calculate bulk modulus numerically from volume, not pressure
    according to test this differs from analytical result by 1.e-5

    :param v: unit-cell volume in A^3
    :param v0: unit-cell volume in A^3 at 1 bar
    :param k0: bulk modulus at reference conditions
    :param k0p: pressure derivative of bulk modulus at reference conditions
    :param precision: precision for numerical calc (default = 1.e-5 * v0)
    :return: dP/dV

def vinet_k_num(v, v0, k0, k0p, precision=1.e-5):
    """
    calculate bulk modulus numerically from volume, not pressure
    according to test this differs from analytical result by 1.e-5

    :param v: unit-cell volume in A^3
    :param v0: unit-cell volume in A^3 at 1 bar
    :param k0: bulk modulus at reference conditions
    :param k0p: pressure derivative of bulk modulus at reference conditions
    :param precision: precision for numerical calc (default = 1.e-5 * v0)
    :return: dP/dV
    """
    return -1. * v * vinet_dPdV(v, v0, k0, k0p, precision=precision)