A critical review of asymptotic numerical methods

Mathematical and numerical connections between polynomial extrapolation and Padé approximants: applications in structural mechanics

“…They lie on classical asymptotic numerical method [10,13] prediction path starting from ( = 0; W = 0). The ÿrst one, denoted point 1, is selected on the path from Padà e approximants at order 25 and the second one is selected on the polynomial (series) path at order 5.…”

“…In the present paper, we consider only a speciÿc VPA [10], where all the fractions have a common denominator and with the same degrees for the numerators and the denominator. The interest of this procedure has been established by a number of practical applications for iterative algorithms and also for path following techniques; see [14] and references therein.…”

“…Introducing ( where P k ( ) is a polynomial of degree k. Second, the N − 1 polynoms P N −n ( ) (16n6N − 1) are replaced by N − 1 rational fractions having the same denominator N −1 ( ), as detailed in [3,4,10]. The ÿnal expression is given by…”

“…A speciÿc sub-class of polynomial extrapolation, called the modiÿed minimal polynomial extrapolation (MMPE) has been evaluated recently [9]. In view of applications in non-linear mechanics, a family of vectorial Padà e approximants (VPA) has been proposed [10]. In the latter approach, the sequence V S N ( ) is accelerated and not only V S N (1) = S N .…”

“…First, it is established that, with a speciÿc choice of the projection basis, MMPE and VPA are mathematically equivalent. In numerical applications, the convergence of such vectorial sequences can be a ected by numerical instabilities [10]. That is why these algorithms have also been evaluated on the basis of a practical numerical application.…”