##
**Partial Padé prediction.**
*(English)*
Zbl 0938.65002

The operation of a sequence transformation may often be referred to numbers placed in columns, each of which is associated with its index \( \tau(r) \), the \( \tau(r)\) \((r \geq 0) \) forming a strictly increasing sequence of nonnegative integers. The first column with index \( \tau(0) = 0 \) contains the successive members \( s(m)\) \((m \geq 0) \) of the sequence \( \mathcal S \) under transformation. The column having index \( \tau(r) \) contains transformed values \( t(r,m|s(m),s(m+1),\ldots,s(m+\tau(r)) \) for \( m \geq 0 \) obtained, in particular, from \( \tau(r) + 1 \) successive members of \( \mathcal S \). The further columns contain auxiliary numbers used in the construction of the transformed numbers. When \( \tau(r) = r\) \((r \geq 0) \) there are no further columns; when \( \tau(r) = 2r\) \((r \geq 0) \) the further columns are interlaced with their counterparts. It often occurs that when \( \mathcal S \) has a special structure, the columns of numbers exhibit a corresponding pattern; for example, for some fixed \( n \geq 0 \) all entries in the column index \( \tau(n) \) are the same, as is so, with \( \tau(n) = 2n \), for the \( \varepsilon \)- and \( \rho \)-algorithms; certain linear processes and the \(q\)-\(d\) algorithm produce other patterns. This property serves as the basis of a sequence extrapolation method: in the case of the \( \varepsilon \)-algorithm, the number \( t(r,0|s(0),\ldots,s(2r)) \) is determined from \( s(0),\ldots,s(2r) \), the column with index \( 2r \) is filled with copies of it, the transformation is put into reverse to construct \( s'(m)\) \((m > 2r) \). The extrapolated sequence \( \mathcal S' \) of terms \( s'(m)\) \((m \geq 0) \) is that sequence of special structure whose first \( 2r + 1 \) members \( s'(m)\) \((0 \leq m \leq 2r) \) agree with those of \( \mathcal S \).

The progressive use of the \( \varepsilon \)-algorithm in this way is considered. It is remarked that if the algorithm is applied to the sequence \( \mathcal S' \) to produce numbers in the column \( 2r \), they are all the same. It is also remarked that the members of \( \mathcal S' \) satisfy a linear recurrence relationship of order \( r + 1 \) with constant coefficients (inhomogeneous if the constant entries are nonzero). An equivalent but more cumbersome and unstable method based upon the use of approximating fractions is considered. Conditions sufficing to ensure that \( \mathcal S \) and \( \mathcal S' \) have the same limit are stated.

Much of the material has been known for some time. In particular, the reviewer has given a stability analysis (not mentioned by the authors) of the progressive use of a number of algorithms [Numer. Math. 1, 142-149 (1959; Zbl 0087.32502)]. It is a curious fact that sequences which induce instability of the \( \varepsilon \)-algorithm in its forward mode of use (i.e. to construct a sequence of numbers with fixed \( m \) and increasing \( r \)) may be extrapolated in a completely stable way. Again, the \( \rho \)-algorithm, whose instability in forward use must be monitored carefully, may be used as a completely stable extrapolation device.

The progressive use of the \( \varepsilon \)-algorithm in this way is considered. It is remarked that if the algorithm is applied to the sequence \( \mathcal S' \) to produce numbers in the column \( 2r \), they are all the same. It is also remarked that the members of \( \mathcal S' \) satisfy a linear recurrence relationship of order \( r + 1 \) with constant coefficients (inhomogeneous if the constant entries are nonzero). An equivalent but more cumbersome and unstable method based upon the use of approximating fractions is considered. Conditions sufficing to ensure that \( \mathcal S \) and \( \mathcal S' \) have the same limit are stated.

Much of the material has been known for some time. In particular, the reviewer has given a stability analysis (not mentioned by the authors) of the progressive use of a number of algorithms [Numer. Math. 1, 142-149 (1959; Zbl 0087.32502)]. It is a curious fact that sequences which induce instability of the \( \varepsilon \)-algorithm in its forward mode of use (i.e. to construct a sequence of numbers with fixed \( m \) and increasing \( r \)) may be extrapolated in a completely stable way. Again, the \( \rho \)-algorithm, whose instability in forward use must be monitored carefully, may be used as a completely stable extrapolation device.

Reviewer: P.Wynn (México)

### MSC:

65B05 | Extrapolation to the limit, deferred corrections |

41A21 | Padé approximation |

40A05 | Convergence and divergence of series and sequences |