コード例 #1
0
ファイル: libmpi.py プロジェクト: rkern/sympy-rkern
def mpi_pow(s, t, prec):
    ta, tb = t
    if ta == tb and ta not in (finf, fninf):
        if ta == from_int(to_int(ta)):
            return mpi_pow_int(s, to_int(ta), prec)
        if ta == fhalf:
            return mpi_sqrt(s, prec)
    u = mpi_log(s, prec + 20)
    v = mpi_mul(u, t, prec + 20)
    return mpi_exp(v, prec)
コード例 #2
0
def mpi_pow(s, t, prec):
    ta, tb = t
    if ta == tb and ta not in (finf, fninf):
        if ta == from_int(to_int(ta)):
            return mpi_pow_int(s, to_int(ta), prec)
        if ta == fhalf:
            return mpi_sqrt(s, prec)
    u = mpi_log(s, prec + 20)
    v = mpi_mul(u, t, prec + 20)
    return mpi_exp(v, prec)
コード例 #3
0
ファイル: gammazeta.py プロジェクト: ryanGT/sympy
def mpf_psi0(x, prec, rnd=round_fast):
    """
    Computation of the digamma function (psi function of order 0)
    of a real argument.
    """
    sign, man, exp, bc = x
    wp = prec + 10
    if not man:
        if x == finf:
            return x
        if x == fninf or x == fnan:
            return fnan
    if x == fzero or (exp >= 0 and sign):
        raise ValueError("polygamma pole")
    # Reflection formula
    if sign and exp + bc > 3:
        c, s = mpf_cos_sin_pi(x, wp)
        q = mpf_mul(mpf_div(c, s, wp), mpf_pi(wp), wp)
        p = mpf_psi0(mpf_sub(fone, x, wp), wp)
        return mpf_sub(p, q, prec, rnd)
    # The logarithmic term is accurate enough
    if (not sign) and bc + exp > wp:
        return mpf_log(mpf_sub(x, fone, wp), prec, rnd)
    # Initial recurrence to obtain a large enough x
    m = to_int(x)
    n = int(0.11 * wp) + 2
    s = MP_ZERO
    x = to_fixed(x, wp)
    one = MP_ONE << wp
    if m < n:
        for k in xrange(m, n):
            s -= (one << wp) // x
            x += one
    x -= one
    # Logarithmic term
    s += to_fixed(mpf_log(from_man_exp(x, -wp, wp), wp), wp)
    # Endpoint term in Euler-Maclaurin expansion
    s += (one << wp) // (2 * x)
    # Euler-Maclaurin remainder sum
    x2 = (x * x) >> wp
    t = one
    prev = 0
    k = 1
    while 1:
        t = (t * x2) >> wp
        bsign, bman, bexp, bbc = mpf_bernoulli(2 * k, wp)
        offset = bexp + 2 * wp
        if offset >= 0:
            term = (bman << offset) // (t * (2 * k))
        else:
            term = (bman >> (-offset)) // (t * (2 * k))
        if k & 1:
            s -= term
        else:
            s += term
        if k > 2 and term >= prev:
            break
        prev = term
        k += 1
    return from_man_exp(s, -wp, wp, rnd)
コード例 #4
0
ファイル: gammazeta.py プロジェクト: jcockayne/sympy-rkern
def mpc_zeta(s, prec, rnd):
    re, im = s
    wp = prec + 20
    n = int(wp/2.54 + 5)
    n += int(0.9*abs(to_int(im)))
    d = borwein_coefficients(n)
    ref = to_fixed(re, wp)
    imf = to_fixed(im, wp)
    tre = MP_ZERO
    tim = MP_ZERO
    one = MP_ONE << wp
    one_2wp = MP_ONE << (2*wp)
    critical_line = re == fhalf
    for k in xrange(n):
        log = log_int_fixed(k+1, wp)
        # A square root is much cheaper than an exp
        if critical_line:
            w = one_2wp // sqrt_fixed((k+1) << wp, wp)
        else:
            w = to_fixed(mpf_exp(from_man_exp(-ref*log, -2*wp), wp), wp)
        if k & 1:
            w *= (d[n] - d[k])
        else:
            w *= (d[k] - d[n])
        wre, wim = cos_sin(from_man_exp(-imf * log_int_fixed(k+1, wp), -2*wp), wp)
        tre += (w * to_fixed(wre, wp)) >> wp
        tim += (w * to_fixed(wim, wp)) >> wp
    tre //= (-d[n])
    tim //= (-d[n])
    tre = from_man_exp(tre, -wp, wp)
    tim = from_man_exp(tim, -wp, wp)
    q = mpc_sub(mpc_one, mpc_pow(mpc_two, mpc_sub(mpc_one, s, wp), wp), wp)
    return mpc_div((tre, tim), q, prec, rnd)
コード例 #5
0
ファイル: gammazeta.py プロジェクト: fperez/sympy
def mpc_zeta(s, prec, rnd=round_fast, alt=0):
    re, im = s
    if im == fzero:
        return mpf_zeta(re, prec, rnd, alt), fzero
    wp = prec + 20
    # Reflection formula. To be rigorous, we should reflect to the left of
    # re = 1/2 (see comments for mpf_zeta), but this leads to unnecessary
    # slowdown for interesting values of s
    if mpf_lt(re, fzero):
        # XXX: could use the separate refl. formula for Dirichlet eta
        if alt:
            q = mpc_sub(mpc_one, mpc_pow(mpc_two, mpc_sub(mpc_one, s, wp),
                wp), wp)
            return mpc_mul(mpc_zeta(s, wp), q, prec, rnd)
        # XXX: -1 should be done exactly
        y = mpc_sub(mpc_one, s, 10*wp)
        a = mpc_gamma(y, wp)
        b = mpc_zeta(y, wp)
        c = mpc_sin_pi(mpc_shift(s, -1), wp)
        rsign, rman, rexp, rbc = re
        isign, iman, iexp, ibc = im
        mag = max(rexp+rbc, iexp+ibc)
        wp2 = wp + mag
        pi = mpf_pi(wp+wp2)
        pi2 = (mpf_shift(pi, 1), fzero)
        d = mpc_div_mpf(mpc_pow(pi2, s, wp2), pi, wp2)
        return mpc_mul(a,mpc_mul(b,mpc_mul(c,d,wp),wp),prec,rnd)
    n = int(wp/2.54 + 5)
    n += int(0.9*abs(to_int(im)))
    d = borwein_coefficients(n)
    ref = to_fixed(re, wp)
    imf = to_fixed(im, wp)
    tre = MP_ZERO
    tim = MP_ZERO
    one = MP_ONE << wp
    one_2wp = MP_ONE << (2*wp)
    critical_line = re == fhalf
    for k in xrange(n):
        log = log_int_fixed(k+1, wp)
        # A square root is much cheaper than an exp
        if critical_line:
            w = one_2wp // sqrt_fixed((k+1) << wp, wp)
        else:
            w = to_fixed(mpf_exp(from_man_exp(-ref*log, -2*wp), wp), wp)
        if k & 1:
            w *= (d[n] - d[k])
        else:
            w *= (d[k] - d[n])
        wre, wim = cos_sin(from_man_exp(-imf * log_int_fixed(k+1, wp), -2*wp), wp)
        tre += (w * to_fixed(wre, wp)) >> wp
        tim += (w * to_fixed(wim, wp)) >> wp
    tre //= (-d[n])
    tim //= (-d[n])
    tre = from_man_exp(tre, -wp, wp)
    tim = from_man_exp(tim, -wp, wp)
    if alt:
        return mpc_pos((tre, tim), prec, rnd)
    else:
        q = mpc_sub(mpc_one, mpc_pow(mpc_two, mpc_sub(mpc_one, s, wp), wp), wp)
        return mpc_div((tre, tim), q, prec, rnd)
コード例 #6
0
def mpc_zeta(s, prec, rnd):
    re, im = s
    wp = prec + 20
    n = int(wp / 2.54 + 5)
    n += int(0.9 * abs(to_int(im)))
    d = borwein_coefficients(n)
    ref = to_fixed(re, wp)
    imf = to_fixed(im, wp)
    tre = MP_ZERO
    tim = MP_ZERO
    one = MP_ONE << wp
    one_2wp = MP_ONE << (2 * wp)
    critical_line = re == fhalf
    for k in xrange(n):
        log = log_int_fixed(k + 1, wp)
        # A square root is much cheaper than an exp
        if critical_line:
            w = one_2wp // sqrt_fixed((k + 1) << wp, wp)
        else:
            w = to_fixed(mpf_exp(from_man_exp(-ref * log, -2 * wp), wp), wp)
        if k & 1:
            w *= (d[n] - d[k])
        else:
            w *= (d[k] - d[n])
        wre, wim = cos_sin(
            from_man_exp(-imf * log_int_fixed(k + 1, wp), -2 * wp), wp)
        tre += (w * to_fixed(wre, wp)) >> wp
        tim += (w * to_fixed(wim, wp)) >> wp
    tre //= (-d[n])
    tim //= (-d[n])
    tre = from_man_exp(tre, -wp, wp)
    tim = from_man_exp(tim, -wp, wp)
    q = mpc_sub(mpc_one, mpc_pow(mpc_two, mpc_sub(mpc_one, s, wp), wp), wp)
    return mpc_div((tre, tim), q, prec, rnd)
コード例 #7
0
def mpc_psi(m, z, prec, rnd=round_fast):
    """
    Computation of the polygamma function of arbitrary integer order
    m >= 0, for a complex argument z.
    """
    if m == 0:
        return mpc_psi0(z, prec, rnd)
    re, im = z
    wp = prec + 20
    sign, man, exp, bc = re
    if not man:
        if re == finf and im == fzero:
            return (fzero, fzero)
        if re == fnan:
            return fnan
    # Recurrence
    w = to_int(re)
    n = int(0.4 * wp + 4 * m)
    s = mpc_zero
    if w < n:
        for k in xrange(w, n):
            t = mpc_pow_int(z, -m - 1, wp)
            s = mpc_add(s, t, wp)
            z = mpc_add_mpf(z, fone, wp)
    zm = mpc_pow_int(z, -m, wp)
    z2 = mpc_pow_int(z, -2, wp)
    # 1/m*(z+N)^m
    integral_term = mpc_div_mpf(zm, from_int(m), wp)
    s = mpc_add(s, integral_term, wp)
    # 1/2*(z+N)^(-(m+1))
    s = mpc_add(s, mpc_mul_mpf(mpc_div(zm, z, wp), fhalf, wp), wp)
    a = m + 1
    b = 2
    k = 1
    # Important: we want to sum up to the *relative* error,
    # not the absolute error, because psi^(m)(z) might be tiny
    magn = mpc_abs(s, 10)
    magn = magn[2] + magn[3]
    eps = mpf_shift(fone, magn - wp + 2)
    while 1:
        zm = mpc_mul(zm, z2, wp)
        bern = mpf_bernoulli(2 * k, wp)
        scal = mpf_mul_int(bern, a, wp)
        scal = mpf_div(scal, from_int(b), wp)
        term = mpc_mul_mpf(zm, scal, wp)
        s = mpc_add(s, term, wp)
        szterm = mpc_abs(term, 10)
        if k > 2 and mpf_le(szterm, eps):
            break
        #print k, to_str(szterm, 10), to_str(eps, 10)
        a *= (m + 2 * k) * (m + 2 * k + 1)
        b *= (2 * k + 1) * (2 * k + 2)
        k += 1
    # Scale and sign factor
    v = mpc_mul_mpf(s, mpf_gamma(from_int(m + 1), wp), prec, rnd)
    if not (m & 1):
        v = mpf_neg(v[0]), mpf_neg(v[1])
    return v
コード例 #8
0
def mpc_psi(m, z, prec, rnd=round_fast):
    """
    Computation of the polygamma function of arbitrary integer order
    m >= 0, for a complex argument z.
    """
    if m == 0:
        return mpc_psi0(z, prec, rnd)
    re, im = z
    wp = prec + 20
    sign, man, exp, bc = re
    if not man:
        if re == finf and im == fzero:
            return (fzero, fzero)
        if re == fnan:
            return fnan
    # Recurrence
    w = to_int(re)
    n = int(0.4*wp + 4*m)
    s = mpc_zero
    if w < n:
        for k in xrange(w, n):
            t = mpc_pow_int(z, -m-1, wp)
            s = mpc_add(s, t, wp)
            z = mpc_add_mpf(z, fone, wp)
    zm = mpc_pow_int(z, -m, wp)
    z2 = mpc_pow_int(z, -2, wp)
    # 1/m*(z+N)^m
    integral_term = mpc_div_mpf(zm, from_int(m), wp)
    s = mpc_add(s, integral_term, wp)
    # 1/2*(z+N)^(-(m+1))
    s = mpc_add(s, mpc_mul_mpf(mpc_div(zm, z, wp), fhalf, wp), wp)
    a = m + 1
    b = 2
    k = 1
    # Important: we want to sum up to the *relative* error,
    # not the absolute error, because psi^(m)(z) might be tiny
    magn = mpc_abs(s, 10)
    magn = magn[2]+magn[3]
    eps = mpf_shift(fone, magn-wp+2)
    while 1:
        zm = mpc_mul(zm, z2, wp)
        bern = mpf_bernoulli(2*k, wp)
        scal = mpf_mul_int(bern, a, wp)
        scal = mpf_div(scal, from_int(b), wp)
        term = mpc_mul_mpf(zm, scal, wp)
        s = mpc_add(s, term, wp)
        szterm = mpc_abs(term, 10)
        if k > 2 and mpf_le(szterm, eps):
            break
        #print k, to_str(szterm, 10), to_str(eps, 10)
        a *= (m+2*k)*(m+2*k+1)
        b *= (2*k+1)*(2*k+2)
        k += 1
    # Scale and sign factor
    v = mpc_mul_mpf(s, mpf_gamma(from_int(m+1), wp), prec, rnd)
    if not (m & 1):
        v = mpf_neg(v[0]), mpf_neg(v[1])
    return v
コード例 #9
0
def mpf_psi0(x, prec, rnd=round_fast):
    """
    Computation of the digamma function (psi function of order 0)
    of a real argument.
    """
    sign, man, exp, bc = x
    wp = prec + 10
    if not man:
        if x == finf: return x
        if x == fninf or x == fnan: return fnan
    if x == fzero or (exp >= 0 and sign):
        raise ValueError("polygamma pole")
    # Reflection formula
    if sign and exp + bc > 3:
        c, s = mpf_cos_sin_pi(x, wp)
        q = mpf_mul(mpf_div(c, s, wp), mpf_pi(wp), wp)
        p = mpf_psi0(mpf_sub(fone, x, wp), wp)
        return mpf_sub(p, q, prec, rnd)
    # The logarithmic term is accurate enough
    if (not sign) and bc + exp > wp:
        return mpf_log(mpf_sub(x, fone, wp), prec, rnd)
    # Initial recurrence to obtain a large enough x
    m = to_int(x)
    n = int(0.11 * wp) + 2
    s = MP_ZERO
    x = to_fixed(x, wp)
    one = MP_ONE << wp
    if m < n:
        for k in xrange(m, n):
            s -= (one << wp) // x
            x += one
    x -= one
    # Logarithmic term
    s += to_fixed(mpf_log(from_man_exp(x, -wp, wp), wp), wp)
    # Endpoint term in Euler-Maclaurin expansion
    s += (one << wp) // (2 * x)
    # Euler-Maclaurin remainder sum
    x2 = (x * x) >> wp
    t = one
    prev = 0
    k = 1
    while 1:
        t = (t * x2) >> wp
        bsign, bman, bexp, bbc = mpf_bernoulli(2 * k, wp)
        offset = (bexp + 2 * wp)
        if offset >= 0: term = (bman << offset) // (t * (2 * k))
        else: term = (bman >> (-offset)) // (t * (2 * k))
        if k & 1: s -= term
        else: s += term
        if k > 2 and term >= prev:
            break
        prev = term
        k += 1
    return from_man_exp(s, -wp, wp, rnd)
コード例 #10
0
