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Cross rules of some extrapolation algorithms. (English) Zbl 1215.65005

Many algorithms involve variables which can be set in a two-dimensional array, the position of a variable being indicated by two attached integer suffixes, \(m \geq 0\) and \(r \geq 0\) say; variables corresponding to a constant value of \(m\) lie on the same diagonal while those corresponding to a constant \(r\)-value lie in a column; there is a connection between variables having attached suffix values of the form \((m+1,r-1)\), \((m,r)\), \((m+1,r)\) and \((m,r+1)\), such variables lying at the vertices of a lozenge in the two dimensional array.
The paper lists a number of algorithms – the \(\varepsilon\)-, \(q\)-, \(d\)-, and \(\omega\)-algorithms and others – of this type. The relationships involving variables lying at the vertices of four conveniently contiguous lozenges may possibly be used to derive a single relationship connecting variables having attached suffix values of the form \((m+2,2r-2)\), \((m,r)\), \((m+1,r)\), \((m+2,r)\) and \((m,r+2)\), such variables lying at the centre and extremities of a cross in the array. In this way the reviewer derived from the \(\varepsilon\)-algorithm a cross relationship connecting approximating fractions (Näherungsbrüchen) which might be set with others derived by Frobenius. The paper lists a number of such relationships. The variables may be transformed by replacing \(m\) by \(t+mh\), suitably rearranging the relationship connecting variables lying at the vertices of a lozenge and letting \(h\) tend to zero. In this way it may occur that a difference-differential relationship can be produced. Again, elimination between three such difference-differential relationships may produce a single collapsed cross form. The paper lists some such relationships. In passing, the confluent form of the \(q\)-\(d\)-algorithm is transcribed as the Toda-equation and it is remarked that the \(\varepsilon\)-algorithm may be reformulated as the discrete Korteweg-de Vries lattice equation.
The references might have been extended. For example, the reviewer’s [Calcolo 15, No. 4, Suppl., 1–103 (1978; Zbl 0531.40002)], [Arch. Ration. Mech. Anal. 28, 83–148 (1968; Zbl 0162.37202)], [J. Reine Angew. Math. 285, 181–208 (1976; Zbl 0326.40005)] contain further structural and convergence results.
Again, extension of the material might also have been possible. For example, there is a further difference-differential relationship for continued fraction coefficients [Proc. Lond. Math. Soc., III. Ser. 23, 283–300 (1971; Zbl 0221.40005)]. Also, it might have been remarked that not only difference-differential relationships but also differential-differential relationships ( i.e., partial differential equations ) have been derived from algorithms [Z. angew. Math. Phys. 15, 273–289 (1964; Zbl 0252.65096)]. In particular, the Padé table leads to such an equation: [C. R. Acad. Sci., Paris, Sér. A 278, 847–850 (1974; Zbl 0276.35015)].
To conclude, it is remarked that the closed expressions upon which the scalar \(\varepsilon\)-algorithm is based were initially given by J. R. Schmidt [Philos. Mag., VII. Ser. 32, 369–383 (1941; Zbl 0061.27109)] and not as might be supposed from the paper under review and some of its predecessors. The academic political advantages that might accrue from an alternative attribution are now fading: the record should perhaps be corrected.

MSC:

65B05 Extrapolation to the limit, deferred corrections
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