Volume 9, Issue 2
A Non-Krylov Subspace Method for Solving Large and Sparse Linear System of Equations

Wujian Peng & Qun Lin

Numer. Math. Theor. Meth. Appl., 9 (2016), pp. 289-314.

Published online: 2016-09

Preview Purchase PDF 6 3195
Export citation
  • Abstract

Most current prevalent iterative methods can be classified into the socalled extended Krylov subspace methods, a class of iterative methods which do not fall into this category are also proposed in this paper. Comparing with traditional Krylov subspace methods which always depend on the matrix-vector multiplication with a fixed matrix, the newly introduced methods (the so-called (progressively) accumulated projection methods, or AP (PAP) for short) use a projection matrix which varies in every iteration to form a subspace from which an approximate solution is sought. More importantly, an accelerative approach (called APAP) is introduced to improve the convergence of PAP method. Numerical experiments demonstrate some surprisingly improved convergence behaviors. Comparisons between benchmark extended Krylov subspace methods (Block Jacobi and GMRES) are made and one can also see remarkable advantage of APAP in some examples. APAP is also used to solve systems with extremely ill-conditioned coefficient matrix (the Hilbert matrix) and numerical experiments shows that it can bring very satisfactory results even when the size of system is up to a few thousands.

  • Keywords

  • AMS Subject Headings

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{NMTMA-9-289, author = {}, title = {A Non-Krylov Subspace Method for Solving Large and Sparse Linear System of Equations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2016}, volume = {9}, number = {2}, pages = {289--314}, abstract = {

Most current prevalent iterative methods can be classified into the socalled extended Krylov subspace methods, a class of iterative methods which do not fall into this category are also proposed in this paper. Comparing with traditional Krylov subspace methods which always depend on the matrix-vector multiplication with a fixed matrix, the newly introduced methods (the so-called (progressively) accumulated projection methods, or AP (PAP) for short) use a projection matrix which varies in every iteration to form a subspace from which an approximate solution is sought. More importantly, an accelerative approach (called APAP) is introduced to improve the convergence of PAP method. Numerical experiments demonstrate some surprisingly improved convergence behaviors. Comparisons between benchmark extended Krylov subspace methods (Block Jacobi and GMRES) are made and one can also see remarkable advantage of APAP in some examples. APAP is also used to solve systems with extremely ill-conditioned coefficient matrix (the Hilbert matrix) and numerical experiments shows that it can bring very satisfactory results even when the size of system is up to a few thousands.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2016.y14014}, url = {http://global-sci.org/intro/article_detail/nmtma/12378.html} }
TY - JOUR T1 - A Non-Krylov Subspace Method for Solving Large and Sparse Linear System of Equations JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 289 EP - 314 PY - 2016 DA - 2016/09 SN - 9 DO - http://doi.org/10.4208/nmtma.2016.y14014 UR - https://global-sci.org/intro/article_detail/nmtma/12378.html KW - AB -

Most current prevalent iterative methods can be classified into the socalled extended Krylov subspace methods, a class of iterative methods which do not fall into this category are also proposed in this paper. Comparing with traditional Krylov subspace methods which always depend on the matrix-vector multiplication with a fixed matrix, the newly introduced methods (the so-called (progressively) accumulated projection methods, or AP (PAP) for short) use a projection matrix which varies in every iteration to form a subspace from which an approximate solution is sought. More importantly, an accelerative approach (called APAP) is introduced to improve the convergence of PAP method. Numerical experiments demonstrate some surprisingly improved convergence behaviors. Comparisons between benchmark extended Krylov subspace methods (Block Jacobi and GMRES) are made and one can also see remarkable advantage of APAP in some examples. APAP is also used to solve systems with extremely ill-conditioned coefficient matrix (the Hilbert matrix) and numerical experiments shows that it can bring very satisfactory results even when the size of system is up to a few thousands.

Wujian Peng & Qun Lin. (2020). A Non-Krylov Subspace Method for Solving Large and Sparse Linear System of Equations. Numerical Mathematics: Theory, Methods and Applications. 9 (2). 289-314. doi:10.4208/nmtma.2016.y14014
Copy to clipboard
The citation has been copied to your clipboard