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836 lines
26 KiB
836 lines
26 KiB
5 months ago
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"""
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Low-level functions for complex arithmetic.
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"""
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import sys
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from .backend import MPZ, MPZ_ZERO, MPZ_ONE, MPZ_TWO, BACKEND
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from .libmpf import (\
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round_floor, round_ceiling, round_down, round_up,
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round_nearest, round_fast, bitcount,
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bctable, normalize, normalize1, reciprocal_rnd, rshift, lshift, giant_steps,
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negative_rnd,
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to_str, to_fixed, from_man_exp, from_float, to_float, from_int, to_int,
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fzero, fone, ftwo, fhalf, finf, fninf, fnan, fnone,
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mpf_abs, mpf_pos, mpf_neg, mpf_add, mpf_sub, mpf_mul,
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mpf_div, mpf_mul_int, mpf_shift, mpf_sqrt, mpf_hypot,
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mpf_rdiv_int, mpf_floor, mpf_ceil, mpf_nint, mpf_frac,
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mpf_sign, mpf_hash,
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ComplexResult
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)
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from .libelefun import (\
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mpf_pi, mpf_exp, mpf_log, mpf_cos_sin, mpf_cosh_sinh, mpf_tan, mpf_pow_int,
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mpf_log_hypot,
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mpf_cos_sin_pi, mpf_phi,
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mpf_cos, mpf_sin, mpf_cos_pi, mpf_sin_pi,
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mpf_atan, mpf_atan2, mpf_cosh, mpf_sinh, mpf_tanh,
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mpf_asin, mpf_acos, mpf_acosh, mpf_nthroot, mpf_fibonacci
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)
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# An mpc value is a (real, imag) tuple
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mpc_one = fone, fzero
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mpc_zero = fzero, fzero
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mpc_two = ftwo, fzero
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mpc_half = (fhalf, fzero)
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_infs = (finf, fninf)
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_infs_nan = (finf, fninf, fnan)
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def mpc_is_inf(z):
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"""Check if either real or imaginary part is infinite"""
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re, im = z
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if re in _infs: return True
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if im in _infs: return True
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return False
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def mpc_is_infnan(z):
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"""Check if either real or imaginary part is infinite or nan"""
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re, im = z
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if re in _infs_nan: return True
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if im in _infs_nan: return True
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return False
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def mpc_to_str(z, dps, **kwargs):
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re, im = z
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rs = to_str(re, dps)
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if im[0]:
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return rs + " - " + to_str(mpf_neg(im), dps, **kwargs) + "j"
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else:
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return rs + " + " + to_str(im, dps, **kwargs) + "j"
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def mpc_to_complex(z, strict=False, rnd=round_fast):
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re, im = z
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return complex(to_float(re, strict, rnd), to_float(im, strict, rnd))
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def mpc_hash(z):
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if sys.version_info >= (3, 2):
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re, im = z
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h = mpf_hash(re) + sys.hash_info.imag * mpf_hash(im)
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# Need to reduce either module 2^32 or 2^64
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h = h % (2**sys.hash_info.width)
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return int(h)
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else:
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try:
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return hash(mpc_to_complex(z, strict=True))
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except OverflowError:
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return hash(z)
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def mpc_conjugate(z, prec, rnd=round_fast):
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re, im = z
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return re, mpf_neg(im, prec, rnd)
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def mpc_is_nonzero(z):
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return z != mpc_zero
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def mpc_add(z, w, prec, rnd=round_fast):
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a, b = z
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c, d = w
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return mpf_add(a, c, prec, rnd), mpf_add(b, d, prec, rnd)
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def mpc_add_mpf(z, x, prec, rnd=round_fast):
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a, b = z
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return mpf_add(a, x, prec, rnd), b
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def mpc_sub(z, w, prec=0, rnd=round_fast):
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a, b = z
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c, d = w
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return mpf_sub(a, c, prec, rnd), mpf_sub(b, d, prec, rnd)
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def mpc_sub_mpf(z, p, prec=0, rnd=round_fast):
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a, b = z
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return mpf_sub(a, p, prec, rnd), b
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def mpc_pos(z, prec, rnd=round_fast):
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a, b = z
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return mpf_pos(a, prec, rnd), mpf_pos(b, prec, rnd)
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def mpc_neg(z, prec=None, rnd=round_fast):
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a, b = z
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return mpf_neg(a, prec, rnd), mpf_neg(b, prec, rnd)
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def mpc_shift(z, n):
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a, b = z
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return mpf_shift(a, n), mpf_shift(b, n)
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def mpc_abs(z, prec, rnd=round_fast):
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"""Absolute value of a complex number, |a+bi|.
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Returns an mpf value."""
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a, b = z
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return mpf_hypot(a, b, prec, rnd)
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def mpc_arg(z, prec, rnd=round_fast):
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"""Argument of a complex number. Returns an mpf value."""
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a, b = z
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return mpf_atan2(b, a, prec, rnd)
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def mpc_floor(z, prec, rnd=round_fast):
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a, b = z
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return mpf_floor(a, prec, rnd), mpf_floor(b, prec, rnd)
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def mpc_ceil(z, prec, rnd=round_fast):
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a, b = z
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return mpf_ceil(a, prec, rnd), mpf_ceil(b, prec, rnd)
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def mpc_nint(z, prec, rnd=round_fast):
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a, b = z
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return mpf_nint(a, prec, rnd), mpf_nint(b, prec, rnd)
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def mpc_frac(z, prec, rnd=round_fast):
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a, b = z
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return mpf_frac(a, prec, rnd), mpf_frac(b, prec, rnd)
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def mpc_mul(z, w, prec, rnd=round_fast):
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"""
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Complex multiplication.
