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一类空间分数阶扩散方程经过有限差分离散后所得到的离散线性方程组的系数矩阵是两个对角矩阵与Toeplitz型矩阵的乘积之和.在本文中,对于几乎各向同性的二维或三维空间分数阶扩散方程的离散线性方程组,采用预处理Krylov子空间迭代方法,我们利用其系数矩阵的特殊结构和具体性质构造了一类分块快速正则Hermite分裂预处理子.通过理论分析,我们证明了所对应的预处理矩阵的特征值大部分都聚集于1的附近.数值实验也表明,这类分块快速正则Hermite分裂预处理子可以明显地加快广义极小残量(GMRES)方法和稳定化的双共轭梯度(BiCGSTAB)方法等Krylov子空间迭代方法的收敛速度. 相似文献
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对半无界区域上的三阶方程提出了Laguerre-Petrov-Galerkin谱逼近方法,选取了相同的试探空间和检验空间.通过构造该空间上的基函数,离散问题所对应的线性系统的系数矩阵是半稀疏的.数值算例验证了该方法的有效性和高精度. 相似文献
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变系数线性系统谱的新不等式及其对部分变元稳定性的应用 总被引:2,自引:0,他引:2
本文利用一个非二次型李雅普诺夫函数,对变系数线性系统给出了一个新的谱不等式.不同于Wazewski不等式,我们避免了需要计算时变矩阵特征值的困难.然后我们利用新的谱不等式,讨论了变系数线性系统对部分变元的指数稳定性,得到了更实用的结果. 相似文献
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系统地论证了二次自伴矩阵多项式特征值,特征向量的性质.给出了二次自伴矩阵多项式特征值与任一非零向量所对应的二次多项式根之间的大小关系;精确地给出了二次自伴矩阵多项式是负定时参数的界;简化了二次自伴矩阵多项式的符号特征是正(负)的特征值对应特征向量间可以是线性无关等定理的证明. 相似文献
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1.前言 关于系数用偶阶导数表示的(2n 1)次样条函数,使用性能较好,但其存在性与唯一性迄今尚未给出证明.这种样条函数与一般的奇次多项式样条函数一样,当n≥2时,方程组的系数矩阵已不具有明显的主对角元素占优,致使J.H.Ahlberg等人说,直接依赖系数矩阵的性质来证明多项式样条函数的存在性是十分困难的,即便是对于 相似文献
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关于解一维抛物型方程组的差分格式 总被引:2,自引:2,他引:0
Caapck曾经研究过解多维抛物型方程组的经济格式.用他的方法解一维问题时,是将抛物型方程组的系数矩阵写成一个下三角形矩阵和一个上三角形矩阵之和,然后采用分数步长法求解.如果未知函数的个数为M,则对于每一个时间步长,需要用2M次追赶法.格式的收敛速度为Ο(τ~(1/2) h~2),这里τ和h分别为时间和空间步长.本文提出一种解一维抛物型方程组的绝对稳定的差分格式.对于每一个时间步长,求解差分方程组只要用M次追赶法,它的收敛速度为Ο(τ h~2)。 相似文献
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考虑终值数据条件下一维空间-时间分数阶变系数对流扩散方程中同时确定空间微分阶数与时间微分阶数的反问题.基于对空间-时间分数阶导数的离散,给出求解正问题的一个隐式差分格式,通过对系数矩阵谱半径的估计,证明差分格式的无条件稳定性和收敛性.联合最佳摄动量算法和同伦方法引入同伦正则化算法,应用一种单调下降的Sigmoid型传输函数作为同伦参数,对所提微分阶数反问题进行精确数据与扰动数据情形下的数值反演.结果表明同伦正则化算法对于空间-时问分数阶反常扩散的参数反演问题是有效的. 相似文献
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The concept of coefficient shift matrix is introduced to represent delay variables in block pulse series. The optimal control of a linear delay system with quadratic performance index is then studied via block pulse functions, which convert the problems into the minimization of a quadratic form with linear algebraic equation constraints. The solution of the two-point boundary-value problem with both delay and advanced arguments is circumvented. The control variable obtained is piecewise constant. 相似文献
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We consider solving eigenvalue problems or model reduction problems for a quadratic matrix polynomial Iλ2 − Aλ − B with large and sparse A and B. We propose new Arnoldi and Lanczos type processes which operate on the same space as A and B live and construct projections of A and B to produce a quadratic matrix polynomial with the coefficient matrices of much smaller size, which is used to approximate the original problem. We shall apply the new processes to solve eigenvalue problems and model reductions of a second order linear input-output system and discuss convergence properties. Our new processes are also extendable to cover a general matrix polynomial of any degree. 相似文献
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E. Golpar-Raboky N. Mahdavi-Amiri 《Journal of Optimization Theory and Applications》2012,152(1):75-96
Classes of integer Abaffy–Broyden–Spedicato (ABS) methods have recently been introduced for solving linear systems of Diophantine
equations. Each method provides the general integer solution of the system by computing an integer solution and an integer
matrix, named Abaffian, with rows generating the integer null space of the coefficient matrix. The Smith normal form of a
general rectangular integer matrix is a diagonal matrix, obtained by elementary nonsingular (unimodular) operations. Here,
we present a class of algorithms for computing the Smith normal form of an integer matrix. In doing this, we propose new ideas
to develop a new class of extended integer ABS algorithms generating an integer basis for the integer null space of the matrix.
