In the paper, we convert a single nonlinear equation to a system consisting of two equations. While a quasi-linear term is added on the first equation, the nonlinear term in the second equation is decomposed at two sides through a weight parameter. After performing a linearization, an iterative scheme is derived, which is proven of third-order convergence for certain parameters. An affine quasi-linear transformation in the plane is established, and the condition for the spectral radius being smaller than one for the convergence of the iterative scheme is derived. By using the splitting method, we can further identify a sufficient condition for the convergence of the iterative scheme. Then, we develop a step-wisely quasi-linear transformation technique to solve nonlinear equations. Proper values of the parameters are qualified by the derived inequalities for both iterative schemes, which accelerate the convergence speed. The performances of the proposed iterative schemes are assessed by numerical tests, whose advantages are fast convergence, saving the function evaluation per iteration and without needing the differential of the given function.

在本文中，我们将单个非线性方程转换为由两个方程组成的系统。在第一个方程上增加了一个拟线性项，而第二个方程中的非线性项通过一个权重参数在两侧分解。执行线性化后，推导出迭代方案，证明某些参数的三阶收敛性。建立了平面内的仿射拟线性变换，推导了迭代方案收敛的谱半径小于1的条件。通过使用分裂方法，我们可以进一步确定迭代方案收敛的充分条件。然后，我们开发了一种逐步拟线性变换技术来求解非线性方程。参数的适当值由两个迭代方案的导出不等式限定，这加快了收敛速度。所提出的迭代方案的性能通过数值测试进行评估，其优点是收敛速度快，节省了每次迭代的函数评估，并且不需要给定函数的微分。