ファイル: libmpc.py プロジェクト: jcockayne/sympy-rkern
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
        # XXX: workaround bug in gmpy
        if abs(re) < 0.1: re = 0
        if abs(im) < 0.1: im = 0
        re = MP_BASE(re)
        im = MP_BASE(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
コード例 #11
0
ファイル: libmpc.py プロジェクト: jcockayne/sympy-rkern
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
        # XXX: workaround bug in gmpy
        if abs(re) < 0.1: re = 0
        if abs(im) < 0.1: im = 0
        re = MP_BASE(re)
        im = MP_BASE(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
コード例 #12
0
def mpc_psi0(z, prec, rnd=round_fast):
    """
    Computation of the digamma function (psi function of order 0)
    of a complex argument.
    """
    re, im = z
    # Fall back to the real case
    if im == fzero:
        return (mpf_psi0(re, prec, rnd), fzero)
    wp = prec + 20
    sign, man, exp, bc = re
    # Reflection formula
    if sign and exp + bc > 3:
        c = mpc_cos_pi(z, wp)
        s = mpc_sin_pi(z, wp)
        q = mpc_mul(mpc_div(c, s, wp), (mpf_pi(wp), fzero), wp)
        p = mpc_psi0(mpc_sub(mpc_one, z, wp), wp)
        return mpc_sub(p, q, prec, rnd)
    # Just the logarithmic term
    if (not sign) and bc + exp > wp:
        return mpc_log(mpc_sub(z, mpc_one, wp), prec, rnd)
    # Initial recurrence to obtain a large enough z
    w = to_int(re)
    n = int(0.11 * wp) + 2
    s = mpc_zero
    if w < n:
        for k in xrange(w, n):
            s = mpc_sub(s, mpc_div(mpc_one, z, wp), wp)
            z = mpc_add_mpf(z, fone, wp)
    z = mpc_sub(z, mpc_one, wp)
    # Logarithmic and endpoint term
    s = mpc_add(s, mpc_log(z, wp), wp)
    s = mpc_add(s, mpc_div(mpc_half, z, wp), wp)
    # Euler-Maclaurin remainder sum
    z2 = mpc_mul(z, z, wp)
    t = mpc_one
    prev = mpc_zero
    k = 1
    eps = mpf_shift(fone, -wp + 2)
    while 1:
        t = mpc_mul(t, z2, wp)
        bern = mpf_bernoulli(2 * k, wp)
        term = mpc_div((bern, fzero), mpc_mul_int(t, 2 * k, wp), wp)
        s = mpc_sub(s, term, wp)
        szterm = mpc_abs(term, 10)
        if k > 2 and mpf_le(szterm, eps):
            break
        prev = term
        k += 1
    return s
コード例 #13
0
ファイル: gammazeta.py プロジェクト: jcockayne/sympy-rkern
def mpf_zeta(s, prec, rnd=round_fast):
    sign, man, exp, bc = s
    if not man:
        if s == fzero:
            return mpf_neg(fhalf)
        if s == finf:
            return fone
        return fnan
    wp = prec + 20
    # First term vanishes?
    if (not sign) and (exp + bc > (math.log(wp,2) + 2)):
        if rnd in (round_up, round_ceiling):
            return mpf_add(fone, mpf_shift(fone,-wp-10), prec, rnd)
        return fone
    elif exp >= 0:
        return mpf_zeta_int(to_int(s), prec, rnd)
    # Less than 0.5?
    if sign or (exp+bc) < 0:
        # XXX: -1 should be done exactly
        y = mpf_sub(fone, s, 10*wp)
        a = mpf_gamma(y, wp)
        b = mpf_zeta(y, wp)
        c = mpf_sin_pi(mpf_shift(s, -1), wp)
        wp2 = wp + (exp+bc)
        pi = mpf_pi(wp+wp2)
        d = mpf_div(mpf_pow(mpf_shift(pi, 1), s, wp2), pi, wp2)
        return mpf_mul(a,mpf_mul(b,mpf_mul(c,d,wp),wp),prec,rnd)
    t = MP_ZERO
    #wp += 16 - (prec & 15)
    # Use Borwein's algorithm
    n = int(wp/2.54 + 5)
    d = borwein_coefficients(n)
    t = MP_ZERO
    sf = to_fixed(s, wp)
    for k in xrange(n):
        u = from_man_exp(-sf*log_int_fixed(k+1, wp), -2*wp, wp)
        esign, eman, eexp, ebc = mpf_exp(u, wp)
        offset = eexp + wp
        if offset >= 0:
            w = ((d[k] - d[n]) * eman) << offset
        else:
            w = ((d[k] - d[n]) * eman) >> (-offset)
        if k & 1:
            t -= w
        else:
            t += w
    t = t // (-d[n])
    t = from_man_exp(t, -wp, wp)
    q = mpf_sub(fone, mpf_pow(ftwo, mpf_sub(fone, s, wp), wp), wp)
    return mpf_div(t, q, prec, rnd)
コード例 #14
0
def mpf_zeta(s, prec, rnd=round_fast):
    sign, man, exp, bc = s
    if not man:
        if s == fzero:
            return mpf_neg(fhalf)
        if s == finf:
            return fone
        return fnan
    wp = prec + 20
    # First term vanishes?
    if (not sign) and (exp + bc > (math.log(wp, 2) + 2)):
        if rnd in (round_up, round_ceiling):
            return mpf_add(fone, mpf_shift(fone, -wp - 10), prec, rnd)
        return fone
    elif exp >= 0:
        return mpf_zeta_int(to_int(s), prec, rnd)
    # Less than 0.5?
    if sign or (exp + bc) < 0:
        # XXX: -1 should be done exactly
        y = mpf_sub(fone, s, 10 * wp)
        a = mpf_gamma(y, wp)
        b = mpf_zeta(y, wp)
        c = mpf_sin_pi(mpf_shift(s, -1), wp)
        wp2 = wp + (exp + bc)
        pi = mpf_pi(wp + wp2)
        d = mpf_div(mpf_pow(mpf_shift(pi, 1), s, wp2), pi, wp2)
        return mpf_mul(a, mpf_mul(b, mpf_mul(c, d, wp), wp), prec, rnd)
    t = MP_ZERO
    #wp += 16 - (prec & 15)
    # Use Borwein's algorithm
    n = int(wp / 2.54 + 5)
    d = borwein_coefficients(n)
    t = MP_ZERO
    sf = to_fixed(s, wp)
    for k in xrange(n):
        u = from_man_exp(-sf * log_int_fixed(k + 1, wp), -2 * wp, wp)
        esign, eman, eexp, ebc = mpf_exp(u, wp)
        offset = eexp + wp
        if offset >= 0:
            w = ((d[k] - d[n]) * eman) << offset
        else:
            w = ((d[k] - d[n]) * eman) >> (-offset)
        if k & 1:
            t -= w
        else:
            t += w
    t = t // (-d[n])
    t = from_man_exp(t, -wp, wp)
    q = mpf_sub(fone, mpf_pow(ftwo, mpf_sub(fone, s, wp), wp), wp)
    return mpf_div(t, q, prec, rnd)
コード例 #15
0
def mpc_psi0(z, prec, rnd=round_fast):
    """
    Computation of the digamma function (psi function of order 0)
    of a complex argument.
    """
    re, im = z
    # Fall back to the real case
    if im == fzero:
        return (mpf_psi0(re, prec, rnd), fzero)
    wp = prec + 20
    sign, man, exp, bc = re
    # Reflection formula
    if sign and exp+bc > 3:
        c = mpc_cos_pi(z, wp)
        s = mpc_sin_pi(z, wp)
        q = mpc_mul_mpf(mpc_div(c, s, wp), mpf_pi(wp), wp)
        p = mpc_psi0(mpc_sub(mpc_one, z, wp), wp)
        return mpc_sub(p, q, prec, rnd)
    # Just the logarithmic term
    if (not sign) and bc + exp > wp:
        return mpc_log(mpc_sub(z, mpc_one, wp), prec, rnd)
    # Initial recurrence to obtain a large enough z
    w = to_int(re)
    n = int(0.11*wp) + 2
    s = mpc_zero
    if w < n:
        for k in xrange(w, n):
            s = mpc_sub(s, mpc_reciprocal(z, wp), wp)
            z = mpc_add_mpf(z, fone, wp)
    z = mpc_sub(z, mpc_one, wp)
    # Logarithmic and endpoint term
    s = mpc_add(s, mpc_log(z, wp), wp)
    s = mpc_add(s, mpc_div(mpc_half, z, wp), wp)
    # Euler-Maclaurin remainder sum
    z2 = mpc_square(z, wp)
    t = mpc_one
    prev = mpc_zero
    k = 1
    eps = mpf_shift(fone, -wp+2)
    while 1:
        t = mpc_mul(t, z2, wp)
        bern = mpf_bernoulli(2*k, wp)
        term = mpc_mpf_div(bern, mpc_mul_int(t, 2*k, wp), wp)
        s = mpc_sub(s, term, wp)
        szterm = mpc_abs(term, 10)
        if k > 2 and mpf_le(szterm, eps):
            break
        prev = term
        k += 1
    return s
コード例 #16
0
ファイル: libhyper.py プロジェクト: Aang/sympy
def mpf_ei(x, prec, rnd=round_fast, e1=False):
    if e1:
        x = mpf_neg(x)
    sign, man, exp, bc = x
    if e1 and not sign:
        if x == fzero:
            return finf
        raise ComplexResult("E1(x) for x < 0")
    if man:
        xabs = 0, man, exp, bc
        xmag = exp+bc
        wp = prec + 20
        can_use_asymp = xmag > wp
        if not can_use_asymp:
            if exp >= 0:
                xabsint = man << exp
            else:
                xabsint = man >> (-exp)
            can_use_asymp = xabsint > int(wp*0.693) + 10
        if can_use_asymp:
            if xmag > wp:
                v = fone
            else:
                v = from_man_exp(ei_asymptotic(to_fixed(x, wp), wp), -wp)
            v = mpf_mul(v, mpf_exp(x, wp), wp)
            v = mpf_div(v, x, prec, rnd)
        else:
            wp += 2*int(to_int(xabs))
            u = to_fixed(x, wp)
            v = ei_taylor(u, wp) + euler_fixed(wp)
            t1 = from_man_exp(v,-wp)
            t2 = mpf_log(xabs,wp)
            v = mpf_add(t1, t2, prec, rnd)
    else:
        if x == fzero: v = fninf
        elif x == finf: v = finf
        elif x == fninf: v = fzero
        else: v = fnan
    if e1:
        v = mpf_neg(v)
    return v
コード例 #17
0
def mpf_ei(x, prec, rnd=round_fast, e1=False):
    if e1:
        x = mpf_neg(x)
    sign, man, exp, bc = x
    if e1 and not sign:
        if x == fzero:
            return finf
        raise ComplexResult("E1(x) for x < 0")
    if man:
        xabs = 0, man, exp, bc
        xmag = exp + bc
        wp = prec + 20
        can_use_asymp = xmag > wp
        if not can_use_asymp:
            if exp >= 0:
                xabsint = man << exp
            else:
                xabsint = man >> (-exp)
            can_use_asymp = xabsint > int(wp * 0.693) + 10
        if can_use_asymp:
            if xmag > wp:
                v = fone
            else:
                v = from_man_exp(ei_asymptotic(to_fixed(x, wp), wp), -wp)
            v = mpf_mul(v, mpf_exp(x, wp), wp)
            v = mpf_div(v, x, prec, rnd)
        else:
            wp += 2 * int(to_int(xabs))
            u = to_fixed(x, wp)
            v = ei_taylor(u, wp) + euler_fixed(wp)
            t1 = from_man_exp(v, -wp)
            t2 = mpf_log(xabs, wp)
            v = mpf_add(t1, t2, prec, rnd)
    else:
        if x == fzero: v = fninf
        elif x == finf: v = finf
        elif x == fninf: v = fzero
        else: v = fnan
    if e1:
        v = mpf_neg(v)
    return v
コード例 #18
0
def nthroot_fixed(y, n, prec, exp1):
    start = 50
    try:
        y1 = rshift(y, prec - n*start)
        r = MP_BASE(int(y1**(1.0/n)))
    except OverflowError:
        y1 = from_int(y1, start)
        fn = from_int(n)
        fn = mpf_rdiv_int(1, fn, start)
        r = mpf_pow(y1, fn, start)
        r = to_int(r)
    extra = 10
    extra1 = n
    prevp = start
    for p in giant_steps(start, prec+extra):
        pm, pe = int_pow_fixed(r, n-1, prevp)
        r2 = rshift(pm, (n-1)*prevp - p - pe - extra1)
        B = lshift(y, 2*p-prec+extra1)//r2
        r = (B + (n-1) * lshift(r, p-prevp))//n
        prevp = p
    return r
コード例 #19
0
ファイル: libelefun.py プロジェクト: jcockayne/sympy-rkern
def nthroot_fixed(y, n, prec, exp1):
    start = 50
    try:
        y1 = rshift(y, prec - n*start)
        r = MP_BASE(y1**(1.0/n))
    except OverflowError:
        y1 = from_int(y1, start)
        fn = from_int(n)
        fn = mpf_rdiv_int(1, fn, start)
        r = mpf_pow(y1, fn, start)
        r = to_int(r)
    extra = 10
    extra1 = n
    prevp = start
    for p in giant_steps(start, prec+extra):
        pm, pe = int_pow_fixed(r, n-1, prevp)
        r2 = rshift(pm, (n-1)*prevp - p - pe - extra1)
        B = lshift(y, 2*p-prec+extra1)//r2
        r = (B + (n-1) * lshift(r, p-prevp))//n
        prevp = p
    return r
コード例 #20
0
ファイル: libhyper.py プロジェクト: laudehenri/rSymPy
def mpf_erfc(x, prec, rnd=round_fast):
    sign, man, exp, bc = x
    if not man:
        if x == fzero: return fone
        if x == finf: return fzero
        if x == fninf: return ftwo
        return fnan
    wp = prec + 20
    mag = bc + exp
    # Preserve full accuracy when exponent grows huge
    wp += max(0, 2 * mag)
    regular_erf = sign or mag < 2
    if regular_erf or not erfc_check_series(x, wp):
        if regular_erf:
            return mpf_sub(fone, mpf_erf(x, prec + 10, negative_rnd[rnd]),
                           prec, rnd)
        # 1-erf(x) ~ exp(-x^2), increase prec to deal with cancellation
        n = to_int(x)
        return mpf_sub(fone, mpf_erf(x, prec + int(n**2 * 1.44) + 10), prec,
                       rnd)
    s = term = MP_ONE << wp
    term_prev = 0
    t = (2 * to_fixed(x, wp)**2) >> wp
    k = 1
    while 1:
        term = ((term * (2 * k - 1)) << wp) // t
        if k > 4 and term > term_prev or not term:
            break
        if k & 1:
            s -= term
        else:
            s += term
        term_prev = term
        #print k, to_str(from_man_exp(term, -wp, 50), 10)
        k += 1
    s = (s << wp) // sqrt_fixed(pi_fixed(wp), wp)
    s = from_man_exp(s, -wp, wp)
    z = mpf_exp(mpf_neg(mpf_mul(x, x, wp), wp), wp)
    y = mpf_div(mpf_mul(z, s, wp), x, prec, rnd)
    return y
コード例 #21
0
def bernfrac(n):
    """
    Computes integers (p,q) such that p/q = B_n exactly, where
    B_n denotes the nth Bernoulli number.