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Returns the real and imaginary part of (a+bi)*(c+di), rounded to
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the specified precision. The rounding mode applies to the real and
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imaginary parts separately.
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"""
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a, b = z
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c, d = w
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p = mpf_mul(a, c)
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q = mpf_mul(b, d)
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r = mpf_mul(a, d)
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s = mpf_mul(b, c)
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re = mpf_sub(p, q, prec, rnd)
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im = mpf_add(r, s, prec, rnd)
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return re, im
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def mpc_square(z, prec, rnd=round_fast):
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# (a+b*I)**2 == a**2 - b**2 + 2*I*a*b
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a, b = z
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p = mpf_mul(a,a)
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q = mpf_mul(b,b)
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r = mpf_mul(a,b, prec, rnd)
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re = mpf_sub(p, q, prec, rnd)
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im = mpf_shift(r, 1)
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return re, im
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def mpc_mul_mpf(z, p, prec, rnd=round_fast):
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a, b = z
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re = mpf_mul(a, p, prec, rnd)
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im = mpf_mul(b, p, prec, rnd)
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return re, im
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def mpc_mul_imag_mpf(z, x, prec, rnd=round_fast):
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"""
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Multiply the mpc value z by I*x where x is an mpf value.
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"""
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a, b = z
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re = mpf_neg(mpf_mul(b, x, prec, rnd))
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im = mpf_mul(a, x, prec, rnd)
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return re, im
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def mpc_mul_int(z, n, prec, rnd=round_fast):
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a, b = z
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re = mpf_mul_int(a, n, prec, rnd)
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im = mpf_mul_int(b, n, prec, rnd)
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return re, im
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def mpc_div(z, w, prec, rnd=round_fast):
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a, b = z
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c, d = w
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wp = prec + 10
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# mag = c*c + d*d
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mag = mpf_add(mpf_mul(c, c), mpf_mul(d, d), wp)
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# (a*c+b*d)/mag, (b*c-a*d)/mag
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t = mpf_add(mpf_mul(a,c), mpf_mul(b,d), wp)
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u = mpf_sub(mpf_mul(b,c), mpf_mul(a,d), wp)
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return mpf_div(t,mag,prec,rnd), mpf_div(u,mag,prec,rnd)
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def mpc_div_mpf(z, p, prec, rnd=round_fast):
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"""Calculate z/p where p is real"""
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a, b = z
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re = mpf_div(a, p, prec, rnd)
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im = mpf_div(b, p, prec, rnd)
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return re, im
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def mpc_reciprocal(z, prec, rnd=round_fast):
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"""Calculate 1/z efficiently"""
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a, b = z
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m = mpf_add(mpf_mul(a,a),mpf_mul(b,b),prec+10)
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re = mpf_div(a, m, prec, rnd)
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im = mpf_neg(mpf_div(b, m, prec, rnd))
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return re, im
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def mpc_mpf_div(p, z, prec, rnd=round_fast):
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"""Calculate p/z where p is real efficiently"""
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a, b = z
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m = mpf_add(mpf_mul(a,a),mpf_mul(b,b), prec+10)
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re = mpf_div(mpf_mul(a,p), m, prec, rnd)
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im = mpf_div(mpf_neg(mpf_mul(b,p)), m, prec, rnd)
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return re, im
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def complex_int_pow(a, b, n):
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"""Complex integer power: computes (a+b*I)**n exactly for
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nonnegative n (a and b must be Python ints)."""
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wre = 1
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wim = 0
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while n:
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if n & 1:
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wre, wim = wre*a - wim*b, wim*a + wre*b
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n -= 1
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a, b = a*a - b*b, 2*a*b
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n //= 2
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return wre, wim
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def mpc_pow(z, w, prec, rnd=round_fast):
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if w[1] == fzero:
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return mpc_pow_mpf(z, w[0], prec, rnd)
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return mpc_exp(mpc_mul(mpc_log(z, prec+10), w, prec+10), prec, rnd)
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def mpc_pow_mpf(z, p, prec, rnd=round_fast):
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psign, pman, pexp, pbc = p
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if pexp >= 0:
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return mpc_pow_int(z, (-1)**psign * (pman<<pexp), prec, rnd)
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if pexp == -1:
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sqrtz = mpc_sqrt(z, prec+10)
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return mpc_pow_int(sqrtz, (-1)**psign * pman, prec, rnd)
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return mpc_exp(mpc_mul_mpf(mpc_log(z, prec+10), p, prec+10), prec, rnd)
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def mpc_pow_int(z, n, prec, rnd=round_fast):
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a, b = z
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if b == fzero:
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return mpf_pow_int(a, n, prec, rnd), fzero
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if a == fzero:
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v = mpf_pow_int(b, n, prec, rnd)
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n %= 4
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if n == 0:
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return v, fzero
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elif n == 1:
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return fzero, v
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elif n == 2:
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return mpf_neg(v), fzero
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elif n == 3:
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return fzero, mpf_neg(v)
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if n == 0: return mpc_one
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if n == 1: return mpc_pos(z, prec, rnd)
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if n == 2: return mpc_square(z, prec, rnd)
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if n == -1: return mpc_reciprocal(z, prec, rnd)
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if n < 0: return mpc_reciprocal(mpc_pow_int(z, -n, prec+4), prec, rnd)
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asign, aman, aexp, abc = a
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bsign, bman, bexp, bbc = b
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if asign: aman = -aman
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if bsign: bman = -bman
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de = aexp - bexp
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abs_de = abs(de)
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exact_size = n*(abs_de + max(abc, bbc))
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if exact_size < 10000:
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if de > 0:
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aman <<= de
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aexp = bexp
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else:
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bman <<= (-de)
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bexp = aexp
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re, im = complex_int_pow(aman, bman, n)
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re = from_man_exp(re, int(n*aexp), prec, rnd)
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im = from_man_exp(im, int(n*bexp), prec, rnd)
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return re, im
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return mpc_exp(mpc_mul_int(mpc_log(z, prec+10), n, prec+10), prec, rnd)
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def mpc_sqrt(z, prec, rnd=round_fast):
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"""Complex square root (principal branch).
