A new iteration method with cubic convergence to solve nonlinear algebraic equations. (English) Zbl 1093.65047
Summary: A new iteration scheme is proposed to solve the roots of an algebraic equation \(f(x)=0\). Given an initial guess, \(x_0\), the roots of the equation can be obtained using the following iteration scheme:
\[
x_{n+1}= x_n+ \frac {-f'(x_n)\pm \sqrt{f^{\prime2}(x_n)- 2f(x_n)f''(x_n)}} {f''(x_n)}.
\]
This iteration scheme has unique convergence characteristics different from the well-known Newton’s method. It is shown that this iteration method has cubic local convergence in the neighborhood of the root. Using this scheme, real or complex roots for specific algebraic equations can be found. Because there are two iteration directions, for a given initial guess, two solutions can be found for certain algebraic equations with multiple roots. Examples are presented and compared with other methods.
This iteration scheme has unique convergence characteristics different from the well-known Newton’s method. It is shown that this iteration method has cubic local convergence in the neighborhood of the root. Using this scheme, real or complex roots for specific algebraic equations can be found. Because there are two iteration directions, for a given initial guess, two solutions can be found for certain algebraic equations with multiple roots. Examples are presented and compared with other methods.
MSC:
| 65H05 | Numerical computation of solutions to single equations |
Keywords:
iteration method; nonlinear algebraic equation; Newton’s method; numerical examples; cubic local convergenceReferences:
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