    Use bernoulli(n) to get a floating-point approximation
    instead of the exact fraction (much faster for large n).
    """
    n = int(n)
    if n < 3:
        return [(1, 1), (-1, 2), (1, 6)][n]
    if n & 1:
        return (0, 1)
    q = 1
    for k in list_primes(n + 1):
        if not (n % (k - 1)):
            q *= k
    prec = bernoulli_size(n) + int(math.log(q, 2)) + 20
    b = mpf_bernoulli(n, prec)
    p = mpf_mul(b, from_int(q))
    pint = to_int(p, round_nearest)
    return (pint, q)
コード例 #22
0
def mpf_fibonacci(x, prec, rnd=round_fast):
    sign, man, exp, bc = x
    if not man:
        if x == fninf:
            return fnan
        return x
    # F(2^n) ~= 2^(2^n)
    size = abs(exp+bc)
    if exp >= 0:
        # Exact
        if size < 10 or size <= bitcount(prec):
            return from_int(ifib(to_int(x)), prec, rnd)
    # Use the modified Binet formula
    wp = prec + size + 20
    a = mpf_phi(wp)
    b = mpf_add(mpf_shift(a, 1), fnone, wp)
    u = mpf_pow(a, x, wp)
    v = mpf_cos_pi(x, wp)
    v = mpf_div(v, u, wp)
    u = mpf_sub(u, v, wp)
    u = mpf_div(u, b, prec, rnd)
    return u
コード例 #23
0
ファイル: gammazeta.py プロジェクト: jcockayne/sympy-rkern
def bernfrac(n):
    """
    Computes integers (p,q) such that p/q = B_n exactly, where
    B_n denotes the nth Bernoulli number.

