Chebyshev Iteration

In numerical linear algebra, the Chebyshev iteration is an iterative method for determining the solutions of a system of linear equations. The method is named after Russian mathematician Pafnuty Chebyshev.

Chebyshev iteration avoids the computation of inner products as is necessary for the other nonstationary methods. For some distributed-memory architectures these inner products are a bottleneck with respect to efficiency. The price one pays for avoiding inner products is that the method requires enough knowledge about spectrum of the coefficient matrix A, that is an upper estimate for the upper eigenvalue and lower estimate for the lower eigenvalue. There are modifications of the method for nonsymmetric matrices A.

Example code in MatLab

function [x] =  SolChebyshev002(A,b,x0,iterNum,lMax,lMin)
 
  d=(lMax+lMin)/2;
  c=(lMax-lMin)/2;
  preCond=eye(size(A)); %preconditioner
  x=x0;
  r=b-A*x;
 
  for i = 1:iterNum % size(A,1)
      z = linsolve(preCond,r);
      if (i==1)
          p=z;
          alpha=2/d;
      else
          beta=(c*alpha/2)^2;
          alpha=1/(d-beta);
          p=z+beta*p;
      end;
 
      x=x+alpha*p;
      r=b-A*x; %(=r-alpha*A*p)
      if (norm(r)<1e-15), break; end; %stop if necessary
  end;
end

External links

  • Templates for the Solution of Linear Systems
  • Chebyshev Iteration. From MathWorld

See also

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.