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A uniform approach for Hermite Padé and simultaneous Padé approximants and their matrix-type generalizations

Bernhard Beckermann, George Labahn
1992 Numerical Algorithms  
This includes descriptions of vector and matrix Padé approximants along with generalizations of simultaneous and Hermite Padé approximants.  ...  This paper introduces the notion of a power Hermite Padé approximant, a generalization of the classical scalar Hermite Padé approximant.  ...  example, to derive algorithms where both types of approximants are computed simultaneously.  ... 
doi:10.1007/bf02141914 fatcat:5wyrxru4dndxtmhwhtfyb47rha

Page 1563 of Mathematical Reviews Vol. , Issue 93c [page]

1993 Mathematical Reviews  
The algorithm provides a method for obtaining ma- trix Padé, matrix Hermite Padé and matrix simultaneous Padé approximants.  ...  The recurrence relation results in a fast algorithm for calculating a Padé-Hermite approximant of any given type.  ... 

Fraction-free computation of simultaneous padé approximants

Bernhard Beckermann, George Labahn
2009 Proceedings of the 2009 international symposium on Symbolic and algebraic computation - ISSAC '09  
How to Compute Simultaneous Padé Approximants (1) linear system of equations for coefficients of U i (z), V (z) system has 'special' structured coefficient matrix fast algorithms take advantage of 'special  ...  ' structure (2) represent as a vector Hermite-Padé problem find order basis for problem fast algorithms order bases find all solutions of these approx problems (3) todays : solve problem via order bases  ... 
doi:10.1145/1576702.1576708 dblp:conf/issac/BeckermannL09 fatcat:k5laijlphzgmzibdo5ak3y2ip4

Page 5067 of Mathematical Reviews Vol. , Issue 93i [page]

1993 Mathematical Reviews  
This includes descriptions of vector and matrix Padé approximants along with generalizations of simultaneous and Hermite Padé approximants.  ...  {For the entire collection see MR 93g:65008.} 93i1:65026 65D15 41A21 Beckermann, Bernhard (D-HANN-AM); Labahn, George (3-WTRL-C) A uniform approach for Hermite Padé and simultaneous Padé approximants and  ... 

Page 3499 of Mathematical Reviews Vol. , Issue 93g [page]

1993 Mathematical Reviews  
rational Hermite interpolation (29-44); Bernhard Beckermann and George Labahn, A uniform approach for Hermite Padé and simultaneous Padé approximants and their matrix-type generalizations (45-54); T.  ...  Lu- binsky, Convergence of simultaneous Hermite-Padé approximants to the n-tuple of g-hypergeometric series {1®1((23)); z)}"_, (185- 192); Mariano Gasca and José J.  ... 

Page 3577 of Mathematical Reviews Vol. , Issue 95f [page]

1995 Mathematical Reviews  
and Hermite Padé approximants [Numer.  ...  The considerations in this paper are based on this generalized form of the classical scalar Hermite Padé approximation problem, power Hermite Padé approximation.  ... 

Page 1662 of Mathematical Reviews Vol. , Issue 97C [page]

1997 Mathematical Reviews  
The article proposes an algorithm that solves two dual vector Padé-Hermite problems at the same time, returning dual basis matrices of the respective solution spaces.”  ...  We also show that the classical duality between Padé-Hermite problems of type I and type II is a special case of vector Padé-Hermite duality.  ... 

Page 4456 of Mathematical Reviews Vol. , Issue 97G [page]

1997 Mathematical Reviews  
Simultaneous Padé (SP) approximation is a dual problem.  ...  and simultaneous Padé systems.  ... 