    Use bernoulli(n) to get a floating-point approximation
    instead of the exact fraction (much faster for large n).
    """
    n = int(n)
    if n < 3:
        return [(1, 1), (-1, 2), (1, 6)][n]
    if n & 1:
        return (0, 1)
    q = 1
    for k in list_primes(n+1):
        if not (n % (k-1)):
            q *= k
    prec = bernoulli_size(n) + int(math.log(q,2)) + 20
    b = mpf_bernoulli(n, prec)
    p = mpf_mul(b, from_int(q))
    pint = to_int(p, round_nearest)
    return (pint, q)
コード例 #24
0
ファイル: libhyper.py プロジェクト: Aang/sympy
def mpf_erfc(x, prec, rnd=round_fast):
    sign, man, exp, bc = x
    if not man:
        if x == fzero: return fone
        if x == finf: return fzero
        if x == fninf: return ftwo
        return fnan
    wp = prec + 20
    mag = bc+exp
    # Preserve full accuracy when exponent grows huge
    wp += max(0, 2*mag)
    regular_erf = sign or mag < 2
    if regular_erf or not erfc_check_series(x, wp):
        if regular_erf:
            return mpf_sub(fone, mpf_erf(x, prec+10, negative_rnd[rnd]), prec, rnd)
        # 1-erf(x) ~ exp(-x^2), increase prec to deal with cancellation
        n = to_int(x)+1
        return mpf_sub(fone, mpf_erf(x, prec + int(n**2*1.44) + 10), prec, rnd)
    s = term = MPZ_ONE << wp
    term_prev = 0
    t = (2 * to_fixed(x, wp) ** 2) >> wp
    k = 1
    while 1:
        term = ((term * (2*k - 1)) << wp) // t
        if k > 4 and term > term_prev or not term:
            break
        if k & 1:
            s -= term
        else:
            s += term
        term_prev = term
        #print k, to_str(from_man_exp(term, -wp, 50), 10)
        k += 1
    s = (s << wp) // sqrt_fixed(pi_fixed(wp), wp)
    s = from_man_exp(s, -wp, wp)
    z = mpf_exp(mpf_neg(mpf_mul(x,x,wp),wp),wp)
    y = mpf_div(mpf_mul(z, s, wp), x, prec, rnd)
    return y
コード例 #25
0
ファイル: mptypes.py プロジェクト: gnulinooks/sympy
 def __int__(s): return int(to_int(s._mpf_))
 def __long__(s): return long(to_int(s._mpf_))
コード例 #26
0
def mpc_ei(z, prec, rnd=round_fast, e1=False):
    if e1:
        z = mpc_neg(z)
    a, b = z
    asign, aman, aexp, abc = a
    bsign, bman, bexp, bbc = b
    if b == fzero:
        if e1:
            x = mpf_neg(mpf_ei(a, prec, rnd))
            if not asign:
                y = mpf_neg(mpf_pi(prec, rnd))
            else:
                y = fzero
            return x, y
        else:
            return mpf_ei(a, prec, rnd), fzero
    if a != fzero:
        if not aman or not bman:
            return (fnan, fnan)
    wp = prec + 40
    amag = aexp + abc
    bmag = bexp + bbc
    zmag = max(amag, bmag)
    can_use_asymp = zmag > wp
    if not can_use_asymp:
        zabsint = abs(to_int(a)) + abs(to_int(b))
        can_use_asymp = zabsint > int(wp * 0.693) + 20
    try:
        if can_use_asymp:
            if zmag > wp:
                v = fone, fzero
            else:
                zre = to_fixed(a, wp)
                zim = to_fixed(b, wp)
                vre, vim = complex_ei_asymptotic(zre, zim, wp)
                v = from_man_exp(vre, -wp), from_man_exp(vim, -wp)
            v = mpc_mul(v, mpc_exp(z, wp), wp)
            v = mpc_div(v, z, wp)
            if e1:
                v = mpc_neg(v, prec, rnd)
            else:
                x, y = v
                if bsign:
                    v = mpf_pos(x, prec, rnd), mpf_sub(y, mpf_pi(wp), prec,
                                                       rnd)
                else:
                    v = mpf_pos(x, prec, rnd), mpf_add(y, mpf_pi(wp), prec,
                                                       rnd)
            return v
    except NoConvergence:
        pass
    #wp += 2*max(0,zmag)
    wp += 2 * int(to_int(mpc_abs(z, 5)))
    zre = to_fixed(a, wp)
    zim = to_fixed(b, wp)
    vre, vim = complex_ei_taylor(zre, zim, wp)
    vre += euler_fixed(wp)
    v = from_man_exp(vre, -wp), from_man_exp(vim, -wp)
    if e1:
        u = mpc_log(mpc_neg(z), wp)
    else:
        u = mpc_log(z, wp)
    v = mpc_add(v, u, prec, rnd)
    if e1:
        v = mpc_neg(v)
    return v
コード例 #27
0
ファイル: libhyper.py プロジェクト: laudehenri/rSymPy
def erfc_check_series(x, prec):
    n = to_int(x)
    if n**2 * 1.44 > prec:
        return True
    return False
コード例 #28
0
ファイル: mptypes.py プロジェクト: gnulinooks/sympy
 def __long__(s): return long(to_int(s._mpf_))
 def __float__(s): return to_float(s._mpf_)
コード例 #29
0
ファイル: gammazeta.py プロジェクト: laudehenri/rSymPy
def bernfrac(n):
    r"""
    Returns a tuple of integers `(p, q)` such that `p/q = B_n` exactly,
    where `B_n` denotes the `n`-th Bernoulli number. The fraction is
    always reduced to lowest terms. Note that for `n > 1` and `n` odd,
    `B_n = 0`, and `(0, 1)` is returned.

