共查询到19条相似文献,搜索用时 547 毫秒
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求解非线性方程组的一种新的全局收敛的Levenberg-Marquardt算法 总被引:10,自引:0,他引:10
本文提出了求解非线性方程组的一种新的全局收敛的Levenberg-Marquardt算法,即μk=ακ(θ||F_k|| (1-θ)||J_k~TF_k||),θ∈[0,1],其中ακ利用信赖域技巧来修正.在不必假设雅可比矩阵非奇异的局部误差界条件下,证明了该算法是全局收敛和局部二次收敛的.数值试验表明该算法能有效地求解奇异非线性方程组问题. 相似文献
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略论奇异性和病态及有关问题 总被引:1,自引:0,他引:1
何旭初 《高等学校计算数学学报》1984,(1)
奇异性和病态这两个概念,在计算数学中有广泛的实际背景。在建立各类计算问题的有效算法时,奇异性和病态是产生困难的重要原因。例如,在解非线性方程组和非线性最优化问题中的牛顿法或拟牛顿法中的雅古比矩阵、海塞矩阵或近似海塞矩阵若发生奇 相似文献
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是在对完全对称雅可比矩阵及相应次对称矩阵对比研究的基础上,导出了完全次对称雅可比矩阵的特征值和相应特征向量之间的某些十分有趣的性质. 相似文献
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拟牛顿法是求解非线性方程组的一类有效方法.相较于经典的牛顿法,拟牛顿法不需要计算Jacobian矩阵且仍具有超线性收敛性.本文基于BFGS和DFP的迭代公式,构造了新的充分下降方向.将该搜索方向和投影技术相结合,本文提出了无导数低存储的投影算法求解带凸约束的非线性单调方程组并证明了该算法是全局且R-线性收敛的.最后,将该算法用于求解压缩感知问题.实验结果表明,本文所提出的算法具有良好的计算效率和稳定性. 相似文献
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汪秉宏 《数学物理学报(A辑)》1988,(4)
对于二维可逆保面积DeVogelaere映象给出n次迭代线性雅可比矩阵和对称偶周期轨道线性雅可比矩阵的一般表示式,从而导出对称周期轨道线性雅可比矩阵的一般结构。证明了一般的可逆保面积映象具有相同的结构,并从所得的一般结构讨论了可逆保面积映象对称周期轨道分岐的一般行为。 相似文献
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信赖域方法是求解非线性方程组的一种重要方法.本文研究了求解非线性方程组的信赖域半径趋于零的信赖域算法在Jacobi矩阵Hölderian连续条件下的全局收敛性质,以及其在Hölderian局部误差界和Jacobi矩阵Hölderian连续条件下的收敛速度. 相似文献
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本文应用迭代法求解一类有限维非线性问题,该方法是求解线性问题的雅可比迭代法在非线性问题上的推广,且此迭代方法具有几何收敛性质. 相似文献
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对于求解非线性方程组F (x) =0的Broyden秩1方法的计算格式提出一种修正算法,尝试利用矩阵的奇异值分解求解迭代方程组,并且配合使用加速技巧,从而大大提高了算法的安全性和收敛速度.数值算例表明了新算法的有效性. 相似文献
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《Optimization》2012,61(4):981-992
In this paper, we consider a trust-region method for solving nonlinear equations which employs a new nonmonotone technique. A strong nonmonotone strategy and a weaker nonmonotone strategy can be obtained by choosing the parameter adaptively. Thus, the disadvantages of the traditional nonmonotone strategy can be avoided. It does not need to compute the Jacobian matrix at every iteration, so that the workload and time are decreased. Theoretical analysis indicates that the new algorithm preserves the global convergence under classical assumptions. Moreover, superlinear and quadratic convergence are established under suitable conditions. Numerical experiments show the efficiency and effectiveness of the proposed method for solving nonlinear equations. 相似文献
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M. Soledad Aronna J. Frédéric Bonnans Pierre Martinon 《Journal of Optimization Theory and Applications》2013,158(2):419-459
In this article, we propose a shooting algorithm for a class of optimal control problems for which all control variables appear linearly. The shooting system has, in the general case, more equations than unknowns and the Gauss–Newton method is used to compute a zero of the shooting function. This shooting algorithm is locally quadratically convergent, if the derivative of the shooting function is one-to-one at the solution. The main result of this paper is to show that the latter holds whenever a sufficient condition for weak optimality is satisfied. We note that this condition is very close to a second order necessary condition. For the case when the shooting system can be reduced to one having the same number of unknowns and equations (square system), we prove that the mentioned sufficient condition guarantees the stability of the optimal solution under small perturbations and the invertibility of the Jacobian matrix of the shooting function associated with the perturbed problem. We present numerical tests that validate our method. 相似文献
