A Preconditioned Precise Integration Method for Solving Ill-Conditioned Linear Equations
-
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.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.
-
[1] BENZI M. Preconditioning techniques for large linear systems: a survey[J]. Journal of Computational Physics,2002,182(2): 418-477. [2] 唐丽, 李鹏飞. 主元加权迭代法求解病态线性方程组[J]. 科学技术与工程, 2012,12(2): 381-383.(TANG Li, LI Pengfei. A pivot element weighting iterative method for solving ill-conditioned linear equations[J]. Science Technology and Engineering,2012,12(2): 381-383.(in Chinese)) [3] LIU C S. Optimally generalized regularization methods for solving linear inverse problems[J]. Computers Materials & Continua,2012,29(2): 103-127. [4] 胡圣荣, 戴纳新. 病态线性方程组新解法:增广方程组法[J]. 华南农业大学学报, 2009,30(1): 119-121.(HU Shengrong, DAI Naxin. A novel method for solving ill-conditioned linear system: augmented system method[J]. Journal of South China Agricultural University,2009,30(1): 119-121.(in Chinese)) [5] 于春肖, 苑润浩. 预处理ICCG法求解稀疏病态方程组[J]. 河北大学学报(自然科学版), 2014,34(1): 1-6.(YU Chunxiao, YUAN Runhao. Preconditioning ICCG method for solving sparse ill-conditioned linear equations[J]. Journal of Hebei University(Natural Science Edition ), 2014,34(1): 1-6.(in Chinese)) [6] 霍志周, 熊登, 张剑锋. 预条件共轭梯度法在地震数据重建方法中的应用[J]. 地球物理学报, 2013,56(4): 1321-1330.(HUO Zhizhou, XIONG Deng, ZHANG Jianfeng. Application of the preconditioned conjugate gradient method to reconstruction of seismic data[J]. Chinse Journal of Geophysics,2013,56(4): 1321-1330.(in Chinese)) [7] 李秀艳, 韩倩, 汪剑鸣, 等. 基于改进共轭梯度法的ERT图像重建[J]. 仪器仪表学报, 2016,37(7): 1673-1679.(LI Xiuyan, HAN Qian, WANG Jianming,et al. ERT image reconstruction based on improved CG method[J]. Chinese Journal of Scientific Instrument,2016,37(7): 1673-1679.(in Chinese)) [8] VAJARGAH B F, MORADI M. Diagonal scaling of ill-conditioned matrixes by genetic algorithm[J]. Journal of Applied Mathematics Statistics & Informatics,2012,8(1): 49-53. [9] LIU C S. A two-side equilibration method to reduce the condition number of an ill-posed linear system[J]. Computer Modeling in Engineering & Sciences,2013,91(1): 17-42. [10] KU C Y. A novel method for solving ill-conditioned systems of linear equations with extreme physical property contrasts[J]. Computer Modeling in Engineering & Sciences,2013,96(9): 409-434. [11] 胡圣荣, 罗锡文. 病态线性方程组的新解法:误差转移法[J]. 华南农业大学学报, 2001,22(4): 92-94.(HU Shengrong, LUO Xiwen. A new method for solving ill-conditioned linear systems[J].Journal of South China Agricultural University,2001,22(4): 92-94.(in Chinese)) [12] 钟万勰. 结构动力方程的精细时程积分法[J]. 大连理工大学学报, 1994,34(2): 131-136.(ZHOUG Wanxie. One precise time-integration method for structural dynamic equations[J]. Journal of Dalian University of Technology,1994,34(2): 131-136.(in Chinese)) [13] 富明慧, 刘祚秋, 林敬华. 一种广义精细积分法[J]. 力学学报, 2007,39(5): 672-677.(FU Minghui, LIU Zuoqiu, LIN Jinghua. A generalized precise time step integration method[J]. Acta Mechanica Sinica,2007,39(5): 672-677.(in Chinese)) [14] 高强, 吴锋, 张洪武, 等. 大规模动力系统改进的快速精细积分方法[J]. 计算力学学报, 2011,28(4): 493-498.(GAO Qiang, WU Feng, ZHANG Hongwu, et al. A fast precise integration method for large-scale dynamic structures[J]. Chinese Journal of Computational Mechanics,2011,28(4): 493-498.(in Chinese)) [15] 富明慧, 张文志. 病态代数方程的精细积分解法[J]. 计算力学学报, 2011,28(4): 530-534.(FU Minghui, ZHANG Wenzhi. Precise integration method for solving ill-conditioned algebraic equations[J]. Chinese Journal of Computational Mechanics,2011,28(4): 530-534.(in Chinese))
点击查看大图
计量
- 文章访问数: 1261
- HTML全文浏览量: 154
- PDF下载量: 1274
- 被引次数: 0