特殊块三对角Toeplitz线性方程组的精化迭代法及收敛性
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摘要
基于迭代精化的基本思想,利用系数矩阵的特殊结构,提出了求解特殊块三对角Toeplitz线性方程组的方法—精化迭代法,它大大提高了解的精确值。该方法的特点是方法简单、稳定性好、解精度高、收敛速度快。最后,将此方法应用于三次均匀B样条曲面拟合,数值实验体现了其高效性。
Based on the basic idea of iterative refinement and the special structure of coefficient matrices,a method called the refined iterative method for solving special block tridiagonal Toeplitz linear systems of equations is proposed,which could greatly improve the accuracy of exact solution.The method is characterized by its simple scheme,numerical stability,high precision and fast convergence. Finally,we apply this method to surface-fitting of cubic uniform B-spline surface and the numerical experiment shows its high efficiency.
Based on the basic idea of iterative refinement and the special structure of coefficient matrices,a method called the refined iterative method for solving special block tridiagonal Toeplitz linear systems of equations is proposed,which could greatly improve the accuracy of exact solution.The method is characterized by its simple scheme,numerical stability,high precision and fast convergence. Finally,we apply this method to surface-fitting of cubic uniform B-spline surface and the numerical experiment shows its high efficiency.
引文
[1]徐仲,张凯院,路全.TOEPLITZ矩阵类的快速算法[M].西安:西北工业大学出版社,1999:29-141.
[2]FALLAT S M,JOHNSON C R. Totally nonnegative matrices[M]. Princeton:Princeton University Press,2011:1-6.
[3]DEMMEL J W.Numerical linear algebra[M].北京:人民邮电出版社,2007:51-54.
[4]CARSON E,HIGHAM N J.A new analysis of iterative refinement and its application to accurate solution of ill-conditioned sparse linear systems[J].Siam journal on scientific computing,2017,39(6):A2834-A2856.
[5]CARSON E,HIGHAM N J. Accelerating the solution of linear systems by iterative refinement in three precisions[J].Siam journal on scientific computing,2018,40(2):A817-A847.
[6]CARNICER J M,DELGADO J,PEA J M.Richardson method and totally nonnegative linear systems[J].Linear Algebra and its Applications,2010,433(11/12):2010-2017.
[2]FALLAT S M,JOHNSON C R. Totally nonnegative matrices[M]. Princeton:Princeton University Press,2011:1-6.
[3]DEMMEL J W.Numerical linear algebra[M].北京:人民邮电出版社,2007:51-54.
[4]CARSON E,HIGHAM N J.A new analysis of iterative refinement and its application to accurate solution of ill-conditioned sparse linear systems[J].Siam journal on scientific computing,2017,39(6):A2834-A2856.
[5]CARSON E,HIGHAM N J. Accelerating the solution of linear systems by iterative refinement in three precisions[J].Siam journal on scientific computing,2018,40(2):A817-A847.
[6]CARNICER J M,DELGADO J,PEA J M.Richardson method and totally nonnegative linear systems[J].Linear Algebra and its Applications,2010,433(11/12):2010-2017.