the upper bound for gmres on normal tridiagonal toeplitz linear system

The generalized minimal residual method (gmres) isoften used to solve a large and sparse system ax = b. this paper establisheserror bound for residuals of gmres on solving an n × nnormal tridiagonal toeplitz linear system. this problem has been studiedpreviously by li [r.-c. li, convergence of cg and gmres on atridiagonal toeplitz linear system, bit 47 (3) (2007) 577-599.], for twospecial right-hand sides b = e1 , en . also, li and zhang [r.-c. li, w.zhang, the rate of convergence of gmres on a tridiagonal toeplitzlinear system, numer. math. 112 (2009) 267-293.] for non-symmetricmatrix a, presented upper bound for gmres residuals. but in this paperwe establish the upper bound on normal tridiagonal toeplitz linearsystems for special right-hand sides b = b(l)el, for 1 ? l ? n .