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We have sqrt(a+bi) = sqrt((r+a)/2) + b/sqrt(2*(r+a))*i where
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r = abs(a+bi), when a+bi is not a negative real number."""
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a, b = z
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if b == fzero:
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if a == fzero:
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return (a, b)
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# When a+bi is a negative real number, we get a real sqrt times i
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if a[0]:
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im = mpf_sqrt(mpf_neg(a), prec, rnd)
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return (fzero, im)
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else:
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re = mpf_sqrt(a, prec, rnd)
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return (re, fzero)
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wp = prec+20
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if not a[0]: # case a positive
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t = mpf_add(mpc_abs((a, b), wp), a, wp) # t = abs(a+bi) + a
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u = mpf_shift(t, -1) # u = t/2
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re = mpf_sqrt(u, prec, rnd) # re = sqrt(u)
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v = mpf_shift(t, 1) # v = 2*t
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w = mpf_sqrt(v, wp) # w = sqrt(v)
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im = mpf_div(b, w, prec, rnd) # im = b / w
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else: # case a negative
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t = mpf_sub(mpc_abs((a, b), wp), a, wp) # t = abs(a+bi) - a
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u = mpf_shift(t, -1) # u = t/2
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im = mpf_sqrt(u, prec, rnd) # im = sqrt(u)
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v = mpf_shift(t, 1) # v = 2*t
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|
|
w = mpf_sqrt(v, wp) # w = sqrt(v)
|
||
|
|
re = mpf_div(b, w, prec, rnd) # re = b/w
|
||
|
|
if b[0]:
|
||
|
|
re = mpf_neg(re)
|
||
|
|
im = mpf_neg(im)
|
||
|
|
return re, im
|
||
|
|
|
||
|
|
def mpc_nthroot_fixed(a, b, n, prec):
|
||
|
|
# a, b signed integers at fixed precision prec
|
||
|
|
start = 50
|
||
|
|
a1 = int(rshift(a, prec - n*start))
|
||
|
|
b1 = int(rshift(b, prec - n*start))
|
||
|
|
try:
|
||
|
|
r = (a1 + 1j * b1)**(1.0/n)
|
||
|
|
re = r.real
|
||
|
|
im = r.imag
|
||
|
|
re = MPZ(int(re))
|
||
|
|
im = MPZ(int(im))
|
||
|
|
except OverflowError:
|
||
|
|
a1 = from_int(a1, start)
|
||
|
|
b1 = from_int(b1, start)
|
||
|
|
fn = from_int(n)
|
||
|
|
nth = mpf_rdiv_int(1, fn, start)
|
||
|
|
re, im = mpc_pow((a1, b1), (nth, fzero), start)
|
||
|
|
re = to_int(re)
|
||
|
|
im = to_int(im)
|
||
|
|
extra = 10
|
||
|
|
prevp = start
|
||
|
|
extra1 = n
|
||
|
|
for p in giant_steps(start, prec+extra):
|
||
|
|
# this is slow for large n, unlike int_pow_fixed
|
||
|
|
re2, im2 = complex_int_pow(re, im, n-1)
|
||
|
|
re2 = rshift(re2, (n-1)*prevp - p - extra1)
|
||
|
|
im2 = rshift(im2, (n-1)*prevp - p - extra1)
|
||
|
|
r4 = (re2*re2 + im2*im2) >> (p + extra1)
|
||
|
|
ap = rshift(a, prec - p)
|
||
|
|
bp = rshift(b, prec - p)
|
||
|
|
rec = (ap * re2 + bp * im2) >> p
|
||
|
|
imc = (-ap * im2 + bp * re2) >> p
|
||
|
|
reb = (rec << p) // r4
|
||
|
|
imb = (imc << p) // r4
|
||
|
|
re = (reb + (n-1)*lshift(re, p-prevp))//n
|
||
|
|
im = (imb + (n-1)*lshift(im, p-prevp))//n
|
||
|
|
prevp = p
|
||
|
|
return re, im
|
||
|
|
|
||
|
|
def mpc_nthroot(z, n, prec, rnd=round_fast):
|
||
|
|
"""
|
||
|
|
Complex n-th root.
