The nonlinear conjugate gradient method is an effective technique for solving large-scale
minimizations problems, and has a wide range of applications in various fields, such as mathematics, chemistry,
physics, engineering and medicine. This study presents a novel spectral conjugate gradient algorithm (non-linear
conjugate gradient algorithm), which is derived based on the Hisham–Khalil (KH) and Newton algorithms. Based on
pure conjugacy condition The importance of this research lies in finding an appropriate method to solve all types of
linear and non-linear fuzzy equations because the Buckley and Qu method is ineffective in solving fuzzy equations.
Moreover, the conjugate gradient method does not need a Hessian matrix (second partial derivatives of functions) in
the solution. The descent property of the proposed method is shown provided that the step size at meets the strong
Wolfe conditions. In numerous circumstances, numerical results demonstrate that the proposed technique is more
efficient than the Fletcher–Reeves and KH algorithms in solving fuzzy nonlinear equations.