._im(c1), wp) | |
| snre = s1re = ctx.to_fixed(ctx._re(s1), wp) | |
| snim = s1im = ctx.to_fixed(ctx._im(s1), wp) | |
| if (nd&1): | |
| sre = (a * snre) >> wp | |
| sim = (a * snim) >> wp | |
| else: | |
| sre = (a * cnre) >> wp | |
| sim = (a * cnim) >> wp | |
| n = 2 | |
| while abs(a) > MIN: | |
| b = (b*x2) >> wp | |
| a = (a*b) >> wp | |
| t1 = (cnre*c1re - cnim*c1im - snre*s1re + snim*s1im) >> wp | |
| t2 = (cnre*c1im + cnim*c1re - snre*s1im - snim*s1re) >> wp | |
| t3 = (snre*c1re - snim*c1im + cnre*s1re - cnim*s1im) >> wp | |
| t4 = (snre*c1im + snim*c1re + cnre*s1im + cnim*s1re) >> wp | |
| cnre = t1 | |
| cnim = t2 | |
| snre = t3 | |
| snim = t4 | |
| if (nd&1): | |
| sre += (a * snre * n**nd) >> wp | |
| sim += (a * snim * n**nd) >> wp | |
| else: | |
| sre += (a * cnre * n**nd) >> wp | |
| sim += (a * cnim * n**nd) >> wp | |
| n += 1 | |
| sre = -(sre << (nd+1)) | |
| sim = -(sim << (nd+1)) | |
| sre = ctx.ldexp(sre, -wp) | |
| sim = ctx.ldexp(sim, -wp) | |
| s = ctx.mpc(sre, sim) | |
| # case z and q complex | |
| else: | |
| wp = ctx.prec + extra2 | |
| xre = ctx.to_fixed(ctx._re(q), wp) | |
| xim = ctx.to_fixed(ctx._im(q), wp) | |
| x2re = (xre*xre - xim*xim) >> wp | |
| x2im = (xre*xim) >> (wp - 1) | |
| are = bre = xre | |
| aim = bim = xim | |
| prec0 = ctx.prec | |
| ctx.prec = wp | |
| # cos(2*z), sin(2*z) with z complex | |
| c1, s1 = ctx.cos_sin(2*z) | |
| ctx.prec = prec0 | |
| cnre = c1re = ctx.to_fixed(ctx._re(c1), wp) | |
| cnim = c1im = ctx.to_fixed(ctx._im(c1), wp) | |
| snre = s1re = ctx.to_fixed(ctx._re(s1), wp) | |
| snim = s1im = ctx.to_fixed(ctx._im(s1), wp) | |
| if (nd&1): | |
| sre = (are * snre - aim * snim) >> wp | |
| sim = (aim * snre + are * snim) >> wp | |
| else: | |
| sre = (are * cnre - aim * cnim) >> wp | |
| sim = (aim * cnre + are * cnim) >> wp | |
| n = 2 | |
| while are**2 + aim**2 > MIN: | |
| bre, bim = (bre * x2re - bim * x2im) >> wp, \ | |
| (bre * x2im + bim * x2re) >> wp | |
| are, aim = (are * bre - aim * bim) >> wp, \ | |
| (are * bim + aim * bre) >> wp | |
| t1 = (cnre*c1re - cnim*c1im - snre*s1re + snim*s1im) >> wp | |
| t2 = (cnre*c1im + cnim*c1re - snre*s1im - snim*s1re) >> wp | |
| t3 = (snre*c1re - snim*c1im + cnre*s1re - cnim*s1im) >> wp | |
| t4 = (snre*c1im + snim*c1re + cnre*s1im + cnim*s1re) >> wp | |
| cnre = t1 | |
| cnim = t2 | |
| snre = t3 | |
| snim = t4 | |
| if(nd&1): | |
| sre += ((are * snre - aim * snim) * n**nd) >> wp | |
| sim += ((aim * snre + are * snim) * n**nd) >> wp | |
| else: | |
| sre += ((are * cnre - aim * cnim) * n**nd) >> wp | |
| sim += ((aim * cnre + are * cnim) * n**nd) >> wp | |
