Geodesic
Look-ahead Levinson- and Schur-type recurrences in the Padé table
Electronic transactions on numerical analysis, Tome 2 (1994), pp. 104-129.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: For computing Pad$\acute e$ approximants, we present presumably stable recursive algorithms that follow two adjacent rows of the Pad$\acute e$ table and generalize the well-known classical Levinson and Schur recurrences to the case of a nonnormal Pad$\acute e$ table. Singular blocks in the table are crossed by look-ahead steps. Ill-conditioned Pad$\acute e$ approximants are skipped also. If the size of these lookahead steps is bounded, the recursive computation of an (m, n) Pad$\acute e$ approximant with either the look-ahead Levinson or the look-ahead Schur algorithm requires $O(n2)$ operations. With recursive doubling and fast polynomial multiplication, the cost of the look-ahead Schur algorithm can be reduced to $O(n log2 n)$.
Classification : 41A21, 42A70, 15A06, 30E05, 60G25, 65F05, 65F30
Keywords: pad$\acute e$ approximation, Toeplitz matrix, levinson algorithm, Schur algorithm, lookahead, fast algorithm, biorthogonal polynomials, szegacute$\Acute o$ polynomials 
@article{ETNA_1994__2__a6,
     author = {Gutknecht, Martin H. and Hochbruck, Marlis},
     title = {Look-ahead {Levinson-} and {Schur-type} recurrences in the {Pad\'e} table},
     journal = {Electronic transactions on numerical analysis},
     pages = {104--129},
     publisher = {mathdoc},
     volume = {2},
     year = {1994},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_1994__2__a6/}
}
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Gutknecht, Martin H.; Hochbruck, Marlis. Look-ahead Levinson- and Schur-type recurrences in the Padé table. Electronic transactions on numerical analysis, Tome 2 (1994), pp. 104-129. http://geodesic.mathdoc.fr/item/ETNA_1994__2__a6/