求解病态线性方程组的预处理精细积分法

富明慧; 李勇息

doi:10.21656/1000-0887.380206

求解病态线性方程组的预处理精细积分法

doi: 10.21656/1000-0887.380206

中山大学应用力学与工程系, 广州 510275

基金项目: 国家自然科学基金（11672338；11502172）

详细信息

作者简介:
富明慧（1966—），男，教授，博士，博士生导师(通讯作者. E-mail: stsfmh@mail.sysu.edu.cn).

中图分类号: O242
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出版历程
- 收稿日期: 2017-07-25
- 修回日期: 2017-11-08
- 刊出日期: 2018-04-15

A Preconditioned Precise Integration Method for Solving Ill-Conditioned Linear Equations

Department of Applied Mechanics and Engineering, Sun Yat-sen University, Guangzhou 510275, P.R.China

Funds: The National Natural Science Foundation of China（11672338;11502172）

摘要

摘要: 为降低病态线性方程组系数矩阵的条件数，根据矩阵行（列）均衡的思想，提出行（列）的1范数均衡法，并扩展为范数均衡法.然后，将范数均衡法与精细积分法相结合，给出求解病态线性方程组的范数均衡预处理精细积分法.数值结果表明，经过范数均衡预处理后精细积分法求解病态方程的精度（有效数字增加5个以上）和效率（迭代次数降低15次左右）均能得到显著提高，适用范围在一定程度上也有所扩展.在上述方法中，以1范数均衡预处理精细积分法效果最为显著.
- 范数 /
- 均衡 /
- 预处理 /
- 精细积分法 /
- 病态线性方程组
Abstract: In order to reduce the condition number of the coefficient matrix of ill-conditioned linear equations, according to the equilibration thought for matrices, a 1-norm equilibration method was proposed to properly reduce the condition number of the matrix, and expanded to the norm equilibration methods. Then, the norm equilibration method together with the precise integration method was combined for solving ill-conditioned linear equations. The numerical results confirm that, the accuracy, efficiency and application scope of the preconditioned precise integration method for ill-conditioned linear equations all improve significantly (the number of significant digits increases by more than 5 and the number of iterations decreases by about 15). In these methods, the preconditioned precise integration method of 1-norm equilibration is the best.
- norm /
- equilibration /
- preconditioned technology /
- precise integration method /
- ill-conditioned linear equations

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参考文献(15)

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