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How to derive the multivariate Pade Approximation for $\ln \left( {1 + \frac{x}{y}} \right)$ ? In this case, multivariate mean variable $x$ and variable $y$.

Tuong Nguyen Minh
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  • What do you mean by it? Can you give a reference? The PadeApproximant command of Mathematica deals with one variable only. – user64494 Mar 17 '21 at 11:53
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    How about PadeApproximant[Log[1 + z], {z, 0, {2, 3}}] /. z -> x/y – chris Mar 17 '21 at 12:45
  • In case it helps your search, Wiki reports that an approximant in two variables is called a Chisholm approximant, one in multiple variables a Canterbury approximant. –  Mar 17 '21 at 12:49
  • @MarcoB: There is no agorithm in the Chisholm's article. I'd like to quote "After the revised edition of this paper had been submitted, the author received a copy of work by C. Lutterodt which also described two-variable approximants. Lutterodt's choice of approximants is different from those studied in this paper; in particular, they do not reduce to Padé approximants when one variable is equated to zero". – user64494 Mar 17 '21 at 13:17
  • @chris: Please, pay your attention to the above comment of me. – user64494 Mar 17 '21 at 13:20
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    @user64494 The original paper by Chisholm mentions that the development of an algorithm still had to be done at that time. Since then, algorithms have been developed, e.g. a variation of the epsilon algorithm used for common Padé approximants (Cuyt, Multivariate Padé approximants revisited, 1986). Cuyt herself also reviews the field's progress in How well can the concept of Padé approximant be generalized to the multivariate case?. –  Mar 17 '21 at 14:09
  • @MarcoB: Thank you for the references (only the latter is free), it's kind of you. I'd like to quote the latter: "In general, uniqueness of the general multivariate Pade approximant, in the sense that all rational functions in $[N/D]_E^f$ reduce to the same irreducible form, is not guaranteed, unless the index set $E\setminus N$ supplies a homogeneous system of linearly independent equations". – user64494 Mar 17 '21 at 14:25
  • @TuongNguyenMinh: You still don't formulate the definition of the bivariate Pade approximant. – user64494 Mar 17 '21 at 16:01
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    so does it mean currently it is impossible to do so because of lack of rigour ? – Tuong Nguyen Minh Mar 17 '21 at 22:40

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