A uniform error bound for the overrelaxation methods

✍ Scribed by Xiezhang Li

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 784 KB
• Volume: 254
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(96)00313-8

✦ Synopsis

Let Ax = b be a system of linear equations where A is symmetric and positive definite. Suppose that the associated block Jacobi matrix B is consistently ordered, weekly cyclic of index 2, and convergent [i.e., I_L~ := p(B) < 11. Consider using the overrelaxation methods (SOR, AOR, MSOR, SSOR, or USSOR), x,+i = Tax, + c, for n > 0, to solve the system. We derive a uniform error bound for the overrelaxation methods, Ibr -X"l12 4 1

[1 ++q ++;)I2

x (to + It,1 p~)2116"l12 -2t,(6,, 6",,) 1 +lt,l p~ll6,ll ll6,+,ll + 11~"+1112]~

where II . II = II * 112, 6, = x, -x,,_~, and s( p2> and t( p2) := t, + t, p2 are two coefficients of the corresponding functional equation connecting the eigenvalues A of T, to the eigenvalues /L of B. As special cases of the uniform error bound, we will give two error bounds for the SSOR and USSOR methods.