| n += 1 | |
| sre = -(sre << (nd+1)) | |
| sim = -(sim << (nd+1)) | |
| sre = ctx.ldexp(sre, -wp) | |
| sim = ctx.ldexp(sim, -wp) | |
| s = ctx.mpc(sre, sim) | |
| if (nd&1): | |
| return (-1)**(nd//2) * s | |
| else: | |
| return (-1)**(1 + nd//2) * s | |
| def _jacobi_theta2a(ctx, z, q): | |
| """ | |
| case ctx._im(z) != 0 | |
| theta(2, z, q) = | |
| q**1/4 * Sum(q**(n*n + n) * exp(j*(2*n + 1)*z), n=-inf, inf) | |
| max term for minimum (2*n+1)*log(q).real - 2* ctx._im(z) | |
| n0 = int(ctx._im(z)/log(q).real - 1/2) | |
| theta(2, z, q) = | |
| q**1/4 * Sum(q**(n*n + n) * exp(j*(2*n + 1)*z), n=n0, inf) + | |
| q**1/4 * Sum(q**(n*n + n) * exp(j*(2*n + 1)*z), n, n0-1, -inf) | |
| """ | |
| n = n0 = int(ctx._im(z)/ctx._re(ctx.log(q)) - 1/2) | |
| e2 = ctx.expj(2*z) | |
| e = e0 = ctx.expj((2*n+1)*z) | |
| a = q**(n*n + n) | |
| # leading term | |
| term = a * e | |
| s = term | |
| eps1 = ctx.eps*abs(term) | |
| while 1: | |
| n += 1 | |
| e = e * e2 | |
| term = q**(n*n + n) * e | |
| if abs(term) < eps1: | |
| break | |
| s += term | |
| e = e0 | |
| e2 = ctx.expj(-2*z) | |
| n = n0 | |
| while 1: | |
| n -= 1 | |
| e = e * e2 | |
| term = q**(n*n + n) * e | |
| if abs(term) < eps1: | |
| break | |
| s += term | |
| s = s * ctx.nthroot(q, 4) | |
| return s | |
| def _jacobi_theta3a(ctx, z, q): | |
| """ | |
| case ctx._im(z) != 0 | |
| theta3(z, q) = Sum(q**(n*n) * exp(j*2*n*z), n, -inf, inf) | |
| max term for n*abs(log(q).real) + ctx._im(z) ~= 0 | |
| n0 = int(- ctx._im(z)/abs(log(q).real)) | |
| """ | |
| n = n0 = int(-ctx._im(z)/abs(ctx._re(ctx.log(q)))) | |
| e2 = ctx.expj(2*z) | |
| e = e0 = ctx.expj(2*n*z) | |
| s = term = q**(n*n) * e | |
| eps1 = ctx.eps*abs(term) | |
| while 1: | |
| n += 1 | |
| e = e * e2 | |
| term = q**(n*n) * e | |
| if abs(term) < eps1: | |
| break | |
| s += term | |
| e = e0 | |
| e2 = ctx.expj(-2*z) | |
| n = n0 | |
| while 1: | |
| n -= 1 | |
| e = e * e2 | |
| term = q**(n*n) * e | |
| if abs(term) < eps1: | |
| break | |
| s += term | |
| return s | |
| def _djacobi_theta2a(ctx, z, q, nd): | |
| """ | |
| case ctx._im(z) != 0 | |
| dtheta(2, z, q, nd) = | |
| j* q**1/4 * Sum(q**(n*n + n) * (2*n+1)*exp(j*(2*n + 1)*z), n=-inf, inf) | |
| max term for (2*n0+1)*log(q).real - 2* ctx._im(z) ~= 0 | |
| n0 = int(ctx._im(z)/log(q).real - 1/2) | |
| """ | |
| n = n0 = int(ctx._im(z)/ctx._re(ctx.log(q)) - 1/2) | |
| e2 = ctx.expj(2*z) | |
| e = e0 = ctx.expj((2*n + 1)*z) | |
| a = q**(n*n + n) | |
| # leading term | |
| term = (2*n+1)**nd * a * e | |
| s = term | |
| eps1 = ctx.eps*abs(term) | |
| while 1: | |
| n += 1 | |
| e = e * e2 | |
