Geodesic
Look-ahead Levinson- and Schur-type recurrences in the Padé table
Electronic transactions on numerical analysis, Tome 2 (1994), pp. 104-129
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},
     year = {1994},
     volume = {2},
     zbl = {0852.41012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_1994__2__a6/}
}
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AU  - Gutknecht,  Martin H.
AU  - Hochbruck,  Marlis
TI  - Look-ahead Levinson- and Schur-type recurrences in the Padé table
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PY  - 1994
SP  - 104
EP  - 129
VL  - 2
UR  - http://geodesic.mathdoc.fr/item/ETNA_1994__2__a6/
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%0 Journal Article
%A Gutknecht,  Martin H.
%A Hochbruck,  Marlis
%T Look-ahead Levinson- and Schur-type recurrences in the Padé table
%J Electronic transactions on numerical analysis
%D 1994
%P 104-129
%V 2
%U http://geodesic.mathdoc.fr/item/ETNA_1994__2__a6/
%G en
%F ETNA_1994__2__a6
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/