|
||
|
|
|
||
|
|
Use Newton method as in the real case when it is faster,
|
||
|
|
otherwise use z**(1/n)
|
||
|
|
"""
|
||
|
|
a, b = z
|
||
|
|
if a[0] == 0 and b == fzero:
|
||
|
|
re = mpf_nthroot(a, n, prec, rnd)
|
||
|
|
return (re, fzero)
|
||
|
|
if n < 2:
|
||
|
|
if n == 0:
|
||
|
|
return mpc_one
|
||
|
|
if n == 1:
|
||
|
|
return mpc_pos((a, b), prec, rnd)
|
||
|
|
if n == -1:
|
||
|
|
return mpc_div(mpc_one, (a, b), prec, rnd)
|
||
|
|
inverse = mpc_nthroot((a, b), -n, prec+5, reciprocal_rnd[rnd])
|
||
|
|
return mpc_div(mpc_one, inverse, prec, rnd)
|
||
|
|
if n <= 20:
|
||
|
|
prec2 = int(1.2 * (prec + 10))
|
||
|
|
asign, aman, aexp, abc = a
|
||
|
|
bsign, bman, bexp, bbc = b
|
||
|
|
pf = mpc_abs((a,b), prec)
|
||
|
|
if pf[-2] + pf[-1] > -10 and pf[-2] + pf[-1] < prec:
|
||
|
|
af = to_fixed(a, prec2)
|
||
|
|
bf = to_fixed(b, prec2)
|
||
|
|
re, im = mpc_nthroot_fixed(af, bf, n, prec2)
|
||
|
|
extra = 10
|
||
|
|
re = from_man_exp(re, -prec2-extra, prec2, rnd)
|
||
|
|
im = from_man_exp(im, -prec2-extra, prec2, rnd)
|
||
|
|
return re, im
|
||
|
|
fn = from_int(n)
|
||
|
|
prec2 = prec+10 + 10
|
||
|
|
nth = mpf_rdiv_int(1, fn, prec2)
|
||
|
|
re, im = mpc_pow((a, b), (nth, fzero), prec2, rnd)
|
||
|
|
re = normalize(re[0], re[1], re[2], re[3], prec, rnd)
|
||
|
|
im = normalize(im[0], im[1], im[2], im[3], prec, rnd)
|
||
|
|
return re, im
|
||
|
|
|
||
|
|
def mpc_cbrt(z, prec, rnd=round_fast):
|
||
|
|
"""
|
||
|
|
Complex cubic root.
|
||
|
|
"""
|
||
|
|
return mpc_nthroot(z, 3, prec, rnd)
|
||
|
|
|
||
|
|
def mpc_exp(z, prec, rnd=round_fast):
|
||
|
|
"""
|
||
|
|
Complex exponential function.
|
||
|
|
|
||
|
|
We use the direct formula exp(a+bi) = exp(a) * (cos(b) + sin(b)*i)
|
||
|
|
for the computation. This formula is very nice because it is
|
||
|
|
pefectly stable; since we just do real multiplications, the only
|
||
|
|
numerical errors that can creep in are single-ulp rounding errors.
|
||
|
|
|
||
|
|
The formula is efficient since mpmath's real exp is quite fast and
|
||
|
|
since we can compute cos and sin simultaneously.
|
||
|
|
|
||
|
|
It is no problem if a and b are large; if the implementations of
|
||
|
|
exp/cos/sin are accurate and efficient for all real numbers, then
|
||
|
|
so is this function for all complex numbers.
|
||
|
|
"""
|
||
|
|
a, b = z
|
||
|
|
if a == fzero:
|
||
|
|
return mpf_cos_sin(b, prec, rnd)
|
||
|
|
if b == fzero:
|
||
|
|
return mpf_exp(a, prec, rnd), fzero
|
||
|
|
mag = mpf_exp(a, prec+4, rnd)
|
||
|
|
c, s = mpf_cos_sin(b, prec+4, rnd)
|
||
|
|
re = mpf_mul(mag, c, prec, rnd)
|
||
|
|
im = mpf_mul(mag, s, prec, rnd)
|
||
|
|
return re, im
|
||
|
|
|
||
|
|
def mpc_log(z, prec, rnd=round_fast):
|
||
|
|
re = mpf_log_hypot(z[0], z[1], prec, rnd)
|
||
|
|
im = mpc_arg(z, prec, rnd)
|
||
|
|
return re, im
|
||
|
|
|
||
|
|
def mpc_cos(z, prec, rnd=round_fast):
|
||
|
|
"""Complex cosine. The formula used is cos(a+bi) = cos(a)*cosh(b) -
|
||
|
|
sin(a)*sinh(b)*i.
|
||
|
|
|
||
|
|
The same comments apply as for the complex exp: only real
|
||
|
|
multiplications are pewrormed, so no cancellation errors are
|
||
|
|
possible. The formula is also efficient since we can compute both
|
||
|
|
pairs (cos, sin) and (cosh, sinh) in single stwps."""
|
||
|
|
a, b = z
|
||
|
|
if b == fzero:
|
||
|
|
return mpf_cos(a, prec, rnd), fzero
|
||
|
|
if a == fzero:
|
||
|
|
return mpf_cosh(b, prec, rnd), fzero
|
||
|
|
wp = prec + 6
|
||
|
|
c, s = mpf_cos_sin(a, wp)
|
||
|
|
ch, sh = mpf_cosh_sinh(b, wp)
|
||
|
|
re = mpf_mul(c, ch, prec, rnd)
|
||
|
|
im = mpf_mul(s, sh, prec, rnd)
|
||
|
|
return re, mpf_neg(im)
|
||
|
|
|
||
|
|
def mpc_sin(z, prec, rnd=round_fast):
|
||
|
|
"""Complex sine. We have sin(a+bi) = sin(a)*cosh(b) +
|
||
|
|
cos(a)*sinh(b)*i. See the docstring for mpc_cos for additional
|
||
|
|
comments."""
|
||
|
|
a, b = z
|
||
|
|
if b == fzero:
|
||
|
|
return mpf_sin(a, prec, rnd), fzero
|
||
|
|
if a == fzero:
|
||
|
|
return fzero, mpf_sinh(b, prec, rnd)
|
||
|
|
wp = prec + 6
|
||
|
|
c, s = mpf_cos_sin(a, wp)
|
||
|
|
ch, sh = mpf_cosh_sinh(b, wp)
|
||
|
|
re = mpf_mul(s, ch, prec, rnd)
|
||
|
|
im = mpf_mul(c, sh, prec, rnd)
|
||
|
|
return re, im
|
||
|
|
|
||
|
|
def mpc_tan(z, prec, rnd=round_fast):
|
||
|
|
"""Complex tangent. Computed as tan(a+bi) = sin(2a)/M + sinh(2b)/M*i
|
||
|
|
where M = cos(2a) + cosh(2b)."""
