class MoreMath::NewtonBisection

method.
This class is used to find the root of a function with Newton’s bisection

def bracket(range = -1..1, n = 50, factor = 1.6)

+n+ tries with the scaling +factor+.
method returns nil, if no such bracket around a root could be found after
Return a bracket around a root, starting from the initial +range+. The 
def bracket(range = -1..1, n = 50, factor =  1.6)
  x1, x2 = range.first.to_f, range.last.to_f
  x1 >= x2 and raise ArgumentError, "bad initial range #{range}"
  f1, f2 = @function[x1], @function[x2]
  n.times do
    f1 * f2 < 0 and return x1..x2
    if f1.abs < f2.abs
      f1 = @function[x1 += factor * (x1 - x2)]
    else
      f2 = @function[x2 += factor * (x2 - x1)]
    end
  end
  return
end

def initialize(&function)

Creates a NewtonBisection instance for +function+, a one-argument block. 
def initialize(&function)
  @function = function
end

def solve(range = nil, n = 1 << 16, epsilon = 1E-16)

the +epsilon+ environment.
DivergentException, if no such root could be found after +n+ tries and in
Find the root of function in +range+ and return it. The method raises a 
def solve(range = nil, n = 1 << 16, epsilon = 1E-16)
  if range
    x1, x2 = range.first.to_f, range.last.to_f
    x1 >= x2 and raise ArgumentError, "bad initial range #{range}"
  elsif range = bracket
    x1, x2 = range.first, range.last
  else
    raise DivergentException, "bracket could not be determined"
  end
  f = @function[x1]
  fmid = @function[x2]
  f * fmid >= 0 and raise DivergentException, "root must be bracketed in #{range}"
  root = if f < 0
           dx = x2 - x1
           x1
         else
           dx = x1 - x2
           x2
         end
  n.times do
    fmid = @function[xmid = root + (dx *= 0.5)]
    fmid < 0 and root = xmid
    dx.abs < epsilon or fmid == 0 and return root
  end
  raise DivergentException, "too many iterations (#{n})"
end