基于多项式预处理的特殊双变量矩阵方程异类约束解算法
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摘要
针对共轭梯度法求解双变量矩阵方程异类约束解收敛速度较慢的问题,引入多项式预处理技术,构造了一个预处理矩阵,从而改变了系数矩阵奇异值的分布,使奇异值的比值趋于1,达到提高收敛速度的目的。针对特殊一类双变量矩阵方程异类约束解的求解问题,构造了多项式预处理共轭梯度法,证明了该算法是收敛性的,且具有Q-线性收敛速度。数值实验结果表明,本算法比共轭梯度法收敛速度更快,迭代时间更短。
Aiming at the problem that the convergence of different constrained solutions of bivariate matrix equations is slow due to use the conjugate gradient method, polynomial preprocessing technology is introduced to construct a preprocessing matrix, which changes the distribution of singular values of the coefficient matrix.The ratio of the value tends to 1 for the purpose of improving the speed of convergence. Then,a new algorithm-polynomial preconditioning conjugate gradient method is proposed for solving the spe cial class of bivariate matrix equations with different constraint solutions. It is proved that the algorithm is convergent and has Q-linear convergence rate; The numerical experiment example shows that the algorithm converges faster and has shorter iteration time than the conjugate gradient method.
Aiming at the problem that the convergence of different constrained solutions of bivariate matrix equations is slow due to use the conjugate gradient method, polynomial preprocessing technology is introduced to construct a preprocessing matrix, which changes the distribution of singular values of the coefficient matrix.The ratio of the value tends to 1 for the purpose of improving the speed of convergence. Then,a new algorithm-polynomial preconditioning conjugate gradient method is proposed for solving the spe cial class of bivariate matrix equations with different constraint solutions. It is proved that the algorithm is convergent and has Q-linear convergence rate; The numerical experiment example shows that the algorithm converges faster and has shorter iteration time than the conjugate gradient method.
引文
[1] 刘莉.矩阵方程AXB+CYD=E的中心对称最小二乘解及其最佳逼近[J].兰州理工大学学报,2011,37(6):148-153.
[2] 方玲,廖安平,雷渊.矩阵方程AXB+CYD=E对称最小范数最小二乘解的极小残差法[J].高等学校计算数学学报,2010,32(1):71-81.
[3] ZHAO W,WANG C,GU Y.On the convergence of dependent GFR conjugate gradient method for unconstrained optimization[J].Numerical Algorithms,2017(1/2):1-18.
[4] 温瑞萍,孟国艳,王川龙.求解大型稀疏线性方程组的不完全SAOR预条件共轭梯度法[J].工程数学学报,2007,24(4):712-718.
[5] JIA Z,KANG W.A residual based sparse approximate inverse preconditioning procedure for large sparse linear systems[J].Numerical Linear Algebra with Applications,2015,24(1):259-299.
[6] 李书连,张凯院.矩阵方程组一种异类约束最小二乘解的迭代算法[J].数学杂志,2013,33(3):485-492.
[7] 牛婷婷,张凯院,宁倩芝.一类离散时间代数Riccati矩阵方程异类约束解的双迭代算法[J].工程数学学报,2014(6):847-85.
[8] 宋卫红,张凯院,聂玉峰.离散对偶代数Riccati方程异类约束解的双迭代算法[J].数学物理学报,2014,34(6):1440-1449.
[9] 陈其安.解线性方程组的共轭梯度法预条件研究[D].重庆:重庆大学,2001:39-46.
[2] 方玲,廖安平,雷渊.矩阵方程AXB+CYD=E对称最小范数最小二乘解的极小残差法[J].高等学校计算数学学报,2010,32(1):71-81.
[3] ZHAO W,WANG C,GU Y.On the convergence of dependent GFR conjugate gradient method for unconstrained optimization[J].Numerical Algorithms,2017(1/2):1-18.
[4] 温瑞萍,孟国艳,王川龙.求解大型稀疏线性方程组的不完全SAOR预条件共轭梯度法[J].工程数学学报,2007,24(4):712-718.
[5] JIA Z,KANG W.A residual based sparse approximate inverse preconditioning procedure for large sparse linear systems[J].Numerical Linear Algebra with Applications,2015,24(1):259-299.
[6] 李书连,张凯院.矩阵方程组一种异类约束最小二乘解的迭代算法[J].数学杂志,2013,33(3):485-492.
[7] 牛婷婷,张凯院,宁倩芝.一类离散时间代数Riccati矩阵方程异类约束解的双迭代算法[J].工程数学学报,2014(6):847-85.
[8] 宋卫红,张凯院,聂玉峰.离散对偶代数Riccati方程异类约束解的双迭代算法[J].数学物理学报,2014,34(6):1440-1449.
[9] 陈其安.解线性方程组的共轭梯度法预条件研究[D].重庆:重庆大学,2001:39-46.