| term = (2*n+1)**nd * q**(n*n + n) * e | |
| if abs(term) < eps1: | |
| break | |
| s += term | |
| e = e0 | |
| e2 = ctx.expj(-2*z) | |
| n = n0 | |
| while 1: | |
| n -= 1 | |
| e = e * e2 | |
| term = (2*n+1)**nd * q**(n*n + n) * e | |
| if abs(term) < eps1: | |
| break | |
| s += term | |
| return ctx.j**nd * s * ctx.nthroot(q, 4) | |
| def _djacobi_theta3a(ctx, z, q, nd): | |
| """ | |
| case ctx._im(z) != 0 | |
| djtheta3(z, q, nd) = (2*j)**nd * | |
| Sum(q**(n*n) * n**nd * exp(j*2*n*z), n, -inf, inf) | |
| max term for minimum n*abs(log(q).real) + ctx._im(z) | |
| """ | |
| n = n0 = int(-ctx._im(z)/abs(ctx._re(ctx.log(q)))) | |
| e2 = ctx.expj(2*z) | |
| e = e0 = ctx.expj(2*n*z) | |
| a = q**(n*n) * e | |
| s = term = n**nd * a | |
| if n != 0: | |
| eps1 = ctx.eps*abs(term) | |
| else: | |
| eps1 = ctx.eps*abs(a) | |
| while 1: | |
| n += 1 | |
| e = e * e2 | |
| a = q**(n*n) * e | |
| term = n**nd * a | |
| if n != 0: | |
| aterm = abs(term) | |
| else: | |
| aterm = abs(a) | |
| if aterm < eps1: | |
| break | |
| s += term | |
| e = e0 | |
| e2 = ctx.expj(-2*z) | |
| n = n0 | |
| while 1: | |
| n -= 1 | |
| e = e * e2 | |
| a = q**(n*n) * e | |
| term = n**nd * a | |
| if n != 0: | |
| aterm = abs(term) | |
| else: | |
| aterm = abs(a) | |
| if aterm < eps1: | |
| break | |
| s += term | |
| return (2*ctx.j)**nd * s | |
| def jtheta(ctx, n, z, q, derivative=0): | |
| if derivative: | |
| return ctx._djtheta(n, z, q, derivative) | |
| z = ctx.convert(z) | |
| q = ctx.convert(q) | |
| # Implementation note | |
| # If ctx._im(z) is close to zero, _jacobi_theta2 and _jacobi_theta3 | |
| # are used, | |
| # which compute the series starting from n=0 using fixed precision | |
| # numbers; | |
| # otherwise _jacobi_theta2a and _jacobi_theta3a are used, which compute | |
| # the series starting from n=n0, which is the largest term. | |
| # TODO: write _jacobi_theta2a and _jacobi_theta3a using fixed-point | |
| if abs(q) > ctx.THETA_Q_LIM: | |
| raise ValueError('abs(q) > THETA_Q_LIM = %f' % ctx.THETA_Q_LIM) | |
| extra = 10 | |
| if z: | |
| M = ctx.mag(z) | |
| if M > 5 or (n == 1 and M < -5): | |
| extra += 2*abs(M) | |
| cz = 0.5 | |
| extra2 = 50 | |
| prec0 = ctx.prec | |
| try: | |
| ctx.prec += extra | |
| if n == 1: | |
| if ctx._im(z): | |
| if abs(ctx._im(z)) < cz * abs(ctx._re(ctx.log(q))): | |
| ctx.dps += extra2 | |
| res = ctx._jacobi_theta2(z - ctx.pi/2, q) | |
| else: | |
| ctx.dps += 10 | |
| res = ctx._jacobi_theta2a(z - ctx.pi/2, q) | |
| else: | |
| res = ctx._jacobi_theta2(z - ctx.pi/2, q) | |
| elif n == 2: | |
| if ctx._im(z): | |