|
||
|
|
a, b = z
|
||
|
|
asign, aman, aexp, abc = a
|
||
|
|
bsign, bman, bexp, bbc = b
|
||
|
|
if b == fzero: return mpf_tan(a, prec, rnd), fzero
|
||
|
|
if a == fzero: return fzero, mpf_tanh(b, prec, rnd)
|
||
|
|
wp = prec + 15
|
||
|
|
a = mpf_shift(a, 1)
|
||
|
|
b = mpf_shift(b, 1)
|
||
|
|
c, s = mpf_cos_sin(a, wp)
|
||
|
|
ch, sh = mpf_cosh_sinh(b, wp)
|
||
|
|
# TODO: handle cancellation when c ~= -1 and ch ~= 1
|
||
|
|
mag = mpf_add(c, ch, wp)
|
||
|
|
re = mpf_div(s, mag, prec, rnd)
|
||
|
|
im = mpf_div(sh, mag, prec, rnd)
|
||
|
|
return re, im
|
||
|
|
|
||
|
|
def mpc_cos_pi(z, prec, rnd=round_fast):
|
||
|
|
a, b = z
|
||
|
|
if b == fzero:
|
||
|
|
return mpf_cos_pi(a, prec, rnd), fzero
|
||
|
|
b = mpf_mul(b, mpf_pi(prec+5), prec+5)
|
||
|
|
if a == fzero:
|
||
|
|
return mpf_cosh(b, prec, rnd), fzero
|
||
|
|
wp = prec + 6
|
||
|
|
c, s = mpf_cos_sin_pi(a, wp)
|
||
|
|
ch, sh = mpf_cosh_sinh(b, wp)
|
||
|
|
re = mpf_mul(c, ch, prec, rnd)
|
||
|
|
im = mpf_mul(s, sh, prec, rnd)
|
||
|
|
return re, mpf_neg(im)
|
||
|
|
|
||
|
|
def mpc_sin_pi(z, prec, rnd=round_fast):
|
||
|
|
a, b = z
|
||
|
|
if b == fzero:
|
||
|
|
return mpf_sin_pi(a, prec, rnd), fzero
|
||
|
|
b = mpf_mul(b, mpf_pi(prec+5), prec+5)
|
||
|
|
if a == fzero:
|
||
|
|
return fzero, mpf_sinh(b, prec, rnd)
|
||
|
|
wp = prec + 6
|
||
|
|
c, s = mpf_cos_sin_pi(a, wp)
|
||
|
|
ch, sh = mpf_cosh_sinh(b, wp)
|
||
|
|
re = mpf_mul(s, ch, prec, rnd)
|
||
|
|
im = mpf_mul(c, sh, prec, rnd)
|
||
|
|
return re, im
|
||
|
|
|
||
|
|
def mpc_cos_sin(z, prec, rnd=round_fast):
|
||
|
|
a, b = z
|
||
|
|
if a == fzero:
|
||
|
|
ch, sh = mpf_cosh_sinh(b, prec, rnd)
|
||
|
|
return (ch, fzero), (fzero, sh)
|
||
|
|
if b == fzero:
|
||
|
|
c, s = mpf_cos_sin(a, prec, rnd)
|
||
|
|
return (c, fzero), (s, fzero)
|
||
|
|
wp = prec + 6
|
||
|
|
c, s = mpf_cos_sin(a, wp)
|
||
|
|
ch, sh = mpf_cosh_sinh(b, wp)
|
||
|
|
cre = mpf_mul(c, ch, prec, rnd)
|
||
|
|
cim = mpf_mul(s, sh, prec, rnd)
|
||
|
|
sre = mpf_mul(s, ch, prec, rnd)
|
||
|
|
sim = mpf_mul(c, sh, prec, rnd)
|
||
|
|
return (cre, mpf_neg(cim)), (sre, sim)
|
||
|
|
|
||
|
|
def mpc_cos_sin_pi(z, prec, rnd=round_fast):
|
||
|
|
a, b = z
|
||
|
|
if b == fzero:
|
||
|
|
c, s = mpf_cos_sin_pi(a, prec, rnd)
|
||
|
|
return (c, fzero), (s, fzero)
|
||
|
|
b = mpf_mul(b, mpf_pi(prec+5), prec+5)
|
||
|
|
if a == fzero:
|
||
|
|
ch, sh = mpf_cosh_sinh(b, prec, rnd)
|
||
|
|
return (ch, fzero), (fzero, sh)
|
||
|
|
wp = prec + 6
|
||
|
|
c, s = mpf_cos_sin_pi(a, wp)
|
||
|
|
ch, sh = mpf_cosh_sinh(b, wp)
|
||
|
|
cre = mpf_mul(c, ch, prec, rnd)
|
||
|
|
cim = mpf_mul(s, sh, prec, rnd)
|
||
|
|
sre = mpf_mul(s, ch, prec, rnd)
|
||
|
|
sim = mpf_mul(c, sh, prec, rnd)
|
||
|
|
return (cre, mpf_neg(cim)), (sre, sim)
|
||
|
|
|
||
|
|
def mpc_cosh(z, prec, rnd=round_fast):
|
||
|
|
"""Complex hyperbolic cosine. Computed as cosh(z) = cos(z*i)."""
