γ-条件下非精确牛顿类方法的半局部收敛性

研究了非精确牛顿类方法的收敛性问题.假设非线性算子满足γ-条件,那么可以建立非精确牛顿类方法的半局部收敛条件;并且,给出一个数值例子说明了本文结果的有效性.

对牛顿类方法的讨论

对解非线性和超越方程f(x)=0的"牛顿类"方法xn+1=xn-f(xn)/(αf(xn)+f′(xn))作了进一步的分析,认为参数α的取值范围直接影响公式的收敛速度,从而给出了α取值的依赖性条件,并给出了加速算法和数值算例.

求解逆特征值问题的全局性非精确牛顿类方法

为了研究求解逆特征值问题的全局性算法,利用反幂法获得近似特征向量,提出了一种求解逆特征值问题的全局性非精确牛顿类算法.在一定的条件下,给出了该全局算法的收敛性分析,并且证明了该算法的超线性/二阶收敛性质.最后,通过数值例子进一步验证所提出算法的全局收敛性.

调整右矢量的加速牛顿类迭代法

设计了一个新的牛顿类迭代方法.该迭代法设计了最佳松弛参量并不断调整线性系统的右端矢量,它比牛顿方法的计算量要少,比修正的牛顿方法收敛得快.分析了松弛参量的作用,并给出了最佳参量的计算公式.使用数值例子证明了该方法的优良性质,用衡量指数对比了其他几种迭代法,证明了该方法的优越性.

一种压缩感知信号的快速恢复方法

针对最优化1-范数恢复压缩感知信号过程中的不可解情况,提出一种压缩感知信号快速恢复方法.该方法从最优化0-范数的观点出发,设计了新的目标函数拟合信号0-范数,以避免求解NP问题及不可解情况;在求解过程中提出一种类牛顿法的搜索方向进行求解,使求解速度达到线性速度.实验结果表明,文中方法恢复的成功率高、稳定性强、速度快,适合处理大型数据.

A Fourth-order Covergence Newton-type Method

A fourth-order convergence method of solving roots for nonlinear equation, which is a variant of Newton＇s method given. Its convergence properties is proved. It is at least fourth-order convergence near simple roots and one order convergence near multiple roots. In the end, numerical tests are given and compared with other known Newton and Newton-type methods. The results show that the proposed method has some more advantages than others. It enriches the methods to find the roots of non-linear equations and it is important in both theory and application.

A Class of Third-order Convergence Variants of Newton＇s Method

A class of third-order convergence methods of solving roots for non-linear equation,which are variant Newton＇s method, are given. Their convergence properties are proved. They are at least third order convergence near simple root and one order convergence near multiple roots. In the end, numerical tests are given and compared with other known Newton＇s methods. The results show that the proposed methods have some more advantages than others. They enrich the methods to find the roots of non-linear equations and they are important in both theory and application.