For the Smith normal form, having the need to solve the quadratic Diophantine equation, we present two algorithms for solving
such equations. The first algorithm makes use of a special integer basis for the row space of the matrix, and the second one,
with the intention of controlling the growth of intermediate results and making use of our given conjecture, is based on a
recently proposed integer ABS algorithm. Finally, we report some numerical results on randomly generated test problems showing
a better performance of the second algorithm in controlling the size of the solution. We also report the results obtained
by our proposed algorithm on the Smith normal form and compare them with the ones obtained using Maple, observing a more balanced
distribution of the intermediate components obtained by our algorithm. 相似文献
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求解三维高次拉格朗日有限元方程的代数多重网格法 总被引:5,自引:0,他引:5
本文针对带有间断系数的三维椭圆问题,讨论任意四面体剖分下的二次拉格朗日有限元方程的代数多重网格法.通过分析线性和高次有限元空间之间的关系,我们给出了一种新的网格粗化算法和构造提升算子的代数途径.进一步,我们还对新的代数多重网格法给出了收敛性分析.数值实验表明这种代数多重网格法对求解二次拉格朗日有限元方程是健壮和有效的。 相似文献
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Israel Zang 《Mathematical Programming》1978,15(1):36-52
The gradient path of a real valued differentiable function is given by the solution of a system of differential equations. For a quadratic function the above equations are linear, resulting in a closed form solution. A quasi-Newton type algorithm for minimizing ann-dimensional differentiable function is presented. Each stage of the algorithm consists of a search along an arc corresponding to some local quadratic approximation of the function being minimized. The algorithm uses a matrix approximating the Hessian in order to represent the arc. This matrix is updated each stage and is stored in its Cholesky product form. This simplifies the representation of the arc and the updating process. Quadratic termination properties of the algorithm are discussed as well as its global convergence for a general continuously differentiable function. Numerical experiments indicating the efficiency of the algorithm are presented. 相似文献
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Mario Annunziato & Eleonora Messina 《计算数学(英文版)》2019,37(1):33-47
We study a numerical method for solving a system of Volterra-renewal integral equations
with space fluxes, that represents the Chapman-Kolmogorov equation for a class of
piecewise deterministic stochastic processes. The solution of this equation is related to
the time dependent distribution function of the stochastic process and it is a non-negative
and non-decreasing function of the space. Based on the Bernstein polynomials, we build
up and prove a non-negative and non-decreasing numerical method to solve that equation,
with quadratic convergence order in space. 相似文献
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Canonical Duality Theory and Solutions to Constrained Nonconvex Quadratic Programming 总被引:6,自引:2,他引:4
David Yang Gao 《Journal of Global Optimization》2004,29(4):377-399
This paper presents a perfect duality theory and a complete set of solutions to nonconvex quadratic programming problems subjected to inequality constraints. By use of the canonical dual transformation developed recently, a canonical dual problem is formulated, which is perfectly dual to the primal problem in the sense that they have the same set of KKT points. It is proved that the KKT points depend on the index of the Hessian matrix of the total cost function. The global and local extrema of the nonconvex quadratic function can be identified by the triality theory [11]. Results show that if the global extrema of the nonconvex quadratic function are located on the boundary of the primal feasible space, the dual solutions should be interior points of the dual feasible set, which can be solved by deterministic methods. Certain nonconvex quadratic programming problems in {\open {R}}^{n} can be converted into a dual problem with only one variable. It turns out that a complete set of solutions for quadratic programming over a sphere is obtained as a by-product. Several examples are illustrated. 相似文献
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Eric Schippers 《Journal d'Analyse Mathématique》2006,98(1):249-277
The coefficients of a quadratic differential which is changing under the Loewner flow satisfy a well-known differential system
studied by Schiffer, Schaeffer and Spencer, and others. By work of Roth, this differential system can be interpreted as Hamilton's
equations. We apply the power matrix to interpret this differential system in terms of the coadjoint action of the matrix
group on the dual of its Lie algebra. As an application, we derive a set of integral invariants of Hamilton's equations which
is in a certain sense complete. In function theoretic terms, these are expressions in the coefficients of the quadratic differential
and Loewner map which are independent of the parameter in the Loewner flow. 相似文献