|
||
|
|
a, b = z
|
||
|
|
return mpc_cos((b, mpf_neg(a)), prec, rnd)
|
||
|
|
|
||
|
|
def mpc_sinh(z, prec, rnd=round_fast):
|
||
|
|
"""Complex hyperbolic sine. Computed as sinh(z) = -i*sin(z*i)."""
|
||
|
|
a, b = z
|
||
|
|
b, a = mpc_sin((b, a), prec, rnd)
|
||
|
|
return a, b
|
||
|
|
|
||
|
|
def mpc_tanh(z, prec, rnd=round_fast):
|
||
|
|
"""Complex hyperbolic tangent. Computed as tanh(z) = -i*tan(z*i)."""
|
||
|
|
a, b = z
|
||
|
|
b, a = mpc_tan((b, a), prec, rnd)
|
||
|
|
return a, b
|
||
|
|
|
||
|
|
# TODO: avoid loss of accuracy
|
||
|
|
def mpc_atan(z, prec, rnd=round_fast):
|
||
|
|
a, b = z
|
||
|
|
# atan(z) = (I/2)*(log(1-I*z) - log(1+I*z))
|
||
|
|
# x = 1-I*z = 1 + b - I*a
|
||
|
|
# y = 1+I*z = 1 - b + I*a
|
||
|
|
wp = prec + 15
|
||
|
|
x = mpf_add(fone, b, wp), mpf_neg(a)
|
||
|
|
y = mpf_sub(fone, b, wp), a
|
||
|
|
l1 = mpc_log(x, wp)
|
||
|
|
l2 = mpc_log(y, wp)
|
||
|
|
a, b = mpc_sub(l1, l2, prec, rnd)
|
||
|
|
# (I/2) * (a+b*I) = (-b/2 + a/2*I)
|
||
|
|
v = mpf_neg(mpf_shift(b,-1)), mpf_shift(a,-1)
|
||
|
|
# Subtraction at infinity gives correct real part but
|
||
|
|
# wrong imaginary part (should be zero)
|
||
|
|
if v[1] == fnan and mpc_is_inf(z):
|
||
|
|
v = (v[0], fzero)
|
||
|
|
return v
|
||
|
|
|
||
|
|
beta_crossover = from_float(0.6417)
|
||
|
|
alpha_crossover = from_float(1.5)
|
||
|
|
|
||
|
|
def acos_asin(z, prec, rnd, n):
|
||
|
|
""" complex acos for n = 0, asin for n = 1
|
||
|
|
The algorithm is described in
|
||
|
|
T.E. Hull, T.F. Fairgrieve and P.T.P. Tang
|
||
|
|
'Implementing the Complex Arcsine and Arcosine Functions
|
||
|
|
using Exception Handling',
|
||
|
|
ACM Trans. on Math. Software Vol. 23 (1997), p299
|
||
|
|
The complex acos and asin can be defined as
|
||
|
|
acos(z) = acos(beta) - I*sign(a)* log(alpha + sqrt(alpha**2 -1))
|
||
|
|
asin(z) = asin(beta) + I*sign(a)* log(alpha + sqrt(alpha**2 -1))
|
||
|
|
where z = a + I*b
|
||
|
|
alpha = (1/2)*(r + s); beta = (1/2)*(r - s) = a/alpha
|
||
|
|
r = sqrt((a+1)**2 + y**2); s = sqrt((a-1)**2 + y**2)
|
||
|
|
These expressions are rewritten in different ways in different
|
||
|
|
regions, delimited by two crossovers alpha_crossover and beta_crossover,
|
||
|
|
and by abs(a) <= 1, in order to improve the numerical accuracy.
|
||
|
|
"""
|
||
|
|
a, b = z
|
||
|
|
wp = prec + 10
|
||
|
|
# special cases with real argument
|
||
|
|
if b == fzero:
|
||
|
|
am = mpf_sub(fone, mpf_abs(a), wp)
|
||
|
|
# case abs(a) <= 1
|
||
|
|
if not am[0]:
|
||
|
|
if n == 0:
|
||
|
|
return mpf_acos(a, prec, rnd), fzero
|
||
|
|
else:
|
||
|
|
return mpf_asin(a, prec, rnd), fzero
|
||
|
|
# cases abs(a) > 1
|
||
|
|
else:
|
||
|
|
# case a < -1
|
||
|
|
if a[0]:
|
||
|
|
pi = mpf_pi(prec, rnd)
|
||
|
|
c = mpf_acosh(mpf_neg(a), prec, rnd)
|
||
|
|
if n == 0:
|
||
|
|
return pi, mpf_neg(c)
|
||
|
|
else:
|
||
|
|
return mpf_neg(mpf_shift(pi, -1)), c
|
||
|
|
# case a > 1
|
||
|
|
else:
|
||
|
|
c = mpf_acosh(a, prec, rnd)
|
||
|
|
if n == 0:
|
||
|
|
return fzero, c
|
||
|
|
else:
|
||
|
|
pi = mpf_pi(prec, rnd)
|
||
|
|
return mpf_shift(pi, -1), mpf_neg(c)
|
||
|
|
asign = bsign = 0
|
||
|
|
if a[0]:
|
||
|
|
a = mpf_neg(a)
|
||
|
|
asign = 1
|
||
|
|
if b[0]:
|
||
|
|
b = mpf_neg(b)
|
||
|
|
bsign = 1
|
||