    **Examples**

    The first few Bernoulli numbers are exactly::

        >>> from mpmath import *
        >>> for n in range(15):
        ...     p, q = bernfrac(n)
        ...     print n, "%s/%s" % (p, q)
        ...
        0 1/1
        1 -1/2
        2 1/6
        3 0/1
        4 -1/30
        5 0/1
        6 1/42
        7 0/1
        8 -1/30
        9 0/1
        10 5/66
        11 0/1
        12 -691/2730
        13 0/1
        14 7/6

    This function works for arbitrarily large `n`::

        >>> p, q = bernfrac(10**4)
        >>> print q
        2338224387510
        >>> print len(str(p))
        27692
        >>> mp.dps = 15
        >>> print mpf(p) / q
        -9.04942396360948e+27677
        >>> print bernoulli(10**4)
        -9.04942396360948e+27677

    Note: :func:`bernoulli` computes a floating-point approximation
    directly, without computing the exact fraction first.
    This is much faster for large `n`.

    **Algorithm**

    :func:`bernfrac` works by computing the value of `B_n` numerically
    and then using the von Staudt-Clausen theorem [1] to reconstruct
    the exact fraction. For large `n`, this is significantly faster than
    computing `B_1, B_2, \ldots, B_2` recursively with exact arithmetic.
    The implementation has been tested for `n = 10^m` up to `m = 6`.

    In practice, :func:`bernfrac` appears to be about three times
    slower than the specialized program calcbn.exe [2]

    **References**

    1. MathWorld, von Staudt-Clausen Theorem:
       http://mathworld.wolfram.com/vonStaudt-ClausenTheorem.html

    2. The Bernoulli Number Page:
       http://www.bernoulli.org/

    """
    n = int(n)
    if n < 3:
        return [(1, 1), (-1, 2), (1, 6)][n]
    if n & 1:
        return (0, 1)
    q = 1
    for k in list_primes(n + 1):
        if not (n % (k - 1)):
            q *= k
    prec = bernoulli_size(n) + int(math.log(q, 2)) + 20
    b = mpf_bernoulli(n, prec)
    p = mpf_mul(b, from_int(q))
    pint = to_int(p, round_nearest)
    return (pint, q)
コード例 #30
0
def bernfrac(n):
    r"""
    Returns a tuple of integers `(p, q)` such that `p/q = B_n` exactly,
    where `B_n` denotes the `n`-th Bernoulli number. The fraction is
    always reduced to lowest terms. Note that for `n > 1` and `n` odd,
    `B_n = 0`, and `(0, 1)` is returned.

    **Examples**

    The first few Bernoulli numbers are exactly::

        >>> from mpmath import *
        >>> for n in range(15):
        ...     p, q = bernfrac(n)
        ...     print n, "%s/%s" % (p, q)
        ...
        0 1/1
        1 -1/2
        2 1/6
        3 0/1
        4 -1/30
        5 0/1
        6 1/42
        7 0/1
        8 -1/30
        9 0/1
        10 5/66
        11 0/1
        12 -691/2730
        13 0/1
        14 7/6

    This function works for arbitrarily large `n`::

        >>> p, q = bernfrac(10**4)
        >>> print q
        2338224387510
        >>> print len(str(p))
        27692
        >>> mp.dps = 15
        >>> print mpf(p) / q
        -9.04942396360948e+27677
        >>> print bernoulli(10**4)
        -9.04942396360948e+27677

    Note: :func:`bernoulli` computes a floating-point approximation
    directly, without computing the exact fraction first.
    This is much faster for large `n`.

    **Algorithm**

    :func:`bernfrac` works by computing the value of `B_n` numerically
    and then using the von Staudt-Clausen theorem [1] to reconstruct
    the exact fraction. For large `n`, this is significantly faster than
    computing `B_1, B_2, \ldots, B_2` recursively with exact arithmetic.
    The implementation has been tested for `n = 10^m` up to `m = 6`.

    In practice, :func:`bernfrac` appears to be about three times
    slower than the specialized program calcbn.exe [2]

    **References**

    1. MathWorld, von Staudt-Clausen Theorem:
       http://mathworld.wolfram.com/vonStaudt-ClausenTheorem.html

    2. The Bernoulli Number Page:
       http://www.bernoulli.org/

    """
    n = int(n)
    if n < 3:
        return [(1, 1), (-1, 2), (1, 6)][n]
    if n & 1:
        return (0, 1)
    q = 1
    for k in list_primes(n+1):
        if not (n % (k-1)):
            q *= k
    prec = bernoulli_size(n) + int(math.log(q,2)) + 20
    b = mpf_bernoulli(n, prec)
    p = mpf_mul(b, from_int(q))
    pint = to_int(p, round_nearest)
    return (pint, q)
コード例 #31
0
ファイル: libmpi.py プロジェクト: Aang/sympy
def mpci_gamma(z, prec, type=0):
    (a1,a2), (b1,b2) = z

    # Real case
    if b1 == b2 == fzero and (type != 3 or mpf_gt(a1,fzero)):
        return mpi_gamma(z, prec, type), mpi_zero

    # Estimate precision
    wp = prec+20
    if type != 3:
        amag = a2[2]+a2[3]
        bmag = b2[2]+b2[3]
        if a2 != fzero:
            mag = max(amag, bmag)
        else:
            mag = bmag
        an = abs(to_int(a2))
        bn = abs(to_int(b2))
        absn = max(an, bn)
        gamma_size = max(0,absn*mag)
        wp += bitcount(gamma_size)

    # Assume type != 1
    if type == 1:
        (a1,a2) = mpi_add((a1,a2), mpi_one, wp); z = (a1,a2), (b1,b2)
        type = 0

    # Avoid non-monotonic region near the negative real axis
    if mpf_lt(a1, gamma_min_b):
        if mpi_overlap((b1,b2), (gamma_mono_imag_a, gamma_mono_imag_b)):
            # TODO: reflection formula
            #if mpf_lt(a2, mpf_shift(fone,-1)):
            #    znew = mpci_sub((mpi_one,mpi_zero),z,wp)
            #    ...
            # Recurrence:
            # gamma(z) = gamma(z+1)/z
            znew = mpi_add((a1,a2), mpi_one, wp), (b1,b2)
            if type == 0: return mpci_div(mpci_gamma(znew, prec+2, 0), z, prec)
            if type == 2: return mpci_mul(mpci_gamma(znew, prec+2, 2), z, prec)
            if type == 3: return mpci_sub(mpci_gamma(znew, prec+2, 3), mpci_log(z,prec+2), prec)

    # Use monotonicity (except for a small region close to the
    # origin and near poles)
    # upper half-plane
    if mpf_ge(b1, fzero):
        minre = mpc_loggamma((a1,b2), wp, round_floor)
        maxre = mpc_loggamma((a2,b1), wp, round_ceiling)
        minim = mpc_loggamma((a1,b1), wp, round_floor)
        maxim = mpc_loggamma((a2,b2), wp, round_ceiling)
    # lower half-plane
    elif mpf_le(b2, fzero):
        minre = mpc_loggamma((a1,b1), wp, round_floor)
        maxre = mpc_loggamma((a2,b2), wp, round_ceiling)
        minim = mpc_loggamma((a2,b1), wp, round_floor)
        maxim = mpc_loggamma((a1,b2), wp, round_ceiling)
    # crosses real axis
    else:
        maxre = mpc_loggamma((a2,fzero), wp, round_ceiling)
        # stretches more into the lower half-plane
        if mpf_gt(mpf_neg(b1), b2):
            minre = mpc_loggamma((a1,b1), wp, round_ceiling)
        else:
            minre = mpc_loggamma((a1,b2), wp, round_ceiling)
        minim = mpc_loggamma((a2,b1), wp, round_floor)
        maxim = mpc_loggamma((a2,b2), wp, round_floor)

    w = (minre[0], maxre[0]), (minim[1], maxim[1])
    if type == 3:
        return mpi_pos(w[0], prec), mpi_pos(w[1], prec)
    if type == 2:
        w = mpci_neg(w)
    return mpci_exp(w, prec)
コード例 #32
0
ファイル: libmpi.py プロジェクト: yuri-karadzhov/sympy
def mpci_gamma(z, prec, type=0):
    (a1, a2), (b1, b2) = z

    # Real case
    if b1 == b2 == fzero and (type != 3 or mpf_gt(a1, fzero)):
        return mpi_gamma(z, prec, type), mpi_zero

    # Estimate precision
    wp = prec + 20
    if type != 3:
        amag = a2[2] + a2[3]
        bmag = b2[2] + b2[3]
        if a2 != fzero:
            mag = max(amag, bmag)
        else:
            mag = bmag
        an = abs(to_int(a2))
        bn = abs(to_int(b2))
        absn = max(an, bn)
        gamma_size = max(0, absn * mag)
        wp += bitcount(gamma_size)

    # Assume type != 1
    if type == 1:
        (a1, a2) = mpi_add((a1, a2), mpi_one, wp)
        z = (a1, a2), (b1, b2)
        type = 0

    # Avoid non-monotonic region near the negative real axis
    if mpf_lt(a1, gamma_min_b):
        if mpi_overlap((b1, b2), (gamma_mono_imag_a, gamma_mono_imag_b)):
            # TODO: reflection formula
            #if mpf_lt(a2, mpf_shift(fone,-1)):
            #    znew = mpci_sub((mpi_one,mpi_zero),z,wp)
            #    ...
            # Recurrence:
            # gamma(z) = gamma(z+1)/z
            znew = mpi_add((a1, a2), mpi_one, wp), (b1, b2)
            if type == 0:
                return mpci_div(mpci_gamma(znew, prec + 2, 0), z, prec)
            if type == 2:
                return mpci_mul(mpci_gamma(znew, prec + 2, 2), z, prec)
            if type == 3:
                return mpci_sub(mpci_gamma(znew, prec + 2, 3),
                                mpci_log(z, prec + 2), prec)

