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Solving systems of nonlinear equations when the nonlinearity is expensive

机译:非线性昂贵时求解非线性方程组

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Construction of multi-step iterative method for solving system of nonlinear equations is considered, when the nonlinearity is expensive. The proposed method is divided into a base method and multi-step part. The convergence order of the base method is five, and each step of multi-step part adds additive-factor of five in the convergence order of the base method. The general formula of convergence order is 5(m-2) where m(>= 3) is the step number. For a single instance of the iterative method we only compute two Jacobian and inversion of one Jacobian is required. The direct inversion of Jacobian is avoided by computing LU factors. The computed LU factors are used in the multi-step part for solving five systems of linear equations that make the method computational efficient. The distinctive feature of the underlying multi-step iterative method is the single call to the computationally expensive nonlinear function and thus offers an increment of additive factor of five in the convergence order per single call. The numerical simulations reveal that our proposed iterative method clearly shows better performance, where the computational cost of the involved nonlinear function is higher than the computational cost for solving five lower and upper triangular systems. (C) 2016 Elsevier Ltd. All rights reserved.
机译:当非线性成本很高时,考虑构造求解非线性方程组的多步迭代方法。所提出的方法分为基本方法和多步骤部分。基本方法的收敛阶数为5,多步部分的每个步骤在基本方法的收敛阶数上加5的加法因子。收敛阶数的一般公式为5(m-2),其中m(> = 3)是步数。对于迭代方法的单个实例,我们仅计算两个雅可比行列,并且需要对一个雅可比行列进行求逆。通过计算LU因子可以避免Jacobian的直接反演。计算出的LU因子在多步骤部分中用于求解五个线性方程组,从而使该方法的计算效率更高。基本的多步迭代方法的显着特征是对计算成本很高的非线性函数的单次调用,因此每个单次调用会以收敛顺序将累加因子增加5。数值模拟表明,我们提出的迭代方法明显表现出更好的性能,其中所涉及的非线性函数的计算成本高于求解五个上下三角系统的计算成本。 (C)2016 Elsevier Ltd.保留所有权利。

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