def _to_mpmath(self, prec, allow_ints=True): # mpmath functions accept ints as input errmsg = "cannot convert to mpmath number" if allow_ints and self.is_Integer: return self.p try: re, im, _, _ = evalf(self, prec, {}) if im: return make_mpc((re, im)) else: return make_mpf(re) except NotImplementedError: v = self._eval_evalf(prec) if v is None: raise ValueError(errmsg) if v.is_Float: return make_mpf(v._mpf_) # Number + Number*I is also fine re, im = v.as_real_imag() if allow_ints and re.is_Integer: re = from_int(re.p) elif re.is_Float: re = re._mpf_ else: raise ValueError(errmsg) if allow_ints and im.is_Integer: im = from_int(im.p) elif im.is_Float: im = im._mpf_ else: raise ValueError(errmsg) return make_mpc((re, im))
def _d(n, j, prec, sq23pi, sqrt8): """ Compute the sinh term in the outer sum of the HRR formula. The constants sqrt(2/3*pi) and sqrt(8) must be precomputed. """ j = from_int(j) pi = mpf_pi(prec) a = mpf_div(sq23pi, j, prec) b = mpf_sub(from_int(n), from_rational(1, 24, prec), prec) c = mpf_sqrt(b, prec) ch, sh = mpf_cosh_sinh(mpf_mul(a, c), prec) D = mpf_div(mpf_sqrt(j, prec), mpf_mul(mpf_mul(sqrt8, b), pi), prec) E = mpf_sub(mpf_mul(a, ch), mpf_div(sh, c, prec), prec) return mpf_mul(D, E)
def _create_evalf_table(): global evalf_table evalf_table = { C.Symbol: evalf_symbol, C.Dummy: evalf_symbol, C.Float: lambda x, prec, options: (x._mpf_, None, prec, None), C.Rational: lambda x, prec, options: (from_rational(x.p, x.q, prec), None, prec, None), C.Integer: lambda x, prec, options: (from_int(x.p, prec), None, prec, None), C.Zero: lambda x, prec, options: (None, None, prec, None), C.One: lambda x, prec, options: (fone, None, prec, None), C.Half: lambda x, prec, options: (fhalf, None, prec, None), C.Pi: lambda x, prec, options: (mpf_pi(prec), None, prec, None), C.Exp1: lambda x, prec, options: (mpf_e(prec), None, prec, None), C.ImaginaryUnit: lambda x, prec, options: (None, fone, None, prec), C.NegativeOne: lambda x, prec, options: (fnone, None, prec, None), C.NaN: lambda x, prec, options: (fnan, None, prec, None), C.exp: lambda x, prec, options: evalf_pow(C.Pow(S.Exp1, x.args[0], evaluate=False), prec, options), C.cos: evalf_trig, C.sin: evalf_trig, C.Add: evalf_add, C.Mul: evalf_mul, C.Pow: evalf_pow, C.log: evalf_log, C.atan: evalf_atan, C.Abs: evalf_abs, C.re: evalf_re, C.im: evalf_im, C.floor: evalf_floor, C.ceiling: evalf_ceiling, C.Integral: evalf_integral, C.Sum: evalf_sum, C.Piecewise: evalf_piecewise, C.bernoulli: evalf_bernoulli, }
def npartitions(n, verbose=False): """ Calculate the partition function P(n), i.e. the number of ways that n can be written as a sum of positive integers. P(n) is computed using the Hardy-Ramanujan-Rademacher formula, described e.g. at http://mathworld.wolfram.com/PartitionFunctionP.html The correctness of this implementation has been tested for 10**n up to n = 8. """ n = int(n) if n < 0: return 0 if n <= 5: return [1, 1, 2, 3, 5, 7][n] # Estimate number of bits in p(n). This formula could be tidied pbits = int((math.pi*(2*n/3.)**0.5-math.log(4*n))/math.log(10)+1)*\ math.log(10,2) prec = p = int(pbits * 1.1 + 100) s = fzero M = max(6, int(0.24 * n**0.5 + 4)) sq23pi = mpf_mul(mpf_sqrt(from_rational(2, 3, p), p), mpf_pi(p), p) sqrt8 = mpf_sqrt(from_int(8), p) for q in xrange(1, M): a = A(n, q, p) d = D(n, q, p, sq23pi, sqrt8) s = mpf_add(s, mpf_mul(a, d), prec) if verbose: print "step", q, "of", M, to_str(a, 10), to_str(d, 10) # On average, the terms decrease rapidly in magnitude. Dynamically # reducing the precision greatly improves performance. p = bitcount(abs(to_int(d))) + 50 np = to_int(mpf_add(s, fhalf, prec)) return int(np)