    # Use monotonicity (except for a small region close to the
    # origin and near poles)
    # upper half-plane
    if mpf_ge(b1, fzero):
        minre = mpc_loggamma((a1, b2), wp, round_floor)
        maxre = mpc_loggamma((a2, b1), wp, round_ceiling)
        minim = mpc_loggamma((a1, b1), wp, round_floor)
        maxim = mpc_loggamma((a2, b2), wp, round_ceiling)
    # lower half-plane
    elif mpf_le(b2, fzero):
        minre = mpc_loggamma((a1, b1), wp, round_floor)
        maxre = mpc_loggamma((a2, b2), wp, round_ceiling)
        minim = mpc_loggamma((a2, b1), wp, round_floor)
        maxim = mpc_loggamma((a1, b2), wp, round_ceiling)
    # crosses real axis
    else:
        maxre = mpc_loggamma((a2, fzero), wp, round_ceiling)
        # stretches more into the lower half-plane
        if mpf_gt(mpf_neg(b1), b2):
            minre = mpc_loggamma((a1, b1), wp, round_ceiling)
        else:
            minre = mpc_loggamma((a1, b2), wp, round_ceiling)
        minim = mpc_loggamma((a2, b1), wp, round_floor)
        maxim = mpc_loggamma((a2, b2), wp, round_floor)

    w = (minre[0], maxre[0]), (minim[1], maxim[1])
    if type == 3:
        return mpi_pos(w[0], prec), mpi_pos(w[1], prec)
    if type == 2:
        w = mpci_neg(w)
    return mpci_exp(w, prec)
コード例 #33
0
ファイル: libhyper.py プロジェクト: Aang/sympy
def mpf_expint(n, x, prec, rnd=round_fast, gamma=False):
    """
    E_n(x), n an integer, x real

    With gamma=True, computes Gamma(n,x)   (upper incomplete gamma function)

    Returns (real, None) if real, otherwise (real, imag)
    The imaginary part is an optional branch cut term

    """
    sign, man, exp, bc = x
    if not man:
        if gamma:
            if x == fzero:
                # Actually gamma function pole
                if n <= 0:
                    return finf, None
                return mpf_gamma_int(n, prec, rnd), None
            if x == finf:
                return fzero, None
            # TODO: could return finite imaginary value at -inf
            return fnan, fnan
        else:
            if x == fzero:
                if n > 1:
                    return from_rational(1, n-1, prec, rnd), None
                else:
                    return finf, None
            if x == finf:
                return fzero, None
            return fnan, fnan
    n_orig = n
    if gamma:
        n = 1-n
    wp = prec + 20
    xmag = exp + bc
    # Beware of near-poles
    if xmag < -10:
        raise NotImplementedError
    nmag = bitcount(abs(n))
    have_imag = n > 0 and sign
    negx = mpf_neg(x)
    # Skip series if direct convergence
    if n == 0 or 2*nmag - xmag < -wp:
        if gamma:
            v = mpf_exp(negx, wp)
            re = mpf_mul(v, mpf_pow_int(x, n_orig-1, wp), prec, rnd)
        else:
            v = mpf_exp(negx, wp)
            re = mpf_div(v, x, prec, rnd)
    else:
        # Finite number of terms, or...
        can_use_asymptotic_series = -3*wp < n <= 0
        # ...large enough?
        if not can_use_asymptotic_series:
            xi = abs(to_int(x))
            m = min(max(1, xi-n), 2*wp)
            siz = -n*nmag + (m+n)*bitcount(abs(m+n)) - m*xmag - (144*m//100)
            tol = -wp-10
            can_use_asymptotic_series = siz < tol
        if can_use_asymptotic_series:
            r = ((-MPZ_ONE) << (wp+wp)) // to_fixed(x, wp)
            m = n
            t = r*m
            s = MPZ_ONE << wp
            while m and t:
                s += t
                m += 1
                t = (m*r*t) >> wp
            v = mpf_exp(negx, wp)
            if gamma:
                # ~ exp(-x) * x^(n-1) * (1 + ...)
                v = mpf_mul(v, mpf_pow_int(x, n_orig-1, wp), wp)
            else:
                # ~ exp(-x)/x * (1 + ...)
                v = mpf_div(v, x, wp)
            re = mpf_mul(v, from_man_exp(s, -wp), prec, rnd)
        elif n == 1:
            re = mpf_neg(mpf_ei(negx, prec, rnd))
        elif n > 0 and n < 3*wp:
            T1 = mpf_neg(mpf_ei(negx, wp))
            if gamma:
                if n_orig & 1:
                    T1 = mpf_neg(T1)
            else:
                T1 = mpf_mul(T1, mpf_pow_int(negx, n-1, wp), wp)
            r = t = to_fixed(x, wp)
            facs = [1] * (n-1)
            for k in range(1,n-1):
                facs[k] = facs[k-1] * k
            facs = facs[::-1]
            s = facs[0] << wp
            for k in range(1, n-1):
                if k & 1:
                    s -= facs[k] * t
                else:
                    s += facs[k] * t
                t = (t*r) >> wp
            T2 = from_man_exp(s, -wp, wp)
            T2 = mpf_mul(T2, mpf_exp(negx, wp))
            if gamma:
                T2 = mpf_mul(T2, mpf_pow_int(x, n_orig, wp), wp)
            R = mpf_add(T1, T2)
            re = mpf_div(R, from_int(ifac(n-1)), prec, rnd)
        else:
            raise NotImplementedError
    if have_imag:
        M = from_int(-ifac(n-1))
        if gamma:
            im = mpf_div(mpf_pi(wp), M, prec, rnd)
        else:
            im = mpf_div(mpf_mul(mpf_pi(wp), mpf_pow_int(negx, n_orig-1, wp), wp), M, prec, rnd)
        return re, im
    else:
        return re, None
コード例 #34
0
ファイル: mptypes.py プロジェクト: jcockayne/sympy-rkern
 def __long__(s): return long(to_int(s._mpf_))
 def __float__(s): return to_float(s._mpf_)
コード例 #35
0
ファイル: gammazeta.py プロジェクト: laudehenri/rSymPy
def mpc_zeta(s, prec, rnd=round_fast, alt=0):
    re, im = s
    if im == fzero:
        return mpf_zeta(re, prec, rnd, alt), fzero
    wp = prec + 20
    # Reflection formula. To be rigorous, we should reflect to the left of
    # re = 1/2 (see comments for mpf_zeta), but this leads to unnecessary
    # slowdown for interesting values of s
    if mpf_lt(re, fzero):
        # XXX: could use the separate refl. formula for Dirichlet eta
        if alt:
            q = mpc_sub(mpc_one, mpc_pow(mpc_two, mpc_sub(mpc_one, s, wp), wp),
                        wp)
            return mpc_mul(mpc_zeta(s, wp), q, prec, rnd)
        # XXX: -1 should be done exactly
        y = mpc_sub(mpc_one, s, 10 * wp)
        a = mpc_gamma(y, wp)
        b = mpc_zeta(y, wp)
        c = mpc_sin_pi(mpc_shift(s, -1), wp)
        rsign, rman, rexp, rbc = re
        isign, iman, iexp, ibc = im
        mag = max(rexp + rbc, iexp + ibc)
        wp2 = wp + mag
        pi = mpf_pi(wp + wp2)
        pi2 = (mpf_shift(pi, 1), fzero)
        d = mpc_div_mpf(mpc_pow(pi2, s, wp2), pi, wp2)
        return mpc_mul(a, mpc_mul(b, mpc_mul(c, d, wp), wp), prec, rnd)
    n = int(wp / 2.54 + 5)
    n += int(0.9 * abs(to_int(im)))
    d = borwein_coefficients(n)
    ref = to_fixed(re, wp)
    imf = to_fixed(im, wp)
    tre = MP_ZERO
    tim = MP_ZERO
    one = MP_ONE << wp
    one_2wp = MP_ONE << (2 * wp)
    critical_line = re == fhalf
    for k in xrange(n):
        log = log_int_fixed(k + 1, wp)
        # A square root is much cheaper than an exp
        if critical_line:
            w = one_2wp // sqrt_fixed((k + 1) << wp, wp)
        else:
            w = to_fixed(mpf_exp(from_man_exp(-ref * log, -2 * wp), wp), wp)
        if k & 1:
            w *= (d[n] - d[k])
        else:
            w *= (d[k] - d[n])
        wre, wim = cos_sin(
            from_man_exp(-imf * log_int_fixed(k + 1, wp), -2 * wp), wp)
        tre += (w * to_fixed(wre, wp)) >> wp
        tim += (w * to_fixed(wim, wp)) >> wp
    tre //= (-d[n])
    tim //= (-d[n])
    tre = from_man_exp(tre, -wp, wp)
    tim = from_man_exp(tim, -wp, wp)
    if alt:
        return mpc_pos((tre, tim), prec, rnd)
    else:
        q = mpc_sub(mpc_one, mpc_pow(mpc_two, mpc_sub(mpc_one, s, wp), wp), wp)
        return mpc_div((tre, tim), q, prec, rnd)
コード例 #36
0
def mpf_zeta(s, prec, rnd=round_fast, alt=0):
    sign, man, exp, bc = s
    if not man:
        if s == fzero:
            if alt:
                return fhalf
            else:
                return mpf_neg(fhalf)
        if s == finf:
            return fone
        return fnan
    wp = prec + 20
    # First term vanishes?
    if (not sign) and (exp + bc > (math.log(wp,2) + 2)):
        return mpf_perturb(fone, alt, prec, rnd)
    # Optimize for integer arguments
    elif exp >= 0:
        if alt:
            if s == fone:
                return mpf_ln2(prec, rnd)
            z = mpf_zeta_int(to_int(s), wp, negative_rnd[rnd])
            q = mpf_sub(fone, mpf_pow(ftwo, mpf_sub(fone, s, wp), wp), wp)
            return mpf_mul(z, q, prec, rnd)
        else:
            return mpf_zeta_int(to_int(s), prec, rnd)
    # Negative: use the reflection formula
    # Borwein only proves the accuracy bound for x >= 1/2. However, based on
    # tests, the accuracy without reflection is quite good even some distance
    # to the left of 1/2. XXX: verify this.
    if sign:
        # XXX: could use the separate refl. formula for Dirichlet eta
        if alt:
            q = mpf_sub(fone, mpf_pow(ftwo, mpf_sub(fone, s, wp), wp), wp)
            return mpf_mul(mpf_zeta(s, wp), q, prec, rnd)
        # XXX: -1 should be done exactly
        y = mpf_sub(fone, s, 10*wp)
        a = mpf_gamma(y, wp)
        b = mpf_zeta(y, wp)
        c = mpf_sin_pi(mpf_shift(s, -1), wp)
        wp2 = wp + (exp+bc)
        pi = mpf_pi(wp+wp2)
        d = mpf_div(mpf_pow(mpf_shift(pi, 1), s, wp2), pi, wp2)
        return mpf_mul(a,mpf_mul(b,mpf_mul(c,d,wp),wp),prec,rnd)