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电阻抗成像是一类椭圆方程反问题,本文在三维区域上对其进行数值模拟和分析.对于椭圆方程Neumann边值正问题,本文提出了四面体单元上的一类对称体积元格式,并证明了格式的半正定性及解的存在性;引入单元形状矩阵的概念,简化了系数矩阵的计算;提出了对电阻率进行拼接逼近的方法来降低反问题求解规模,使之与正问题的求解规模相匹配;导出了误差泛函的Jacobi矩阵的计算公式,利用体积元格式的对称性和特殊的电流基向量,将每次迭代中需要求解的正问题的个数降到最低.一系列数值实验的结果验证了数学模型的可靠性和算法的可行性.本文所提出的这些方法,已成功应用于三维电阻抗成像的实际数值模拟. 相似文献
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In this paper, we propose a smoothing Levenberg-Marquardt method for the symmetric cone complementarity problem. Based on a smoothing function, we turn this problem into a system of nonlinear equations and then solve the equations by the method proposed. Under the condition of Lipschitz continuity of the Jacobian matrix and local error bound, the new method is proved to be globally convergent and locally superlinearly/quadratically convergent. Numerical experiments are also employed to show that the method is stable and efficient.
相似文献15.
M. Vuitovich 《Computational Mathematics and Modeling》2010,21(4):409-413
Differentiation with respect to a parameter is proposed as a method for solving a system of nonlinear equations with a poorly
conditioned Jacobian matrix. Singular and polar decompositions of the Jacobian matrix lead to a system of ordinary differential
equations with small parameters multiplying the derivatives. This system is solved by introducing auxiliary linear differential
equations. The numerical solution of a poorly conditioned system is reduced in this case to the approximation of an exponential
with a large negative argument. 相似文献
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O.P. Burdakov 《Numerical Functional Analysis & Optimization》2013,34(2):183-195
Symmetric methods (SS methods) of the secant type are proposed for systems of equations with symmetric Jacobian matrix. The SSI and SS2 methods generate sequences of symmetric matrices J and H which approximate the Jacobian matrix and inverse one, respectively. Rank-two quasi-Newton formulas for updating J and H are derived. The structure of the approximations J and H is better than the structure of the corresponding approximations in the traditional secant method because the SS methods take into account symmetry of the Jacobian matrix. Furthermore, the new methods retain the main properties of the traditional secant method, namely, J and H are consistent approximations to the Jacobian matrix; the SS methods converge superlinearly; the sequential (n + 1)-point SS methods have the R-order at least equal to the positive root of tn+1-1=0. 相似文献
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Inexact Interior-Point Method 总被引:2,自引:0,他引:2
S. Bellavia 《Journal of Optimization Theory and Applications》1998,96(1):109-121
In this paper, we introduce an inexact interior-point algorithm for a constrained system of equations. The formulation of the problem is quite general and includes nonlinear complementarity problems of various kinds. In our convergence theory, we interpret the inexact interior-point method as an inexact Newton method. This enables us to establish a global convergence theory for the proposed algorithm. Under the additional assumption of the invertibility of the Jacobian at the solution, the superlinear convergence of the iteration sequence is proved. 相似文献
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B. S. Goh 《BIT Numerical Mathematics》1978,18(1):84-90
This paper describes some sufficient conditions for global convergence in five differential equation algorithms for solving systems of non-linear algebraic equations involving positive variables. The algorithms are continuous time versions of a modified steepest descent method, Newton's method, a modified Newton's method and two algorithms using the transpose of the Jacobian in place of the inverse of the Jacobian in Newton's method. It is shown that under a set of mildly restrictive conditions the Jacobian transpose algorithm has qualitatively the same convergence as Newton's method. 相似文献