Optimal Acceleration of Convergence

• Autores: Xiaoyan Liu, Arieh Iserles
• Localización: Methods and applications of analysis, ISSN 1073-2772, Vol. 8, Nº 1, 2001, págs. 15-32
• Idioma: inglés
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• Resumen

  • In this paper we investigate the quadratic model bTu+uTCu aTU' which generalises the Shanks transformation and the familiar Padé model, to accelerate the conver-gence of a sequence {uk}kEZP. In the new model, u T =[u0, u 1>~ ,Tln,1, a, b E R"+1 and C is an (n + 1) x (n + 1) Hermitian matrix. (Note that, although u might be complex, we use the transpose uT, rather than the adjoint u*.) Let us assume that the original sequence is obtained from the dynamical system z H f (z), where f (z) = v, + E~ 1 k, ¡3k (z - ú)k is analytic about its fixed point ú E C. Suppose that uo = ú -f a and uk = f°k(uo). It is easy to show that b = 0 and a = 2C1. We give an iterative formula for the construction of the matrices C. Fhrthermore, we discuss the rate of convergence, inclusive of the special case when the fixed point is at oo.