Stone's method

<h2 id="algorithm">Algorithm</h2>
method stone is
    For the linear system Ax = b
    calculate incomplete LU factorization of matrix A
       Ax = (M-N)x = (LU-N)x = b
       Mx(k+1) = Nx(k)+b , with ||M|| >> ||N||
       Mx(k+1) = LUx(k+1) = c(k)
       LUx(k) = L(Ux(k+1)) = Ly(k) = c(k)
    set a guess
       k = 0, x(k)
       r(k)=b - Ax(k)
    while ( ||r(k)||2 ≥ ε ) do
       evaluate new right hand side
          c(k) = Nx(k) + b
       solve Ly(k) = c(k) by forward substitution
          y(k) = L−1c(k)
       solve Ux(k+1) = y(k) by back substitution
          x(k+1) = U−1y(k)
    end while

<h2 id="footnotes">Footnotes</h2>

<ul><li>Stone, H. L. (1968). "Iterative Solution of Implicit Approximations of Multidimensional Partial Differential Equations". <i>SIAM Journal on Numerical Analysis</i>. 5 (3): 530–538. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1968SJNA....5..530S">1968SJNA....5..530S</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1137%2F0705044">10.1137/0705044</a>. <a href="/facts/Hdl_(identifier)/rdebSxmC">hdl</a>:<a href="https://hdl.handle.net/10338.dmlcz%2F104038">10338.dmlcz/104038</a>. - the original article</li>
<li><a href="/facts/Joel_H._Ferziger/yvjpftQ9">Ferziger, J.H.</a> and Peric, M. (2001). <i>Computational Methods for Fluid Dynamics</i>. Springer-Verlag, Berlin. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-42074-6.{{cite book}}:  CS1 maint: multiple names: authors list (link)</li>
<li>Acosta, J.M. (2001). <i>Numerical Algorithms for Three Dimensional Computational Fluid Dynamic Problems. PhD Thesis</i>. Polytechnic University of Catalonia.</li>
<li>This article incorporates text from the article <a href="http://www.cfd-online.com/Wiki/Stone's_method">Stone's_method</a> on <a href="http://www.cfd-online.com/Wiki/Main_Page">CFD-Wiki</a> that is under the <a href="/facts/GFDL/IhVHF3zP">GFDL</a> license.</li></ul>

Stone's method open-in-new

Stone's method