By C. Brezinski, M. Redivo Zaglia

This quantity is a self-contained, exhaustive exposition of the extrapolation tools concept, and of a number of the algorithms and methods for accelerating the convergence of scalar and vector sequences. Many subroutines (written in FORTRAN seventy seven) with directions for his or her use are supplied on a floppy disk so that it will exhibit to these operating with sequences the benefits of using extrapolation tools. Many numerical examples displaying the effectiveness of the techniques and a consequent bankruptcy on functions also are supplied – together with a few by no means earlier than released effects and purposes. even if meant for researchers within the box, and for these utilizing extrapolation equipment for fixing specific difficulties, this quantity additionally presents a precious source for graduate classes at the topic.

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A,b). If there exists c such that lim (Tn-S)/(Sn-S) n—*oo = c, then lim ATn/A5n n—»oo As proved by a counter-example, the reciprocal of this theorem is not true. 22 If (Sn) is monotone and if there exists c, finite or not, such that lim ATn/A5n = c, then lim (Tn - T)/(Sn - S) = c. 23 Assume that there exist a and b with a < 1 < 6 and N such that Vn>N If \Tn -T\ = 0 (\Sn - 5|), then |ATn| = 0 (|ASn|). 24 Assume ^a*limsup|AT n |» = l/R and lim |A5n|" = 1/r. Ifr

There are two possibilities for such an algorithm: its normal form or its progressive form. 8 respectively but let us now explain why ratios of determinants and recursive schemes are obtained. 6 Triangular recursive schemes A unified treatment of sequence transformations in relation with their determinantal expressions and triangular recursive schemes for their implementation was given by Brezinski and Walz [104]. In fact this theory goes far beyond extrapolation methods since it includes in the same framework B-splines, Bernstein polynomials, orthogonal polynomials, divided differences and certainly many other topics.

Thus 6 and d will be almost equal and E will be the difference of two large and badly computed numbers. There is a near-breakdown in the algorithm. After some algebraic manipulations it can be proved that the cross rule of the £-algorithm can be equivalently written as with This rule was shown to be more stable than the rule given above for computing E. It is called a particular rule for the £-algorithm. If C is infinity, it reduces to thus allowing to compute E by jumping over the singularity (or the breakdown).