牛顿迭代法与几种改进格式的效率指数

研究牛顿迭代、牛顿弦截法以及它们的六种改进格式的计算效率,计算了它们的效率指数,得到牛顿迭代、改进牛顿法、弦截法和改进弦截法(即所谓牛顿迭代的P.C格式)、二次插值迭代格式、推广的牛顿迭代法、调和平均牛顿法和中点牛顿法的效率指数分别为0.347/n、0.3662/n、0.4812/n、0.4812/n、0.347/n、0.3662/n、0.3662/n、0.3662/n.我们的结果显示,利用抛物插值多项式推出的迭代格式和改进弦截法并没有真正提高迭代的计算效率.此外,我们还证明了改进弦截法与牛顿弦截法等价,并利用这一结论给出了改进弦截法收敛阶为2.618的一个简化证明.

The Efficiency Indexes of Newton's Method and Its Modified Forms

The computational efficiency of Newton′s method,Newton′s Secant method and their six modified forms have been discussed.We get that the efficiency indexes of Newton′s method,modified Newton′s method,Newton′s Secant method,modified secant method(Newton iteration P.C.form),quadratic interpolation iteration form,generalized Newton′s method, Harmonic mean Newton′s method and midpoint Newton′s method are 0.347/n,0.3662/n,0.4812/n,0.4812/n,0.347/n,0.3662/n,0.3662/n and 0.3662/n respectively.Our results show that quadratic interpolation iteration form and modified secant method can′t really improve the computational efficiency.Moreover,we prove the equivalence of modified secant method and Newton′s Secant method.With this result,we give a simplified proof for the order of convergence of modified secant method as 2.618.