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本文讨论了非线性矩阵方程X-A*X-pA=Q在p>1时正定解存在的必要条件和充分条件以及正定解的一些性质,推导出此方程存在唯一正定解的充分条件,同时构造了数值求解的迭代方法,并且把某些已知的结论推广到任意实数p>1. 相似文献
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矩阵方程X-A*X-q A=Q当q>1时的Hermite正定解 总被引:5,自引:0,他引:5
讨论了矩阵方程X-A*X-qA=Q在q>1时的Hermite正定解的存在性和解的性质,并且构造了两种数值求解的迭代方法.利用数值例子对以上结果进行了说明. 相似文献
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本文基于求线性代数方程组的共轭梯度法的思想,通过特殊的变形与近似处理,建立了求一般线性矩阵方程的双对称最小二乘解的迭代算法,并证明了迭代算法的收敛性。不考虑舍入误差时,迭代算法能够在有限步计算之后得到矩阵方程的双对称最小二乘解;选取特殊的初始矩阵时,还能够求得矩阵方程的极小范数双对称最小二乘解。同时,也能够给出指定矩阵的最佳逼近双对称矩阵。算例表明,迭代算法是有效的。 相似文献
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基于求解线性代数方程组的共轭梯度法,通过对相关矩阵和系数的修改,建立了一种求多矩阵变量矩阵方程异类约束解的修正共轭梯度法.该算法不要求等价线性代数方程组的系数矩阵具备正定性、可逆性或者列满秩性,因此算法总是可行的.利用该算法不仅可以判断矩阵方程的异类约束解是否存在,而且在有异类约束解,不考虑舍入误差时,可在有限步计算后求得矩阵方程的一组异类约束解;选取特殊初始矩阵时,可求得矩阵方程的极小范数异类约束解.另外,还可求得指定矩阵在异类约束解集合中的最佳逼近.算例验证了该算法的有效性. 相似文献
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本文研究在最优控制系统中遇到的离散时间代数Riccati矩阵方程(DTARME)异类约束解的数值计算问题.首先对多变量DTARME中的逆矩阵采用矩阵级数方法进行等价转化,然后采用牛顿算法求多变量DTARME的异类约束解,并采用修正共轭梯度法求由牛顿算法每一步迭代计算导出的线性矩阵方程的异类约束解或者异类约束最小二乘解,建立求多变量DTARME的异类约束解的双迭代算法.双迭代算法仅要求多变量DTARME有异类约束解,不要求它的异类约束解唯一,也不对它的系数矩阵做附加限定.数值算例表明,双迭代算法是有效的. 相似文献
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基于求线性矩阵方程组约束解的修正共轭梯度法,讨论了由Nash均衡对策导出的一类双矩阵变量Riccati矩阵方程组(R-MEs)对称解的数值计算问题.提出用牛顿算法将R-MEs的对称解问题转化为双矩阵变量线性矩阵方程组的对称解或者对称最小二乘解问题,并采用修正共轭梯度法解决后一计算问题,建立了求R-MEs对称解的新型迭代算法.新型迭代算法仅要求R-MEs有对称解,不要求它的对称解唯一,也不对它的系数矩阵做附加限定.数值算例表明,新型迭代算法是有效的. 相似文献
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F. A. Mirza 《International journal for numerical methods in engineering》1984,20(8):1555-1561
A solution technique for indefinite systems of symmetric linear, simultaneous equations, via the Hellinger–Reissner variational principle, is presented. The method utilizes symmetry of the global matrix and its expected real eigenvalues. Premultiplication of the global matrix with itself renders a positive definite matrix, hence enabling the use of any standard equation solver for a positive definite system and requiring only about twice the memory requirement for the original set of equations. Two subroutines, MULT and MULTIP, which are compatible with the sky-line technique, are also listed. 相似文献
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Jing Li Olof B. Widlund 《International journal for numerical methods in engineering》2006,66(2):250-271
The FETI‐DP and BDDC algorithms are reformulated using Block Cholesky factorizations, an approach which can provide a useful framework for the design of domain decomposition algorithms for solving symmetric positive definite linear system of equations. Instead of introducing Lagrange multipliers to enforce the coarse level, primal continuity constraints in these algorithms, a change of variables is used such that each primal constraint corresponds to an explicit degree of freedom. With the new formulation of these algorithms, a simplified proof is provided that the spectra of a pair of FETI‐DP and BDDC algorithms, with the same set of primal constraints, are essentially the same. Numerical experiments for a two‐dimensional Laplace's equation also confirm this result. Copyright © 2005 John Wiley & Sons, Ltd. 相似文献
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The computation of limit and bifurcation points in structural mechanics using iterative preconditioned Lanczos solvers is studied. Contrary to classical implementations of algorithms for the calculation of limit and bifurcation points, which depend in general strongly on observing the diagonal elements of the decomposed matrix – obtained by a Gauß- or Cholesky decomposition – , we use an approach of determining limit and bifurcation points by examination of the subspace spanned by the iteration vectors of the Lanczos solver. Using a multilevel preconditioning with a coarse grid solver may result in a non positive definite preconditioning matrix if the coarse grid matrix is not positive definite in the post-critical solution branch. In that case the iteration has to be performed in the complex vector space. We prove by mathematical induction that all vectors and scalars are either purely real or purely imaginary. Therefore the generalized computation can be performed with about the same number of operations as in the case of a positive definite preconditioning matrix. 相似文献
