Scholarlink

<p>The extended Kaczmarz type methods are often used to solve large inconsistent overdetermined linear systems whose coefficient matrix <i>A</i> is of full column rank. These methods are also applicable to overdetermined linear systems where <i>A</i> is rank-deficient. In this paper, a modified partially randomized extended Kaczmarz method is proposed for solving large inconsistent linear systems with the rank-deficient coefficient matrix. The randomized range finder method proposed by Halko et al. (SIAM Rev 53(2):217–288, 2011) is used to generate a column-orthonormal matrix <i>Q</i> whose range approximates the range of the coefficient matrix <i>A</i> as a preconditioner. Then <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(Q^{\top }A\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>Q</mi> <mi>⊤</mi> </msup> <mi>A</mi> </mrow> </math></EquationSource> </InlineEquation> is a full row rank matrix and it has fewer rows than <i>A</i>. Thus the rows are selected based on residual for the preconditioned linear systems. Also, the convergence of the proposed extended Kaczmarz method is proved and the upper bound of the expected convergence rate is derived. Furthermore, numerical experiments are carried out and show that the proposed partially randomized extended Kaczmarz method is more efficient than the original methods.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

A partially randomized extended Kaczmarz method for solving inconsistent rank-deficient linear systems

  • 期刊论文

  • 发表日期：

    2026年2月3日

查看全文
  • Meng-ying Wu,
  • Zheng-sheng Wang

摘要

The extended Kaczmarz type methods are often used to solve large inconsistent overdetermined linear systems whose coefficient matrix A is of full column rank. These methods are also applicable to overdetermined linear systems where A is rank-deficient. In this paper, a modified partially randomized extended Kaczmarz method is proposed for solving large inconsistent linear systems with the rank-deficient coefficient matrix. The randomized range finder method proposed by Halko et al. (SIAM Rev 53(2):217–288, 2011) is used to generate a column-orthonormal matrix Q whose range approximates the range of the coefficient matrix A as a preconditioner. Then \(Q^{\top }A\) Q ⊤ A is a full row rank matrix and it has fewer rows than A. Thus the rows are selected based on residual for the preconditioned linear systems. Also, the convergence of the proposed extended Kaczmarz method is proved and the upper bound of the expected convergence rate is derived. Furthermore, numerical experiments are carried out and show that the proposed partially randomized extended Kaczmarz method is more efficient than the original methods.