IMPLEMENTATION OF AN ADAPTIVE ALGORITHM FOR RICHARDSON METHOD

被引:20
作者
SAYLOR, PE
SMOLARSKI, DC
机构
[1] ETH,INST WISSENSCHAFTLICHES RECHNEN,DEPT INFORMAT,ZURICH,SWITZERLAND
[2] UNIV SANTA CLARA,DEPT MATH,SANTA CLARA,CA 95053
基金
美国国家科学基金会;
关键词
D O I
10.1016/0024-3795(91)90397-F
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
We discuss the implementation of an adaptive algorithm proposed by one of us. The algorithm is a hybrid of the GMRES method and Richardson's method. Richardson's method (RM) depends on a set of parameters that are computed by minimizing the L2 norm of a polynomial over the convex hull of eigenvalues. Execution of GMRES yields not only an approximate solution but also the approximate convex hull. RM is used to avoid storing and working with a large number of vectors as GMRES often requires. This method is also advantageous for the solution of large problems. We consider several test problems and compare our algorithm primarily with the conjugate-gradient-squared algorithm, but also with GMRES and to CG (applied to the normal equations). For many (test) problems our algorithm takes roughly 50 percent more work than the conjugate-gradient-squared algorithm, although if the matrix is either preconditioned or indefinite, our algorithm is more efficient. However, our algorithm currently imposes an undesirable burden on the user, who is invited to consider a variety of numerical parameters to manipulate, such as the number of steps of RM, in order to enhance performance: the values we suggest are only empirical.
引用
收藏
页码:615 / 646
页数:32
相关论文
共 42 条
  • [1] ANDERSSEN RS, 1972, CS72304 STANF U COMP
  • [2] ADAPTIVE POLYNOMIAL PRECONDITIONING FOR HERMITIAN INDEFINITE LINEAR-SYSTEMS
    ASHBY, SF
    MANTEUFFEL, TA
    SAYLOR, PE
    [J]. BIT, 1989, 29 (04): : 583 - 609
  • [3] ASHBY SF, 1985, UIUCDCSR851203 U ILL
  • [4] AXELSSON O, 1985, BIT, V25, P166
  • [5] MATRIX-FREE METHODS FOR STIFF SYSTEMS OF ODES
    BROWN, PN
    HINDMARSH, AC
    [J]. SIAM JOURNAL ON NUMERICAL ANALYSIS, 1986, 23 (03) : 610 - 638
  • [6] CASTOR J, 1988, COMMUNICATION
  • [7] CHRONOPOULOS A, 1986, UIUCDCSR861267 U ILL
  • [8] ON THE EFFICIENT IMPLEMENTATION OF PRECONDITIONED S-STEP CONJUGATE-GRADIENT METHODS ON MULTIPROCESSORS WITH MEMORY-HIERARCHY
    CHRONOPOULOS, AT
    GEAR, CW
    [J]. PARALLEL COMPUTING, 1989, 11 (01) : 37 - 53
  • [9] S-STEP ITERATIVE METHODS FOR SYMMETRIC LINEAR-SYSTEMS
    CHRONOPOULOS, AT
    GEAR, CW
    [J]. JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 1989, 25 (02) : 153 - 168
  • [10] Eiermann M., 1987, JAHRESBER DTSCH MATH, V89, P1