Improved bracketing parabolic method for numerical solution of nonlinear equations
作者：
Highlights：
• The well-known methods for solving the scalar nonlinear equation f(x)=0 have been criticized, which use the conditions |xi+1−xi|≤ε,|f(xi+1)−f(xi)|≤δ, for stopping the iterative process, where ε,δ are a small positive numbers, i is the iteration number. On the examples of the solution of equations, it is shown that this widespread opinion is erroneous, since from the fact that these conditions are met, it does not follow that |xi+1−x*|≤ε, where x⁎ is the exact solution of the problem.
• An improved parabolic method is proposed, which first uses the condition |xi+1−xi|≤ε, confirms or refines solution, using the condition f(xi+1)·f(xi)≤0 to guarantee accuracy.
• The method is not inferior and even slightly exceeds the methods of Brent, Ridders and Müller, each of which, however, does not always guarantee accuracy when finding the roots of equations.
摘要
•The well-known methods for solving the scalar nonlinear equation f(x)=0 have been criticized, which use the conditions |xi+1−xi|≤ε,|f(xi+1)−f(xi)|≤δ, for stopping the iterative process, where ε,δ are a small positive numbers, i is the iteration number. On the examples of the solution of equations, it is shown that this widespread opinion is erroneous, since from the fact that these conditions are met, it does not follow that |xi+1−x*|≤ε, where x⁎ is the exact solution of the problem.•An improved parabolic method is proposed, which first uses the condition |xi+1−xi|≤ε, confirms or refines solution, using the condition f(xi+1)·f(xi)≤0 to guarantee accuracy.•The method is not inferior and even slightly exceeds the methods of Brent, Ridders and Müller, each of which, however, does not always guarantee accuracy when finding the roots of equations.
论文关键词：Nonlinear equation,Bisection method,Müller method,Brent method,Ridders method,Parabolic method,Improved parabolic method,Speed method
论文评审过程：Received 25 August 2019, Revised 5 January 2021, Accepted 6 January 2021, Available online 18 February 2021, Version of Record 18 February 2021.
论文官网地址：https://doi.org/10.1016/j.amc.2021.125995
