Combined Regula–Falsi and Newton–Raphson Method

Abstract

AbstractThe present work proposes a modified Regula–Falsi method (RFM) by combining with Newton–Raphson method (NRM), to attain some advantages. RFM requires two initial values of \(x\), say \(x_{1}\) and \(x_{2}\), such that \(f\left( {{ }x_{1} } \right).f\left( {{ }x_{2} } \right) < 0\), to solve a nonlinear equation. During the iteration process, \(f\left( x \right)\) changes the sign randomly without any particular pattern. The previous value of \(x\) is replaced accordingly by checking the sign of \(f\left( x \right)\) in RFM. In the proposed modified RFM, only one initial value, \(x_{1}\) is required and \(x_{2}\) is evaluated using \(x_{1}\). The proposed method ensures that the sign of the function \(f\left( x \right)\), alternately, changes from \(+ ve\) to \(- ve\) and vice-versa. It can be easily seen that no successive values of \(f\left( x \right)\) have same sign. Thereby, checking of sign of \(f\left( x \right)\) after every iteration is eliminated in the algorithm, which reduces the computational time. NRM fails under two situations; when the derivative of \(f\left( x \right)\) becomes zero and when alternate values of \(x\) are equal. Under both the situations, NRM fails to converge. The proposed method circumvents these limitations of Newton–Raphson method (NRM). The proposed method has convergence rate of \(\sqrt 3\). In order to verify the proposed methods, two examples have been solved and compared with the RFM and NRMKeywordsNonlinear equationRegula–Falsi methodNewton–Raphson methodConvergence