    # Near pole
    r = mpf_sub(fone, s, wp)
    asign, aman, aexp, abc = mpf_abs(r)
    pole_dist = -2*(aexp+abc)
    if pole_dist > wp:
        if alt:
            return mpf_ln2(prec, rnd)
        else:
            q = mpf_neg(mpf_div(fone, r, wp))
            return mpf_add(q, mpf_euler(wp), prec, rnd)
    else:
        wp += max(0, pole_dist)

    t = MPZ_ZERO
    #wp += 16 - (prec & 15)
    # Use Borwein's algorithm
    n = int(wp/2.54 + 5)
    d = borwein_coefficients(n)
    t = MPZ_ZERO
    sf = to_fixed(s, wp)
    for k in xrange(n):
        u = from_man_exp(-sf*log_int_fixed(k+1, wp), -2*wp, wp)
        esign, eman, eexp, ebc = mpf_exp(u, wp)
        offset = eexp + wp
        if offset >= 0:
            w = ((d[k] - d[n]) * eman) << offset
        else:
            w = ((d[k] - d[n]) * eman) >> (-offset)
        if k & 1:
            t -= w
        else:
            t += w
    t = t // (-d[n])
    t = from_man_exp(t, -wp, wp)
    if alt:
        return mpf_pos(t, prec, rnd)
    else:
        q = mpf_sub(fone, mpf_pow(ftwo, mpf_sub(fone, s, wp), wp), wp)
        return mpf_div(t, q, prec, rnd)
コード例 #37
0
ファイル: gammazeta.py プロジェクト: laudehenri/rSymPy
def mpf_zeta(s, prec, rnd=round_fast, alt=0):
    sign, man, exp, bc = s
    if not man:
        if s == fzero:
            if alt:
                return fhalf
            else:
                return mpf_neg(fhalf)
        if s == finf:
            return fone
        return fnan
    wp = prec + 20
    # First term vanishes?
    if (not sign) and (exp + bc > (math.log(wp, 2) + 2)):
        return mpf_perturb(fone, alt, prec, rnd)
    # Optimize for integer arguments
    elif exp >= 0:
        if alt:
            if s == fone:
                return mpf_ln2(prec, rnd)
            z = mpf_zeta_int(to_int(s), wp, negative_rnd[rnd])
            q = mpf_sub(fone, mpf_pow(ftwo, mpf_sub(fone, s, wp), wp), wp)
            return mpf_mul(z, q, prec, rnd)
        else:
            return mpf_zeta_int(to_int(s), prec, rnd)
    # Negative: use the reflection formula
    # Borwein only proves the accuracy bound for x >= 1/2. However, based on
    # tests, the accuracy without reflection is quite good even some distance
    # to the left of 1/2. XXX: verify this.
    if sign:
        # XXX: could use the separate refl. formula for Dirichlet eta
        if alt:
            q = mpf_sub(fone, mpf_pow(ftwo, mpf_sub(fone, s, wp), wp), wp)
            return mpf_mul(mpf_zeta(s, wp), q, prec, rnd)
        # XXX: -1 should be done exactly
        y = mpf_sub(fone, s, 10 * wp)
        a = mpf_gamma(y, wp)
        b = mpf_zeta(y, wp)
        c = mpf_sin_pi(mpf_shift(s, -1), wp)
        wp2 = wp + (exp + bc)
        pi = mpf_pi(wp + wp2)
        d = mpf_div(mpf_pow(mpf_shift(pi, 1), s, wp2), pi, wp2)
        return mpf_mul(a, mpf_mul(b, mpf_mul(c, d, wp), wp), prec, rnd)
    t = MP_ZERO
    #wp += 16 - (prec & 15)
    # Use Borwein's algorithm
    n = int(wp / 2.54 + 5)
    d = borwein_coefficients(n)
    t = MP_ZERO
    sf = to_fixed(s, wp)
    for k in xrange(n):
        u = from_man_exp(-sf * log_int_fixed(k + 1, wp), -2 * wp, wp)
        esign, eman, eexp, ebc = mpf_exp(u, wp)
        offset = eexp + wp
        if offset >= 0:
            w = ((d[k] - d[n]) * eman) << offset
        else:
            w = ((d[k] - d[n]) * eman) >> (-offset)
        if k & 1:
            t -= w
        else:
            t += w
    t = t // (-d[n])
    t = from_man_exp(t, -wp, wp)
    if alt:
        return mpf_pos(t, prec, rnd)
    else:
        q = mpf_sub(fone, mpf_pow(ftwo, mpf_sub(fone, s, wp), wp), wp)
        return mpf_div(t, q, prec, rnd)
コード例 #38
0
def mpc_zeta(s, prec, rnd=round_fast, alt=0, force=False):
    re, im = s
    if im == fzero:
        return mpf_zeta(re, prec, rnd, alt), fzero

    # slow for large s
    if (not force) and mpf_gt(mpc_abs(s, 10), from_int(prec)):
        raise NotImplementedError

    wp = prec + 20

    # Near pole
    r = mpc_sub(mpc_one, s, wp)
    asign, aman, aexp, abc = mpc_abs(r, 10)
    pole_dist = -2*(aexp+abc)
    if pole_dist > wp:
        if alt:
            q = mpf_ln2(wp)
            y = mpf_mul(q, mpf_euler(wp), wp)
            g = mpf_shift(mpf_mul(q, q, wp), -1)
            g = mpf_sub(y, g)
            z = mpc_mul_mpf(r, mpf_neg(g), wp)
            z = mpc_add_mpf(z, q, wp)
            return mpc_pos(z, prec, rnd)
        else:
            q = mpc_neg(mpc_div(mpc_one, r, wp))
            q = mpc_add_mpf(q, mpf_euler(wp), wp)
            return mpc_pos(q, prec, rnd)
    else:
        wp += max(0, pole_dist)

    # Reflection formula. To be rigorous, we should reflect to the left of
    # re = 1/2 (see comments for mpf_zeta), but this leads to unnecessary
    # slowdown for interesting values of s
    if mpf_lt(re, fzero):
        # XXX: could use the separate refl. formula for Dirichlet eta
        if alt:
            q = mpc_sub(mpc_one, mpc_pow(mpc_two, mpc_sub(mpc_one, s, wp),
                wp), wp)
            return mpc_mul(mpc_zeta(s, wp), q, prec, rnd)
        # XXX: -1 should be done exactly
        y = mpc_sub(mpc_one, s, 10*wp)
        a = mpc_gamma(y, wp)
        b = mpc_zeta(y, wp)
        c = mpc_sin_pi(mpc_shift(s, -1), wp)
        rsign, rman, rexp, rbc = re
        isign, iman, iexp, ibc = im
        mag = max(rexp+rbc, iexp+ibc)
        wp2 = wp + mag
        pi = mpf_pi(wp+wp2)
        pi2 = (mpf_shift(pi, 1), fzero)
        d = mpc_div_mpf(mpc_pow(pi2, s, wp2), pi, wp2)
        return mpc_mul(a,mpc_mul(b,mpc_mul(c,d,wp),wp),prec,rnd)
    n = int(wp/2.54 + 5)
    n += int(0.9*abs(to_int(im)))
    d = borwein_coefficients(n)
    ref = to_fixed(re, wp)
    imf = to_fixed(im, wp)
    tre = MPZ_ZERO
    tim = MPZ_ZERO
    one = MPZ_ONE << wp
    one_2wp = MPZ_ONE << (2*wp)
    critical_line = re == fhalf
    for k in xrange(n):
        log = log_int_fixed(k+1, wp)
        # A square root is much cheaper than an exp
        if critical_line:
            w = one_2wp // sqrt_fixed((k+1) << wp, wp)
        else:
            w = to_fixed(mpf_exp(from_man_exp(-ref*log, -2*wp), wp), wp)
        if k & 1:
            w *= (d[n] - d[k])
        else:
            w *= (d[k] - d[n])
        wre, wim = mpf_cos_sin(from_man_exp(-imf * log, -2*wp), wp)
        tre += (w * to_fixed(wre, wp)) >> wp
        tim += (w * to_fixed(wim, wp)) >> wp
    tre //= (-d[n])
    tim //= (-d[n])
    tre = from_man_exp(tre, -wp, wp)
    tim = from_man_exp(tim, -wp, wp)
    if alt:
        return mpc_pos((tre, tim), prec, rnd)
    else:
        q = mpc_sub(mpc_one, mpc_pow(mpc_two, r, wp), wp)
        return mpc_div((tre, tim), q, prec, rnd)
コード例 #39
0
def mpf_expint(n, x, prec, rnd=round_fast, gamma=False):
    """
    E_n(x), n an integer, x real

    With gamma=True, computes Gamma(n,x)   (upper incomplete gamma function)

    Returns (real, None) if real, otherwise (real, imag)
    The imaginary part is an optional branch cut term

