By C. Brezinski, M. Redivo Zaglia
This quantity is a self-contained, exhaustive exposition of the extrapolation tools idea, and of many of the algorithms and systems 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 on the way to show to these operating with sequences some great benefits of using extrapolation equipment. Many numerical examples displaying the effectiveness of the tactics and a consequent bankruptcy on purposes also are supplied - together with a few by no means sooner than released effects and functions. even if meant for researchers within the box, and for these utilizing extrapolation tools for fixing specific difficulties, this quantity additionally presents a useful source for graduate classes at the topic
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Additional resources for Extrapolation methods : theory and practice
Thus Aitken's Δ 2 process is also optimal in this sense since no other transformation can produce a better acceleration for linear sequences. These results on the degree of acceleration were refined by Trojan . He considered transformations of the form Tn = Fk ( 5 „ , . . , Sn+k) j n = 0,1,... were Fk is a rational function independent of n. Obviously Aitken's Δ 2 process has this form. Let X p , m be the set of sequences such that, Vn Sn-S = - A 5 n + a1(-AS„)p + ... 11. Optimality 45 with a\ φ 0, p > 1 and b a bounded function in a neighborhood of zero.
18. 27 Let α,/3, α',β' be real numbers with 0 < α < β < 1 and Q < a' < β' < 1. ] and Δ5π+ι/Δ5„€[α/ϊ/3']. ). 18. Additional results and tools for the asymptotic comparison of se quences could be found in Brezinski [37, 58]. When accelerating the convergence of sequences we are naturally led to use words like acceleration or speed of convergence. Thus we shall now give precise definitions of such concepts based on kinematical notions and the main corresponding results. The interested reader is referred for more details to Brezinski .
4. Increase k by 1 for solving the next system (if wanted). 5. Set pk = 1. 6. Compute ßk7. If ßk is non-singular, compute 2^+1 and go to step 4. If ßk is singular, border the system by the next row and the next column, increase pk by 1 and go to step 6. Of course a test for checking the numerical singularity of a matrix is needed. The permutation-perturbation method of La Porte and Vignes  is particularly well adapted for that purpose. Instead of using the previous block bordering method it is possible to use a pivoting strategy.
Extrapolation methods : theory and practice by C. Brezinski, M. Redivo Zaglia