Example #1
0
def hypot(ctx, x, y):
    r"""
    Computes the Euclidean norm of the vector `(x, y)`, equal
    to `\sqrt{x^2 + y^2}`. Both `x` and `y` must be real."""
    x = ctx.convert(x)
    y = ctx.convert(y)
    return ctx.make_mpf(libmpf.mpf_hypot(x._mpf_, y._mpf_, *ctx._prec_rounding))
Example #2
0
File: libmpc.py Project: vks/sympy
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
Example #3
0
File: libmpc.py Project: vks/sympy
def mpc_abs(z, prec, rnd=round_fast):
    """Absolute value of a complex number, |a+bi|.
    Returns an mpf value."""
    a, b = z
    return mpf_hypot(a, b, prec, rnd)
Example #4
0
def mpc_sub_mpf((a, b), p, prec, rnd=round_fast):
    return mpf_sub(a, p, prec, rnd), b

def mpc_pos((a, b), prec, rnd=round_fast):
    return mpf_pos(a, prec, rnd), mpf_pos(b, prec, rnd)

def mpc_neg((a, b), prec=None, rnd=round_fast):
    return mpf_neg(a, prec, rnd), mpf_neg(b, prec, rnd)

def mpc_shift((a, b), n):
    return mpf_shift(a, n), mpf_shift(b, n)

def mpc_abs((a, b), prec, rnd=round_fast):
    """Absolute value of a complex number, |a+bi|.
    Returns an mpf value."""
    return mpf_hypot(a, b, prec, rnd)

def mpc_arg((a, b), prec, rnd=round_fast):
    """Argument of a complex number. Returns an mpf value."""
    return mpf_atan2(b, a, prec, rnd)

def mpc_floor((a, b), prec, rnd=round_fast):
    return mpf_floor(a, prec, rnd), mpf_floor(b, prec, rnd)

def mpc_ceil((a, b), prec, rnd=round_fast):
    return mpf_ceil(a, prec, rnd), mpf_ceil(b, prec, rnd)

def mpc_mul((a, b), (c, d), prec, rnd=round_fast):
    """Complex multiplication.

    Returns the real and imaginary part of (a+bi)*(c+di), rounded to
Example #5
0
def hypot(x, y):
    """Returns the Euclidean distance sqrt(x*x + y*y). Both x and y
    must be real."""
    x = convert_lossless(x)
    y = convert_lossless(y)
    return make_mpf(libmpf.mpf_hypot(x._mpf_, y._mpf_, *prec_rounding))