    """
    sign, man, exp, bc = x
    if not man:
        if gamma:
            if x == fzero:
                # Actually gamma function pole
                if n <= 0:
                    return finf
                return mpf_gamma_int(n, prec, rnd)
            if x == finf:
                return fzero, None
            # TODO: could return finite imaginary value at -inf
            return fnan, fnan
        else:
            if x == fzero:
                if n > 1:
                    return from_rational(1, n - 1, prec, rnd), None
                else:
                    return finf, None
            if x == finf:
                return fzero, None
            return fnan, fnan
    n_orig = n
    if gamma:
        n = 1 - n
    wp = prec + 20
    xmag = exp + bc
    # Beware of near-poles
    if xmag < -10:
        raise NotImplementedError
    nmag = bitcount(abs(n))
    have_imag = n > 0 and sign
    negx = mpf_neg(x)
    # Skip series if direct convergence
    if n == 0 or 2 * nmag - xmag < -wp:
        if gamma:
            v = mpf_exp(negx, wp)
            re = mpf_mul(v, mpf_pow_int(x, n_orig - 1, wp), prec, rnd)
        else:
            v = mpf_exp(negx, wp)
            re = mpf_div(v, x, prec, rnd)
    else:
        # Finite number of terms, or...
        can_use_asymptotic_series = -3 * wp < n <= 0
        # ...large enough?
        if not can_use_asymptotic_series:
            xi = abs(to_int(x))
            m = min(max(1, xi - n), 2 * wp)
            siz = -n * nmag + (m + n) * bitcount(abs(m + n)) - m * xmag - (
                144 * m // 100)
            tol = -wp - 10
            can_use_asymptotic_series = siz < tol
        if can_use_asymptotic_series:
            r = ((-MP_ONE) << (wp + wp)) // to_fixed(x, wp)
            m = n
            t = r * m
            s = MP_ONE << wp
            while m and t:
                s += t
                m += 1
                t = (m * r * t) >> wp
            v = mpf_exp(negx, wp)
            if gamma:
                # ~ exp(-x) * x^(n-1) * (1 + ...)
                v = mpf_mul(v, mpf_pow_int(x, n_orig - 1, wp), wp)
            else:
                # ~ exp(-x)/x * (1 + ...)
                v = mpf_div(v, x, wp)
            re = mpf_mul(v, from_man_exp(s, -wp), prec, rnd)
        elif n == 1:
            re = mpf_neg(mpf_ei(negx, prec, rnd))
        elif n > 0 and n < 3 * wp:
            T1 = mpf_neg(mpf_ei(negx, wp))
            if gamma:
                if n_orig & 1:
                    T1 = mpf_neg(T1)
            else:
                T1 = mpf_mul(T1, mpf_pow_int(negx, n - 1, wp), wp)
            r = t = to_fixed(x, wp)
            facs = [1] * (n - 1)
            for k in range(1, n - 1):
                facs[k] = facs[k - 1] * k
            facs = facs[::-1]
            s = facs[0] << wp
            for k in range(1, n - 1):
                if k & 1:
                    s -= facs[k] * t
                else:
                    s += facs[k] * t
                t = (t * r) >> wp
            T2 = from_man_exp(s, -wp, wp)
            T2 = mpf_mul(T2, mpf_exp(negx, wp))
            if gamma:
                T2 = mpf_mul(T2, mpf_pow_int(x, n_orig, wp), wp)
            R = mpf_add(T1, T2)
            re = mpf_div(R, from_int(int_fac(n - 1)), prec, rnd)
        else:
            raise NotImplementedError
    if have_imag:
        M = from_int(-int_fac(n - 1))
        if gamma:
            im = mpf_div(mpf_pi(wp), M, prec, rnd)
        else:
            im = mpf_div(
                mpf_mul(mpf_pi(wp), mpf_pow_int(negx, n_orig - 1, wp), wp), M,
                prec, rnd)
        return re, im
    else:
        return re, None
コード例 #40
0
ファイル: libhyper.py プロジェクト: Aang/sympy
def mpc_ei(z, prec, rnd=round_fast, e1=False):
    if e1:
        z = mpc_neg(z)
    a, b = z
    asign, aman, aexp, abc = a
    bsign, bman, bexp, bbc = b
    if b == fzero:
        if e1:
            x = mpf_neg(mpf_ei(a, prec, rnd))
            if not asign:
                y = mpf_neg(mpf_pi(prec, rnd))
            else:
                y = fzero
            return x, y
        else:
            return mpf_ei(a, prec, rnd), fzero
    if a != fzero:
        if not aman or not bman:
            return (fnan, fnan)
    wp = prec + 40
    amag = aexp+abc
    bmag = bexp+bbc
    zmag = max(amag, bmag)
    can_use_asymp = zmag > wp
    if not can_use_asymp:
        zabsint = abs(to_int(a)) + abs(to_int(b))
        can_use_asymp = zabsint > int(wp*0.693) + 20
    try:
        if can_use_asymp:
            if zmag > wp:
                v = fone, fzero
            else:
                zre = to_fixed(a, wp)
                zim = to_fixed(b, wp)
                vre, vim = complex_ei_asymptotic(zre, zim, wp)
                v = from_man_exp(vre, -wp), from_man_exp(vim, -wp)
            v = mpc_mul(v, mpc_exp(z, wp), wp)
            v = mpc_div(v, z, wp)
            if e1:
                v = mpc_neg(v, prec, rnd)
            else:
                x, y = v
                if bsign:
                    v = mpf_pos(x, prec, rnd), mpf_sub(y, mpf_pi(wp), prec, rnd)
                else:
                    v = mpf_pos(x, prec, rnd), mpf_add(y, mpf_pi(wp), prec, rnd)
            return v
    except NoConvergence:
        pass
    #wp += 2*max(0,zmag)
    wp += 2*int(to_int(mpc_abs(z, 5)))
    zre = to_fixed(a, wp)
    zim = to_fixed(b, wp)
    vre, vim = complex_ei_taylor(zre, zim, wp)
    vre += euler_fixed(wp)
    v = from_man_exp(vre,-wp), from_man_exp(vim,-wp)
    if e1:
        u = mpc_log(mpc_neg(z),wp)
    else:
        u = mpc_log(z,wp)
    v = mpc_add(v, u, prec, rnd)
    if e1:
        v = mpc_neg(v)
    return v
コード例 #41
0
ファイル: libhyper.py プロジェクト: Aang/sympy
def erfc_check_series(x, prec):
    n = to_int(x)
    if n**2 * 1.44 > prec:
        return True
    return False
コード例 #42
0
ファイル: mptypes.py プロジェクト: jcockayne/sympy-rkern
 def __int__(s): return int(to_int(s._mpf_))
 def __long__(s): return long(to_int(s._mpf_))
コード例 #43
0
ファイル: gammazeta.py プロジェクト: xinjie0831/AMC
def mpc_zeta(s, prec, rnd=round_fast, alt=0, force=False):
    re, im = s
    if im == fzero:
        return mpf_zeta(re, prec, rnd, alt), fzero

    # slow for large s
    if (not force) and mpf_gt(mpc_abs(s, 10), from_int(prec)):
        raise NotImplementedError

    wp = prec + 20

    # Near pole
    r = mpc_sub(mpc_one, s, wp)
    asign, aman, aexp, abc = mpc_abs(r, 10)
    pole_dist = -2 * (aexp + abc)
    if pole_dist > wp:
        if alt:
            q = mpf_ln2(wp)
            y = mpf_mul(q, mpf_euler(wp), wp)
            g = mpf_shift(mpf_mul(q, q, wp), -1)
            g = mpf_sub(y, g)
            z = mpc_mul_mpf(r, mpf_neg(g), wp)
            z = mpc_add_mpf(z, q, wp)
            return mpc_pos(z, prec, rnd)
        else:
            q = mpc_neg(mpc_div(mpc_one, r, wp))
            q = mpc_add_mpf(q, mpf_euler(wp), wp)
            return mpc_pos(q, prec, rnd)
    else:
        wp += max(0, pole_dist)

    # Reflection formula. To be rigorous, we should reflect to the left of
    # re = 1/2 (see comments for mpf_zeta), but this leads to unnecessary
    # slowdown for interesting values of s
    if mpf_lt(re, fzero):
        # XXX: could use the separate refl. formula for Dirichlet eta
        if alt:
            q = mpc_sub(mpc_one, mpc_pow(mpc_two, mpc_sub(mpc_one, s, wp), wp),
                        wp)
            return mpc_mul(mpc_zeta(s, wp), q, prec, rnd)
        # XXX: -1 should be done exactly
        y = mpc_sub(mpc_one, s, 10 * wp)
        a = mpc_gamma(y, wp)
        b = mpc_zeta(y, wp)
        c = mpc_sin_pi(mpc_shift(s, -1), wp)
        rsign, rman, rexp, rbc = re
        isign, iman, iexp, ibc = im
        mag = max(rexp + rbc, iexp + ibc)
        wp2 = wp + mag
        pi = mpf_pi(wp + wp2)
        pi2 = (mpf_shift(pi, 1), fzero)
        d = mpc_div_mpf(mpc_pow(pi2, s, wp2), pi, wp2)
        return mpc_mul(a, mpc_mul(b, mpc_mul(c, d, wp), wp), prec, rnd)
    n = int(wp / 2.54 + 5)
    n += int(0.9 * abs(to_int(im)))
    d = borwein_coefficients(n)
    ref = to_fixed(re, wp)
    imf = to_fixed(im, wp)
    tre = MPZ_ZERO
    tim = MPZ_ZERO
    one = MPZ_ONE << wp
    one_2wp = MPZ_ONE << (2 * wp)
    critical_line = re == fhalf
    for k in xrange(n):
        log = log_int_fixed(k + 1, wp)
        # A square root is much cheaper than an exp
        if critical_line:
            w = one_2wp // sqrt_fixed((k + 1) << wp, wp)
        else:
            w = to_fixed(mpf_exp(from_man_exp(-ref * log, -2 * wp), wp), wp)
        if k & 1:
            w *= (d[n] - d[k])
        else:
            w *= (d[k] - d[n])
        wre, wim = mpf_cos_sin(from_man_exp(-imf * log, -2 * wp), wp)
        tre += (w * to_fixed(wre, wp)) >> wp
        tim += (w * to_fixed(wim, wp)) >> wp
    tre //= (-d[n])
    tim //= (-d[n])
    tre = from_man_exp(tre, -wp, wp)
    tim = from_man_exp(tim, -wp, wp)
    if alt:
        return mpc_pos((tre, tim), prec, rnd)
    else:
        q = mpc_sub(mpc_one, mpc_pow(mpc_two, r, wp), wp)
        return mpc_div((tre, tim), q, prec, rnd)