Abstract
An iterative formula based on Newton’s 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 (i.e., ). Newton method is said to fail in certain cases leading to oscillation, divergence to increasingly large number, or offshooting away to another root further from the desired domain or offshooting 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.
Highlights
Iterative procedures for solutions of equations are routinely employed in many science and engineering problems
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 like the traditional Newton method
Newton or secant methods can be combined with bisection to bracket the root by a small interval so that a good initial guess is available for applying Newton method with quadratic convergence
Summary
Iterative procedures for solutions of equations are routinely employed in many science and engineering problems. Newton’s method displays a faster quadratic convergence near the root while it requires evaluation of the function and its derivative at each step of the iteration. Modifications of the Newton method with higher order convergence have been proposed that require evaluation of a function and its derivatives. Newton or secant methods can be combined with bisection to bracket the root by a small interval so that a good initial guess is available for applying Newton method with quadratic convergence. Brent’s method [15] is a root finding algorithm that combines root bracketing, bisection, and inverse quadratic interpolation It is a modification of Dekker’s method to avoid slow convergence when the difference between consecutive estimates of x is arbitrarily small. Being dependent on Newton method for the intermediate step, the method may suffer from the same drawbacks of using the traditional Newton method mentioned above
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