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《山东大学学报(理学版)》 ›› 2024, Vol. 59 ›› Issue (10): 1-9.doi: 10.6040/j.issn.1671-9352.0.2023.225

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一类复对称线性方程组的块三角分裂及其预处理迭代算法

王洋()   

  1. 吉林师范大学数学与计算机学院, 吉林 四平 136100
  • 收稿日期:2023-05-19 出版日期:2024-10-20 发布日期:2024-10-10
  • 作者简介:王洋(1982—), 女, 副教授, 硕士生导师, 博士, 研究方向为数值代数及其应用. E-mail: yyang3721@163.com
  • 基金资助:
    吉林省社会科学基金资助项目(2021B136)

Block triangular splitting and its preconditioning iterative algorithms for a class of complex symmetric linear systems

Yang WANG()   

  1. College of Mathematics and Computer, Jilin Normal University, Siping 136100, Jilin, China
  • Received:2023-05-19 Online:2024-10-20 Published:2024-10-10

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摘要：

基于2×2块矩阵的块三角分裂, 提出求解复对称线性方程组的块三角分裂(BTS)迭代方法及其预处理迭代方法(PBTS)。理论分析表明, 在迭代参数α满足一定的条件下, BTS和PBTS迭代方法是收敛性的, 给出两类方法迭代格式中理论最优参数的计算方法。数值实验结果证明BTS迭代方法和PBTS迭代方法的有效性和优越性。

关键词: 复对称线性方程组, 收敛性分析, 块三角分裂, 预处理

Abstract:

Based on block triangular splitting for block 2×2 matrix, the block triangular splitting(BTS) iteration method and the preconditioned block triangular splitting (PBTS) iteration method for a class of complex symmetric linear system are proposed. Theoretical analysis shows that the BTS and PBTS methods converge under certain conditions. The optimal iteration parameters of these two methods are obtained. Numerical experiments demonstrate the effectiveness and superiority of the BTS method and the PBTS iterative methods.

Key words: complex symmetric linear system, convergence analysis, block triangular splitting, preconditioning

中图分类号: 

  • O242.2

图1

例1中迭代矩阵谱半径的比较"

图2

例2中迭代矩阵谱半径的比较"

图3

例3中迭代矩阵谱半径的比较"

表1

例1中6种迭代方法的理论最优参数值"

算法 n×n 8×8 16×16 32×32 64×64
GSOR αopt 0.6168 0.5516 0.4967 0.4590
SBTS αopt 6.5880/0.5411 8.4153/0.5316 10.6630/0.5250 12.7600/0.5204
NSBTS αopt 3.5645 4.4735 5.5938 6.6402
PGSOR αopt 0.9944 0.9908 0.9877 0.9855
ω* 0.7017 0.6578 0.6239 0.6025
PSBTS αopt 1.1753/0.8702 1.2344/0.8404 1.1866/0.8641 1.2050/0.8544
ω* 0.7018 0.6577 0.6239 0.6025
PNSBTS αopt 1.0116 1.0187 1.0253 1.0298
ω* 0.7018 0.6577 0.6239 0.6025

表2

例2中6种迭代方法的理论最优参数值"

算法 n×n 8×8 16×16 32×32 64×64
GSOR αopt 0.9636 0.9083 0.7764 0.5661
SBTS αopt 1.3747/0.7858 1.7470/0.7005 2.8820/0.6050 6.8786/0.5319
NSBTS αopt 1.0803 1.2238 1.7435 3.7080
PGSOR αopt 0.9937 0.9820 0.9556 0.9183
ω* 4.5036 3.0020 1.9780 1.4366
PSBTS αopt 1.1278/0.8982 1.2339/0.8406 1.4284/0.7693 1.6760/0.7126
ω* 4.5036 3.0020 1.9783 1.4366
PNSBTS αopt 1.0256 1.0373 1.0988 1.1943
ω* 4.5036 3.0020 1.9783 1.4366

表3

例3中6种迭代方法的理论最优参数值"

算法 n×n 8×8 16×16 32×32 64×64
GSOR αopt 0.4507 0.4553 0.4567 0.4571
SBTS αopt 12.3030/0.5212 11.9860/0.5217 11.8980/0.5219 11.8749/0.5219
NSBTS αopt 6.412 6.2540 6.2101 6.1984
PGSOR αopt 0.9035 0.8978 0.8962 0.8957
ω* 1.2536 1.3081 1.3236 1.3278
PSBTS αopt 1.7784/0.6956 1.8194/0.6894 1.8309/0.6878 1.8339/0.6874
ω* 1.2536 1.3081 1.3236 1.3278
PNSBTS αopt 1.4727 1.5072 1.2594 1.2607
ω* 1.2536 1.3081 1.3236 1.3278

表4

例1迭代步数(IT)迭代时间(CPU)的比较"

算法 n×n 8×8 16×16 32×32 64×64
GSOR IT/CPU 14/0.005 15/0.023 16/0.047 16/0.215
SBTS IT/CPU 13/0.004 17/0.020 20/0.112 20/0.508
NSBTS IT/CPU 13/0.007 17/0.020 20/0.054 20/0.267
PGSOR IT/CPU 3/0.001 3/0.005 3/0.008 3/0.040
PSBTS IT/CPU 3/0.003 3/0.008 3/0.020 3/0.056
PNSBTS IT/CPU 3/0.001 3/0.002 3/0.009 3/0.042

表5

例2迭代步数(IT)迭代时间(CPU)的比较"

算法 n×n 8×8 16×16 32×32 64×64
GSOR IT/CPU 7/0.003 9/0.008 14/0.068 26/0.546
SBTS IT/CPU 7/0.004 11/0.016 22/0.238 60/2.652
NSBTS IT/CPU 7/0.003 11/0.009 22/0.100 60/1.310
PGSOR IT/CPU 5/0.003 6/0.006 8/0.044 9/0.168
PSBTS IT/CPU 6/0.004 6/0.014 8/0.075 10/0.427
PNSBTS IT/CPU 6/0.002 6/0.004 8/0.037 10/0.236

表6

例3迭代步数(IT)迭代时间(CPU)的比较"

算法 n×n 8×8 16×16 32×32 64×64
GSOR IT/CPU 34/0.007 33/0.020 31/0.086 30/0.362
SBTS IT/CPU 95/0.035 92/0.127 93/0.476 96/2.281
NSBTS IT/CPU 96/0.026 93/0.056 93/0.254 95/1.141
PGSOR IT/CPU 9/0.002 9/0.012 9/0.035 9/0.114
PSBTS IT/CPU 10/0.006 10/0.027 11/0.065 11/0.267
PNSBTS IT/CPU 10/0.003 10/0.010 11/0.030 12/0.145
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