| if abs(ctx._im(z)) < cz * abs(ctx._re(ctx.log(q))): | |
| ctx.dps += extra2 | |
| res = ctx._jacobi_theta2(z, q) | |
| else: | |
| ctx.dps += 10 | |
| res = ctx._jacobi_theta2a(z, q) | |
| else: | |
| res = ctx._jacobi_theta2(z, q) | |
| elif n == 3: | |
| if ctx._im(z): | |
| if abs(ctx._im(z)) < cz * abs(ctx._re(ctx.log(q))): | |
| ctx.dps += extra2 | |
| res = ctx._jacobi_theta3(z, q) | |
| else: | |
| ctx.dps += 10 | |
| res = ctx._jacobi_theta3a(z, q) | |
| else: | |
| res = ctx._jacobi_theta3(z, q) | |
| elif n == 4: | |
| if ctx._im(z): | |
| if abs(ctx._im(z)) < cz * abs(ctx._re(ctx.log(q))): | |
| ctx.dps += extra2 | |
| res = ctx._jacobi_theta3(z, -q) | |
| else: | |
| ctx.dps += 10 | |
| res = ctx._jacobi_theta3a(z, -q) | |
| else: | |
| res = ctx._jacobi_theta3(z, -q) | |
| else: | |
| raise ValueError | |
| finally: | |
| ctx.prec = prec0 | |
| return res | |
| def _djtheta(ctx, n, z, q, derivative=1): | |
| z = ctx.convert(z) | |
| q = ctx.convert(q) | |
| nd = int(derivative) | |
| if abs(q) > ctx.THETA_Q_LIM: | |
| raise ValueError('abs(q) > THETA_Q_LIM = %f' % ctx.THETA_Q_LIM) | |
| extra = 10 + ctx.prec * nd // 10 | |
| if z: | |
| M = ctx.mag(z) | |
| if M > 5 or (n != 1 and M < -5): | |
| extra += 2*abs(M) | |
| cz = 0.5 | |
| extra2 = 50 | |
| prec0 = ctx.prec | |
| try: | |
| ctx.prec += extra | |
| if n == 1: | |
| if ctx._im(z): | |
| if abs(ctx._im(z)) < cz * abs(ctx._re(ctx.log(q))): | |
| ctx.dps += extra2 | |
| res = ctx._djacobi_theta2(z - ctx.pi/2, q, nd) | |
| else: | |
| ctx.dps += 10 | |
| res = ctx._djacobi_theta2a(z - ctx.pi/2, q, nd) | |
| else: | |
| res = ctx._djacobi_theta2(z - ctx.pi/2, q, nd) | |
| elif n == 2: | |
| if ctx._im(z): | |
| if abs(ctx._im(z)) < cz * abs(ctx._re(ctx.log(q))): | |
| ctx.dps += extra2 | |
| res = ctx._djacobi_theta2(z, q, nd) | |
| else: | |
| ctx.dps += 10 | |
| res = ctx._djacobi_theta2a(z, q, nd) | |
| else: | |
| res = ctx._djacobi_theta2(z, q, nd) | |
| elif n == 3: | |
| if ctx._im(z): | |
| if abs(ctx._im(z)) < cz * abs(ctx._re(ctx.log(q))): | |
| ctx.dps += extra2 | |
| res = ctx._djacobi_theta3(z, q, nd) | |
| else: | |
| ctx.dps += 10 | |
| res = ctx._djacobi_theta3a(z, q, nd) | |
| else: | |
| res = ctx._djacobi_theta3(z, q, nd) | |
| elif n == 4: | |
| if ctx._im(z): | |
| if abs(ctx._im(z)) < cz * abs(ctx._re(ctx.log(q))): | |
| ctx.dps += extra2 | |
| res = ctx._djacobi_theta3(z, -q, nd) | |
| else: | |
| ctx.dps += 10 | |
| res = ctx._djacobi_theta3a(z, -q, nd) | |
| else: | |
| res = ctx._djacobi_theta3(z, -q, nd) | |
| else: | |
| raise ValueError | |
| finally: | |
| ctx.prec = prec0 | |
| return +res | |