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求解正定线性方程组的外推的PSS迭代方法

吴思婷 鲍亮 黄景宣

吴思婷, 鲍亮, 黄景宣. 求解正定线性方程组的外推的PSS迭代方法[J]. 华东理工大学学报(自然科学版). doi: 10.14135/j.cnki.1006-3080.20210312001
引用本文: 吴思婷, 鲍亮, 黄景宣. 求解正定线性方程组的外推的PSS迭代方法[J]. 华东理工大学学报(自然科学版). doi: 10.14135/j.cnki.1006-3080.20210312001
An extrapolated PSS iterative method for positive definite linear systems[J]. Journal of East China University of Science and Technology. doi: 10.14135/j.cnki.1006-3080.20210312001
Citation: An extrapolated PSS iterative method for positive definite linear systems[J]. Journal of East China University of Science and Technology. doi: 10.14135/j.cnki.1006-3080.20210312001

求解正定线性方程组的外推的PSS迭代方法

doi: 10.14135/j.cnki.1006-3080.20210312001
详细信息
    作者简介:

    吴思婷(1996-),女,浙江温州,硕士研究生,主要研究方向:数值代数及其应用。E-mail:wusiting96@163.com

    通讯作者:

    鲍 亮,E-mail:lbao@ecust.edu.cn

  • 中图分类号: O241.6

An extrapolated PSS iterative method for positive definite linear systems

  • 摘要: 为了更高效地求解大型稀疏正定线性方程组,提出了一种外推的正定和反Hermitian迭代方法。新方法首先对系数矩阵进行正定和反Hermitian分裂,再构造出了一种新的非对称二步迭代格式。然后理论分析了新方法的收敛性,并给出了新方法收敛的充要条件。最后数值实验表明,通过参数值的选择,新方法比PSS迭代方法和EHSS迭代方法具有更快的收敛速度和更小的迭代次数。并且选择了合适的参数值后,新方法的收敛效率可以大大提高。

     

  • 图  1  表示例1取$qh = 100,1000$$N = 512$时,EPSS和PSS迭代方法的残量下降速度比较

    Figure  1.  The comparison of the residuals speed in EPSS and PSS for example 1when $qh = 100,1000$,$N = 512$

    图  2  表示例2取$q = 1000$,$N = 12,14$时,EPSS、PSS和EHSS迭代方法的残量下降速度比较

    Figure  2.  The comparison of the residuals speed in EPSS, PSS and EHSS for example 2when $q = 1000$ and $N = 12,14$

    图  3  为例1取$qh = 100,1000$$N = 512$时,EPSS和PSS迭代方法的谱半径变化,分别固定$\omega = 0.6,0.7$

    Figure  3.  The comparison of the spectral radius in EPSS and PSS for example 1 when $qh = 100,1000$ and $N = 512$ with respectively fixed $\omega = 0.6,0.7$

    图  4  为例1取$qh = 100,1000$$N = 256$时,EPSS迭代方法中参数$\alpha $$\omega $与谱半径的关系

    Figure  4.  The relationship between the parameters $\alpha $,$\omega $ and the spectral radius in the EGHSS iterative method for example 1 when $qh = 100,1000$ and $N = 256$

    表  1  例4.1 PSS和EPSS迭代方法谱半径,收敛迭代次数以及时间的比较

    Table  1.   Spectral radius, IT and CPU of PSS and EPSS iterative method for example 4.1

    $qh$$N$PSSEPSS
    $\rho $ITCPU$\rho $ITCPU
    10640.3583360.00150.1973230.0010
    1280.4355520.00270.2268370.0016
    2560.5623730.00690.3782470.0034
    5120.66961020.04430.4925650.0081
    100640.70941000.00360.3084140.0012
    1280.76091390.00530.3214150.0017
    2560.81261930.01180.3238190.0021
    5120.86102680.10100.3240280.0051
    1000640.7391430.00570.192980.0010
    1280.8182680.00690.2153100.0015
    2560.90481310.02210.2873220.0019
    5120.95122760.11340.2968150.0043
    下载: 导出CSV

    表  2  例4.2 PSS、EPSS和EHSS迭代方法谱半径,收敛迭代次数以及时间的比较

    Table  2.   Spectral radius, IT and CPU of PSS, EPSS and EHSS iterative method for example 4.2 when $q = 1000$

    N$\omega $$\alpha $$\rho $ITCPU
    PSS83.50.5329220.0469
    102.90.5825250.1406
    122.60.6270300.4375
    142.30.6626330.9219
    EPSS80.13.90.5191210.0156
    100.13.30.5657240.0781
    120.12.90.6094270.3594
    140.12.60.6460310.7969
    EHSS80.13.30.6269330.0313
    100.12.50.6508330.1563
    120.12.20.6869370.4063
    140.14.30.6978381.0625
    下载: 导出CSV
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出版历程
  • 收稿日期:  2021-03-12
  • 网络出版日期:  2021-06-29

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