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A. Jennings G. M. Malik 《International journal for numerical methods in engineering》1978,12(1):141-158
The convergence properties of the conjugate gradient method are discussed in relation to relaxation methods and Chebyshev accelerated Jacobi iteration when applied to the solution of large sets of linear equations which have a sparse, symmetric and positive definite coefficient matrix. The conclusion is reached that its convergence rate is unlikely to be much worse than these methods, and may be considerably better. The conjugate gradient method may either be applied to the basic unscaled or scaled equations or alternatively to various transformed equations. Preconditioning, block elimination and partial elimination methods of transforming equations are considered, and some comparative tests given for six problems. 相似文献
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A lopsided alternating direction iteration (LADI) method and an induced
block diagonal preconditioner for solving block two-by-two generalised saddle point
problems are presented. The convergence of the LADI method is analysed, and the block diagonal preconditioner can accelerate the convergence rates of Krylov subspace iteration methods such as GMRES. Our new preconditioned method only requires a solver for two linear equation sub-systems with symmetric and positive definite coefficient matrices. Numerical experiments show that the GMRES with the new preconditioner is quite effective. 相似文献
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F. Guarracino V. Minutolo L. Nunziante 《Engineering Analysis with Boundary Elements》1992,10(4):283-289
The present paper analyses some particular features of the boundary element method applied to the elastic half-space, which are derived from the choice of a suitable fundamental solution and appropriate discretization. In this way it is possible to get a symmetric and positive definite matrix, which enables a very simple coupling of BEM-FEM procedures.
A way of modelling and solving the classic soil-structure interaction problem, based on coupling Somigliana's equation in discretized form for the elastic half-space with the system which comes from the finite elements treatment of structure, will be developed.
This technique in soil-structure analysis appears interesting because it gives rise to a real simplification of the problem and allows the taking into account of the elastic soil deformability in the structural responses evaluation. 相似文献
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求工程结构最低阶固有频率的一种算法 总被引:2,自引:0,他引:2
本文采用子空间迭代法将工程结构高阶动力系统减缩为低阶动力系统,然后用Collatz包含定理的推广求出该结构系统的最低阶固有频率。 相似文献
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A simple approximated analytical solution is introduced to analyse arbitrary non-uniform transmission lines (NTLs) at low frequencies. First, the differential equations of NTLs are written as a suitable matrix differential equation. Then, the matrix differential equation is solved to obtain the chain parameter matrix of NTLs. Later, the voltage and current of the line are obtained at any point using the obtained chain parameter matrix. Finally, the validation of the introduced solution is studied. 相似文献
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An approximated closed-form analytic solution is introduced for arbitrary inhomogeneous planar layers (IPLs). First, the differential equations of IPLs are written as a suitable matrix differential equation. Then, the matrix differential equation is solved to obtain the chain parameter matrix of IPLs. Afterwards, the electric and magnetic fields at any point and also the reflection and transmission coefficients are obtained using the chain parameter matrix. The validation of the introduced solution is studied finally. 相似文献