  An iterative formula based on Newton Method alone is presented for the
iterative solutions of equations that ensures convergence in cases where the
traditional Newton Method may fail to converge to the desired root. In
addition, the method has super quadratic convergence of order 2.414. Newton
method is said to fail in certain cases leading to oscillation, divergence to
increasingly large number or off-shooting away to another root further from the
desired domain or off shooting to an invalid domain where the function may not
be defined. In addition when the derivative at the iteration point is zero,
Newton method stalls. In most of these cases, hybrids of several methods such
as Newton, bisection and secant methods are suggested as substitute methods and
Newton method is essentially blended with other methods or altogether
abandoned. This paper argues that a solution is still possible in most of these
cases by the application of Newton Method alone without resorting to other
methods and with the same computational effort, two functional evaluations per
iteration, like the traditional Newton method. In addition, the proposed
modified formula based on Newton method has better convergence characteristics
than the traditional Newton method.
