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

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

华东理工大学理学院，上海 200237

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
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出版历程
- 收稿日期: 2021-03-12
- 网络出版日期: 2021-06-29

An extrapolated PSS iterative method for positive definite linear systems

摘要

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。
- 正定线性方程组 /
- PSS迭代方法 /
- EHSS迭代方法 /
- 谱半径 /
- 收敛性
Abstract: Positive definite linear systems arise in many areas of scientific computing and engineering applications, such as solid mechanics, dynamics, nonlinear programming and partial differential equations. It is very meaningful to explore how to efficiently solve the large scale sparse saddle point problem. This paper proposes an extrapolated positive definite and skew-Hermitian (EPSS) iterative method for solving large sparse positive definite linear systems. The new method first splits the coefficient matrix into positive definite matrix and skew-Hermitian matrix, next constructs a new non-symmetric two-step iterative scheme. The new method can not only solve non-Hermitian positive definite linear equations, but also be used for solving Hermitian positive definite linear equations, which greatly accelerates the convergence speed of the iterative method. Then theoretical analysis shows that the new method is convergent. And the necessary and sufficient conditions for the convergence of the new method are given.Moreover the spectral radius of the iterative matrix of the new method is smaller than that of the iterative matrix of the positive definite and skew-Hermitian (PSS) iterative method when selecting appropriate variables. After that numerical experiments are given to show that the new method is efficient and more competitive than PSS iteration method and the extrapolated Hermitian and skew-Hermitian (EHSS) iterative method. Finally, numerical experiments analyze the sensitivity of the parameters in the EPSS iterative method and find the approximate optimal parameters.
- positive definite linear systems /
- PSS iteration method /
- EHSS iteration method /
- spectral radius /
- convergence

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图 1 表示例1取$qh = 100,1000$且$N = 512$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 $qh = 100,1000$,$N = 512$

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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 $N = 12,14$

下载: 全尺寸图片幻灯片

图 3 为例1取$qh = 100,1000$且$N = 512$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 $\omega = 0.6,0.7$

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图 4 为例1取$qh = 100,1000$且$N = 256$时，EPSS迭代方法中参数$\alpha $和$\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 $N = 256$

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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

$qh$	$N$	PSS			EPSS
$qh$	$N$	$\rho $	IT	CPU	$\rho $	IT	CPU
10	64	0.3583	36	0.0015	0.1973	23	0.0010
	128	0.4355	52	0.0027	0.2268	37	0.0016
	256	0.5623	73	0.0069	0.3782	47	0.0034
	512	0.6696	102	0.0443	0.4925	65	0.0081
100	64	0.7094	100	0.0036	0.3084	14	0.0012
	128	0.7609	139	0.0053	0.3214	15	0.0017
	256	0.8126	193	0.0118	0.3238	19	0.0021
	512	0.8610	268	0.1010	0.3240	28	0.0051
1000	64	0.7391	43	0.0057	0.1929	8	0.0010
	128	0.8182	68	0.0069	0.2153	10	0.0015
	256	0.9048	131	0.0221	0.2873	22	0.0019
	512	0.9512	276	0.1134	0.2968	15	0.0043

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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 $	IT	CPU
PSS	8	−	3.5	0.5329	22	0.0469
	10	−	2.9	0.5825	25	0.1406
	12	−	2.6	0.6270	30	0.4375
	14	−	2.3	0.6626	33	0.9219
EPSS	8	0.1	3.9	0.5191	21	0.0156
	10	0.1	3.3	0.5657	24	0.0781
	12	0.1	2.9	0.6094	27	0.3594
	14	0.1	2.6	0.6460	31	0.7969
EHSS	8	0.1	3.3	0.6269	33	0.0313
	10	0.1	2.5	0.6508	33	0.1563
	12	0.1	2.2	0.6869	37	0.4063
	14	0.1	4.3	0.6978	38	1.0625

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