Cajigas Santiago, Jesús

Profile Picture

Filters

20121
Reset filters

Settings

Citations

Publication Search Results

Now showing 1 - 1 of 1
  • Thumbnail Image
    PublicationRestricted
    A new preconditions for solving linear systems with symmetric Z-matrices
    (2012-05) Cajigas Santiago, Jesús; Yong, Xuerong; College of Arts and Sciences - Sciences; Vásquez Urbano, Pedro; Castillo, Paul; Department of Mathematics; González Quevedo, Antonio A.
    There are many preconditioners for linear systems, for example, P = I + Smax developed by Kotakemori in [9] and its extension P˜ developed by Arenas & Yong in [1]. These preconditioners were built to speed up the convergence of the method when solving it. Symmetry is a useful property to preserve when applying a preconditioner. Preserving symmetry is advantageous since there are known methods that ensure convergence if the coefficient matrix is symmetric, among other things. Due to this fact, preconditioners that improve the convergence of the method at the same time that keeps symmetry are interesting and useful. This thesis introduces a new preconditioner, named PSY M, that preserves symmetricity of the coeficient matrix and improves the convergence of the Symmetric Gauss-Seidel method when applied. This new preconditioner was based on the one proposed by Arenas & Yong in [1] using the idea proposed by Kotakemori in [9]. This one is to turn the largest entry (in terms of absolute value) above the main diagonal, by row, into zero when the preconditioner is applied. By applying the new preconditioner, a clear reduction on the number of iterations until convergence and the elapsed time in the Symmetric Gauss-Seidel method is observed. In addition, a reduction of the condition number of the coefficient matrix is noticed. Even though this method is not as general as the one proposed by Arenas & Yong ([1]) or Kotakemori ([9]) when comparing them by considering the coefficient matrix as a symmetric one, the new method shows an improvement. On the other hand, other preconditioners, with the exception of, P˜, are not pose as iterative, the preconditioner proposed for this thesis, PSY M is. The preconditioner P˜ if applied iteratively, transforms the matrix into a lower triangular matrix; meanwhile, PSY M improves this. By applying the preconditioner PSY M iteratively, the matrix turns into a diagonal matrix.