This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages) This article relies excessively on references to primary sources. Please improve this article by adding secondary or tertiary sources. Find sources: "Stone's method" – news · newspapers · books · scholar · JSTOR (October 2015) (Learn how and when to remove this message) This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (September 2013) (Learn how and when to remove this message) (Learn how and when to remove this message)

In numerical analysis, Stone's method, also known as the strongly implicit procedure or SIP, is an algorithm for solving a sparse linear system of equations. The method uses an incomplete LU decomposition, which approximates the exact LU decomposition, to get an iterative solution of the problem. The method is named after Harold S. Stone, who proposed it in 1968.

The LU decomposition is an excellent general-purpose linear equation solver. The biggest disadvantage is that it fails to take advantage of coefficient matrix to be a sparse matrix. The LU decomposition of a sparse matrix is usually not sparse, thus, for a large system of equations, LU decomposition may require a prohibitive amount of memory and number of arithmetical operations.

In the preconditioned iterative methods, if the preconditioner matrix M is a good approximation of coefficient matrix A then the convergence is faster. This brings one to idea of using approximate factorization LU of A as the iteration matrix M.

A version of incomplete lower-upper decomposition method was proposed by Stone in 1968. This method is designed for equation system arising from discretisation of partial differential equations and was firstly used for a pentadiagonal system of equations obtained while solving an elliptic partial differential equation in a two-dimensional space by a finite difference method. The LU approximate decomposition was looked^{[clarification needed]} in the same pentadiagonal form as the original matrix (three diagonals for L and three diagonals for U) as the best match of the seven possible equations for the five unknowns for each row of the matrix.

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

• Stone, H. L. (1968). "Iterative Solution of Implicit Approximations of Multidimensional Partial Differential Equations". SIAM Journal on Numerical Analysis. 5 (3): 530–538. Bibcode:1968SJNA....5..530S. doi:10.1137/0705044. hdl:10338.dmlcz/104038. - the original article
• Ferziger, J.H. and Peric, M. (2001). Computational Methods for Fluid Dynamics. Springer-Verlag, Berlin. ISBN 3-540-42074-6.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Acosta, J.M. (2001). Numerical Algorithms for Three Dimensional Computational Fluid Dynamic Problems. PhD Thesis. Polytechnic University of Catalonia.
• This article incorporates text from the article Stone's_method on CFD-Wiki that is under the GFDL license.