It is generally supposed that from the beginning of his inquiries, Newton applied a special method in his scholarly researches on the nature and his two influential works, Mathematical Principles of Natural Philosophy and Optics, are the product of applying such a.method. However, Einstein has warned us that if want to know the physicists' methods, we should pay attention to their actions not ta their speeches. Keeping his warning in view, the author firstly attempts to evaluate Newton's scientific methodology appeared in the different editions of his Optics. Secondly, he tries to present what Newton has done in reality, which divulges the contradictions between what he has expressed and what he has done, and which reveals that there weren't any set of methodological principles in the beginning.

Evaluating the conformity of Newton’ s methodological statements with his actual practice of science is the prime goal of this article. The importance of the question is that despite much researches about Newton’ s scientific method in the last quarter of a century, there is still not even a rough picture of what that method looks like. I have surmised that part of the explanation for this inscrutability lies in the fact that Newton’ s words and deeds may, for good reasons, be on the whole not quite coherent. Thus, I want to raise and answer this question: Is it possible that a large share of the inscrutability of Newton’ s method rests on the fact that some of Newton’ s proclaimed methodological rules do not quite cohere with his practiced rules. But more so, in some cases, these two kinds of proclaimed and practiced rules may simply be inconsistent. For this, I have mainly relied on scrutiny of Newton’ s two classic works, The Mathematical Principles of Natural Philosophy and Optics, and some of his relevant correspondence.

In this paper, we apply Newton's method to solve a class of integro-differential equations of the Volterra-Fredholm type with nonlocal characteristics, involving almost sectorial operators and Hilfer fractional derivatives. Since these equations play a key role in mathematics, engineering, biology, physics, chemistry, control theory and  economy, finding an appropriate solution is important. By Newton's method, we linearize a nonlinear Volterra-Fredholm integro-differential equation which is equivalent to a category of Volterra-Fredholm integro-differential equations with nonlocal properties, incorporating almost sectorial operators and Hilfer fractional derivatives and nearly sectorial operators. This technique has been shown to be an effective tool for the numerical solution of initial value problems in nonlinear integro-differential equations. The convergence analysis is also investigated.

The computation of the inverse roots of matrices arises in evaluating unsymmetric eigenvalue problems, solving nonlinear matrix equations, computing some matrix functions, control theory and several other areas of applications. It is possible to approximate the matrix inverse pth roots by exploiting a specialized version of Newton's method, but previous researchers have mentioned that some iterations have poor convergence and stability proper- ties. In this work, a stable recursive technique to evaluate an inverse pth root of a given matrix is presented. The scheme is analyzed and its properties are investigated. Computational experiments are also performed to illustrate the strengths and weaknesses of the proposed method.

While many real-world optimization problems typically involve multiple constraints, unconstrained problems hold practical and fundamental significance. They can arise directly in specific applications or as transformed versions of constrained optimization problems.‎ ‎Newton's method‎, ‎a notable numerical technique within the category of line search algorithms, is widely used for function optimization‎. The search direction and step length play crucial roles in this algorithm. ‎This paper introduces an algorithm aimed at enhancing the step length within the Broyden quasi-Newton process‎. ‎Additionally‎, ‎numerical examples are provided to compare the effectiveness of this new method with another approach‎.

It is well known that, one of the useful and rapid methods for a non-linear system of algebraic equations is Newton's method. Newton's method has at least quadratic convergence when the Jacobian is a nonsingular matrix in a neighbor-hood of the solution. In this paper, a di, erential continuation method is presented for solving the nonlinear system of algebraic equations whose Jacobian matrix is singular at the solution. For this purpose, at , rst, an auxiliary equation named the homo-topy equation is constructed. Then, by di, erentiating from the homotopy equation, a system of di, erential equations is replaced instead of the target problem and solved. In other words, the solution of the nonlinear system of algebraic equations with sin-gular Jacobian is transformed to the solution of a system of di, erential equations. Some numerical tests are presented at the end and the computational e, ciency of the method is described.

A new learning strategy is proposed for training of radial basis functions (RBF) network. We apply two different local optimization methods to update the output weights in training process, the gradient method and a combination of the gradient and Newton methods. Numerical results obtained in solving nonlinear integral equations show the excellent performance of the combined gradient method in comparison with gradient method as local back propagation algorithms.