def npartitions(n, verbose=False): """ Calculate the partition function P(n), i.e. the number of ways that n can be written as a sum of positive integers. P(n) is computed using the Hardy-Ramanujan-Rademacher formula, described e.g. at http://mathworld.wolfram.com/PartitionFunctionP.html The correctness of this implementation has been tested for 10**n up to n = 8. """ n = int(n) if n < 0: return 0 if n <= 5: return [1, 1, 2, 3, 5, 7][n] # Estimate number of bits in p(n). This formula could be tidied pbits = int((math.pi*(2*n/3.)**0.5-math.log(4*n))/math.log(10)+1)*\ math.log(10,2) prec = p = int(pbits*1.1 + 100) s = fzero M = max(6, int(0.24*n**0.5+4)) sq23pi = mpf_mul(mpf_sqrt(from_rational(2,3,p), p), mpf_pi(p), p) sqrt8 = mpf_sqrt(from_int(8), p) for q in xrange(1, M): a = A(n,q,p) d = D(n,q,p, sq23pi, sqrt8) s = mpf_add(s, mpf_mul(a, d), prec) if verbose: print "step", q, "of", M, to_str(a, 10), to_str(d, 10) # On average, the terms decrease rapidly in magnitude. Dynamically # reducing the precision greatly improves performance. p = bitcount(abs(to_int(d))) + 50 np = to_int(mpf_add(s, fhalf, prec)) return int(np)
def calc_part(expr, nexpr): nint = int(to_int(nexpr, round_nearest)) expr = C.Add(expr, -nint, evaluate=False) x, _, x_acc, _ = evalf(expr, 10, options) check_target(expr, (x, None, x_acc, None), 3) nint += int(no*(mpf_cmp(x or fzero, fzero) == no)) nint = from_int(nint) return nint, fastlog(nint) + 10
def calc_part(expr, nexpr): nint = int(to_int(nexpr, round_nearest)) expr = C.Add(expr, -nint, evaluate=False) x, _, x_acc, _ = evalf(expr, 10, options) check_target(expr, (x, None, x_acc, None), 3) nint += int(no * (mpf_cmp(x or fzero, fzero) == no)) nint = from_int(nint) return nint, fastlog(nint) + 10
def _convert_tol(ctx, tol): if isinstance(tol, int_types): return from_int(tol) if isinstance(tol, float): return from_float(tol) if hasattr(tol, "_mpf_"): return tol._mpf_ prec, rounding = ctx._prec_rounding if isinstance(tol, basestring): return from_str(tol, prec, rounding) raise ValueError("expected a real number, got %s" % tol)
def calc_part(expr, nexpr): nint = int(to_int(nexpr, rnd)) expr = C.Add(expr, -nint, evaluate=False) x, _, x_acc, _ = evalf(expr, 10, options) try: check_target(expr, (x, None, x_acc, None), 3) except PrecisionExhausted: if not expr.equals(0): raise PrecisionExhausted x = fzero nint += int(no * (mpf_cmp(x or fzero, fzero) == no)) nint = from_int(nint) return nint, fastlog(nint) + 10
def calc_part(expr, nexpr): nint = int(to_int(nexpr, rnd)) expr = C.Add(expr, -nint, evaluate=False) x, _, x_acc, _ = evalf(expr, 10, options) try: check_target(expr, (x, None, x_acc, None), 3) except PrecisionExhausted: if not expr.equals(0): raise PrecisionExhausted x = fzero nint += int(no*(mpf_cmp(x or fzero, fzero) == no)) nint = from_int(nint) return nint, fastlog(nint) + 10
def _create_evalf_table(): global evalf_table evalf_table = { C.Symbol: evalf_symbol, C.Dummy: evalf_symbol, C.Float: lambda x, prec, options: (x._mpf_, None, prec, None), C.Rational: lambda x, prec, options: (from_rational(x.p, x.q, prec), None, prec, None), C.Integer: lambda x, prec, options: (from_int(x.p, prec), None, prec, None), C.Zero: lambda x, prec, options: (None, None, prec, None), C.One: lambda x, prec, options: (fone, None, prec, None), C.Half: lambda x, prec, options: (fhalf, None, prec, None), C.Pi: lambda x, prec, options: (mpf_pi(prec), None, prec, None), C.Exp1: lambda x, prec, options: (mpf_e(prec), None, prec, None), C.ImaginaryUnit: lambda x, prec, options: (None, fone, None, prec), C.NegativeOne: lambda x, prec, options: (fnone, None, prec, None), C.NaN : lambda x, prec, options: (fnan, None, prec, None), C.exp: lambda x, prec, options: evalf_pow(C.Pow(S.Exp1, x.args[0], evaluate=False), prec, options), C.cos: evalf_trig, C.sin: evalf_trig, C.Add: evalf_add, C.Mul: evalf_mul, C.Pow: evalf_pow, C.log: evalf_log, C.atan: evalf_atan, C.Abs: evalf_abs, C.re: evalf_re, C.im: evalf_im, C.floor: evalf_floor, C.ceiling: evalf_ceiling, C.Integral: evalf_integral, C.Sum: evalf_sum, C.Product: evalf_prod, C.Piecewise: evalf_piecewise, C.bernoulli: evalf_bernoulli, }
def _as_mpf_val(self, prec): return mlib.from_int(self.p)