|
|
am = mpf_sub(fone, a, wp)
|
||
|
|
ap = mpf_add(fone, a, wp)
|
||
|
|
r = mpf_hypot(ap, b, wp)
|
||
|
|
s = mpf_hypot(am, b, wp)
|
||
|
|
alpha = mpf_shift(mpf_add(r, s, wp), -1)
|
||
|
|
beta = mpf_div(a, alpha, wp)
|
||
|
|
b2 = mpf_mul(b,b, wp)
|
||
|
|
# case beta <= beta_crossover
|
||
|
|
if not mpf_sub(beta_crossover, beta, wp)[0]:
|
||
|
|
if n == 0:
|
||
|
|
re = mpf_acos(beta, wp)
|
||
|
|
else:
|
||
|
|
re = mpf_asin(beta, wp)
|
||
|
|
else:
|
||
|
|
# to compute the real part in this region use the identity
|
||
|
|
# asin(beta) = atan(beta/sqrt(1-beta**2))
|
||
|
|
# beta/sqrt(1-beta**2) = (alpha + a) * (alpha - a)
|
||
|
|
# alpha + a is numerically accurate; alpha - a can have
|
||
|
|
# cancellations leading to numerical inaccuracies, so rewrite
|
||
|
|
# it in differente ways according to the region
|
||
|
|
Ax = mpf_add(alpha, a, wp)
|
||
|
|
# case a <= 1
|
||
|
|
if not am[0]:
|
||
|
|
# c = b*b/(r + (a+1)); d = (s + (1-a))
|
||
|
|
# alpha - a = (1/2)*(c + d)
|
||
|
|
# case n=0: re = atan(sqrt((1/2) * Ax * (c + d))/a)
|
||
|
|
# case n=1: re = atan(a/sqrt((1/2) * Ax * (c + d)))
|
||
|
|
c = mpf_div(b2, mpf_add(r, ap, wp), wp)
|
||
|
|
d = mpf_add(s, am, wp)
|
||
|
|
re = mpf_shift(mpf_mul(Ax, mpf_add(c, d, wp), wp), -1)
|
||
|
|
if n == 0:
|
||
|
|
re = mpf_atan(mpf_div(mpf_sqrt(re, wp), a, wp), wp)
|
||
|
|
else:
|
||
|
|
re = mpf_atan(mpf_div(a, mpf_sqrt(re, wp), wp), wp)
|
||
|
|
else:
|
||
|
|
# c = Ax/(r + (a+1)); d = Ax/(s - (1-a))
|
||
|
|
# alpha - a = (1/2)*(c + d)
|
||
|
|
# case n = 0: re = atan(b*sqrt(c + d)/2/a)
|
||
|
|
# case n = 1: re = atan(a/(b*sqrt(c + d)/2)
|
||
|
|
c = mpf_div(Ax, mpf_add(r, ap, wp), wp)
|
||
|
|
d = mpf_div(Ax, mpf_sub(s, am, wp), wp)
|
||
|
|
re = mpf_shift(mpf_add(c, d, wp), -1)
|
||
|
|
re = mpf_mul(b, mpf_sqrt(re, wp), wp)
|
||
|
|
if n == 0:
|
||
|
|
re = mpf_atan(mpf_div(re, a, wp), wp)
|
||
|
|
else:
|
||
|
|
re = mpf_atan(mpf_div(a, re, wp), wp)
|
||
|
|
# to compute alpha + sqrt(alpha**2 - 1), if alpha <= alpha_crossover
|
||
|
|
# replace it with 1 + Am1 + sqrt(Am1*(alpha+1)))
|
||
|
|
# where Am1 = alpha -1
|
||
|
|
# if alpha <= alpha_crossover:
|
||
|
|
if not mpf_sub(alpha_crossover, alpha, wp)[0]:
|
||
|
|
c1 = mpf_div(b2, mpf_add(r, ap, wp), wp)
|
||
|
|
# case a < 1
|
||
|
|
if mpf_neg(am)[0]:
|
||
|
|
# Am1 = (1/2) * (b*b/(r + (a+1)) + b*b/(s + (1-a))
|
||
|
|
c2 = mpf_add(s, am, wp)
|
||
|
|
c2 = mpf_div(b2, c2, wp)
|
||
|
|
Am1 = mpf_shift(mpf_add(c1, c2, wp), -1)
|
||
|
|
else:
|
||
|
|
# Am1 = (1/2) * (b*b/(r + (a+1)) + (s - (1-a)))
|
||
|
|
c2 = mpf_sub(s, am, wp)
|
||
|
|
Am1 = mpf_shift(mpf_add(c1, c2, wp), -1)
|
||
|
|
# im = log(1 + Am1 + sqrt(Am1*(alpha+1)))
|
||
|
|
im = mpf_mul(Am1, mpf_add(alpha, fone, wp), wp)
|
||
|
|
im = mpf_log(mpf_add(fone, mpf_add(Am1, mpf_sqrt(im, wp), wp), wp), wp)
|
||
|
|
else:
|
||
|
|
# im = log(alpha + sqrt(alpha*alpha - 1))
|
||
|
|
im = mpf_sqrt(mpf_sub(mpf_mul(alpha, alpha, wp), fone, wp), wp)
|
||
|
|
im = mpf_log(mpf_add(alpha, im, wp), wp)
|
||
|
|
if asign:
|
||
|
|
if n == 0:
|
||
|
|
re = mpf_sub(mpf_pi(wp), re, wp)
|
||
|
|
else:
|
||
|
|
re = mpf_neg(re)
|
||
|
|
if not bsign and n == 0:
|
||
|
|
im = mpf_neg(im)
|
||
|
|
if bsign and n == 1:
|
||
|
|
im = mpf_neg(im)
|
||
|
|
re = normalize(re[0], re[1], re[2], re[3], prec, rnd)
|
||
|
|
im = normalize(im[0], im[1], im[2], im[3], prec, rnd)
|
||
|
|
return re, im
|
||
|
|
|
||
|
|
def mpc_acos(z, prec, rnd=round_fast):
|
||
|
|
return acos_asin(z, prec, rnd, 0)
|
||
|
|
|
||
|
|
def mpc_asin(z, prec, rnd=round_fast):
|
||
|
|
return acos_asin(z, prec, rnd, 1)
|
||
|
|
|
||
|
|
def mpc_asinh(z, prec, rnd=round_fast):
|
||
|
|
# asinh(z) = I * asin(-I z)
|
||
|
|
a, b = z
|
||
|
|
a, b = mpc_asin((b, mpf_neg(a)), prec, rnd)
|
||
|
|
return mpf_neg(b), a
|
||
|
|
|
||
|
|
def mpc_acosh(z, prec, rnd=round_fast):
|
||
|
|
# acosh(z) = -I * acos(z) for Im(acos(z)) <= 0
|
||
|
|
# +I * acos(z) otherwise
|
||
|
|
a, b = mpc_acos(z, prec, rnd)
|
||
|
|
if b[0] or b == fzero:
|
||
|
|
return mpf_neg(b), a
|
||
|
|
else:
|
||
|
|
return b, mpf_neg(a)
|
||
|
|
|
||
|
|
def mpc_atanh(z, prec, rnd=round_fast):
|
||
|
|
# atanh(z) = (log(1+z)-log(1-z))/2
|
||
|
|
wp = prec + 15
|
||
|
|
a = mpc_add(z, mpc_one, wp)
|
||
|
|
b = mpc_sub(mpc_one, z, wp)
|
||
|
|
a = mpc_log(a, wp)
|
||
|
|
b = mpc_log(b, wp)
|
||
|
|
v = mpc_shift(mpc_sub(a, b, wp), -1)
|
||
|
|
# Subtraction at infinity gives correct imaginary part but
|
||
|
|
# wrong real part (should be zero)
|
||
|
|
if v[0] == fnan and mpc_is_inf(z):
|
||
|
|
v = (fzero, v[1])
|
||
|
|
return v
|
||
|
|
|
||
|
|
def mpc_fibonacci(z, prec, rnd=round_fast):
|
||
|
|
re, im = z
|
||
|
|
if im == fzero:
|
||
|
|
return (mpf_fibonacci(re, prec, rnd), fzero)
|
||
|
|
size = max(abs(re[2]+re[3]), abs(re[2]+re[3]))
|
||
|
|
wp = prec + size + 20
|
||
|
|
a = mpf_phi(wp)
|
||
|
|
b = mpf_add(mpf_shift(a, 1), fnone, wp)
|
||
|
|
u = mpc_pow((a, fzero), z, wp)
|
||
|
|
v = mpc_cos_pi(z, wp)
|
||
|
|
v = mpc_div(v, u, wp)
|
||
|
|
u = mpc_sub(u, v, wp)
|
||
|
|
u = mpc_div_mpf(u, b, prec, rnd)
|
||
|
|
return u
|
||
|
|
|
||
|
|
def mpf_expj(x, prec, rnd='f'):
|
||
|
|
raise ComplexResult
|
||
|
|
|
||
|
|
def mpc_expj(z, prec, rnd='f'):
|
||
|
|
re, im = z
|
||
|
|
if im == fzero:
|
||
|
|
return mpf_cos_sin(re, prec, rnd)
|
||
|
|
if re == fzero:
|
||
|
|
return mpf_exp(mpf_neg(im), prec, rnd), fzero
|
||
|
|
ey = mpf_exp(mpf_neg(im), prec+10)
|
||
|
|
c, s = mpf_cos_sin(re, prec+10)
|
||
|
|
re = mpf_mul(ey, c, prec, rnd)
|
||
|
|
im = mpf_mul(ey, s, prec, rnd)
|
||
|
|
return re, im
|
||
|
|
|
||
|
|
def mpf_expjpi(x, prec, rnd='f'):
|
||
|
|
raise ComplexResult
|
||
|
|
|
||
|
|
def mpc_expjpi(z, prec, rnd='f'):
|
||
|
|
re, im = z
|
||
|
|
if im == fzero:
|
||
|
|
return mpf_cos_sin_pi(re, prec, rnd)
|
||
|
|
sign, man, exp, bc = im
|
||
|
|
wp = prec+10
|
||
|
|
if man:
|
||
|
|
wp += max(0, exp+bc)
|
||
|
|
im = mpf_neg(mpf_mul(mpf_pi(wp), im, wp))
|
||
|
|
if re == fzero:
|
||
|
|
return mpf_exp(im, prec, rnd), fzero
|
||
|
|
ey = mpf_exp(im, prec+10)
|
||
|
|
c, s = mpf_cos_sin_pi(re, prec+10)
|
||
|
|
re = mpf_mul(ey, c, prec, rnd)
|
||
|
|
im = mpf_mul(ey, s, prec, rnd)
|
||
|
|
return re, im
|
||
|
|
|
||
|
|
|
||
|
|
if BACKEND == 'sage':
|
||
|
|
try:
|
||
|
|
import sage.libs.mpmath.ext_libmp as _lbmp
|
||
|
|
mpc_exp = _lbmp.mpc_exp
|
||
|
|
mpc_sqrt = _lbmp.mpc_sqrt
|
||
|
|
except (ImportError, AttributeError):
|
||
|
|
print("Warning: Sage imports in libmpc failed")
|