On the theory and computation of nonperfect padé-hermite approximants

Stan Cabay, George Labahn, Bernhard Beckermann
1992 Journal of Computational and Applied Mathematics  
The recurrence relation results in a fast algorithm for calculating a Pad&Hermite approximant of any given type.  ...  For a vector of k + 1 power series we introduce two new types of rational approximations, the weak Pad&Hermite form and the weak Pad&Hermite fraction.  ...  When k = 1, Pad&Hermite approxi-mants reduce to Pade approximants, and the algorithm becomes that of [5] and the scalar algorithm of [16] .  ... 
doi:10.1016/0377-0427(92)90206-d fatcat:dppxuwdjvne7rbws4pivw5tara

A fast, reliable algorithm for calculating Pade´-Hermite forms

S. Cabay, G. Labahn
1989 Proceedings of the ACM-SIGSAM 1989 international symposium on Symbolic and algebraic computation - ISSAC '89  
This complexity is the same as that of other fast algorithms for computing Pad&Hermite approximants.  ...  We present a new fast algorithm for the calculation of a Pad&Hermite form for a vector of power series.  ...  A WPHFr can be interpreted as providing a a set of simultaneous Pade approximants for the quotient power series &(z)/&(z) (cf., de Bruin[2]).  ... 
doi:10.1145/74540.74553 dblp:conf/issac/CabayL89 fatcat:oi3sr3kfjza2lk2lpugv752nwu

Computation of Numerical Padé–Hermite and Simultaneous Padé Systems II: A Weakly Stable Algorithm

Stan Cabay, Anthony R. Jones, George Labahn
1996 SIAM Journal on Matrix Analysis and Applications  
of Padé-Hermite and simultaneous Padé systems and, thereby, also for the numerical inversion of striped and mosaic Sylvester matrices.  ...  The near inverses are expressed in terms of numerical Padé-Hermite systems and simultaneous Padé systems.  ...  Padé-Hermite and simultaneous Padé approximants were introduced by Hermite in the last century and has been widely studied by several authors (for a bibliography, see, for example [1, 2, 3, 4, 15] ).  ... 
doi:10.1137/s0895479894268695 fatcat:fb53lzr55rdqdedw4yl57jmoh4

Page 1099 of Mathematical Reviews Vol. , Issue 82c [page]

1982 Mathematical Reviews  
algorithms for the computation of Padé and c-tables in the presence of blocks, and algorithms for the choice of the best PA when a number of Taylor coefficients of a power series are given.  ...  New elements are presented, most important of which are the algebraic theory of Padé approximation in the presence of blocks, the analysis of the convergence of the e-algorithm via Padé approximation,  ... 

Computation of Numerical Padé–Hermite and Simultaneous Padé Systems I: Near Inversion of Generalized Sylvester Matrices

Stan Cabay, Anthony R. Jones, George Labahn
1996 SIAM Journal on Matrix Analysis and Applications  
of Padé-Hermite and simultaneous Padé systems and, thereby, also for the numerical inversion of striped and mosaic Sylvester matrices.  ...  The near inverses are expressed in terms of numerical Padé-Hermite systems and simultaneous Padé systems.  ...  Padé-Hermite and simultaneous Padé approximants were introduced by Hermite in the last century and has been widely studied by several authors (for a bibliography, see, for example [1, 2, 3, 4, 15] ).  ... 
doi:10.1137/s0895479894268683 fatcat:u72p6zysibgynkwqtdxm3p2mfm

Algorithms for Simultaneous Padé Approximations

Johan Rosenkilde né Nielsen, Arne Storjohann
2016 Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation - ISSAC '16  
We describe how to solve simultaneous Padé approximations over a power series ring K[[x]] for a field K using O (n^ω - 1 d) operations in K, where d is the sought precision and n is the number of power  ...  Our results are made possible by recent breakthroughs in fast computations of minimal approximant bases and Hermite Padé approximations.  ...  The authors would like to thank George Labahn for valuable discussions, and for making us aware of the Hermite-Simultaneous Padé duality.  ... 
doi:10.1145/2930889.2930933 dblp:conf/issac/RosenkildeS16 fatcat:ew5jndxpovg2hijommcgiouuau

Foreword to the proceedings of the OrthoQuad 2014 conference

Adhemar Bultheel, Guillermo López-Lagomasino, Francisco Marcellán, Olav Njåstad, Ramón Orive
2015 Journal of Computational and Applied Mathematics  
Another generalization is Hermite-Padé approximation where a vector of Laurent series is simultaneously approximated by a vector of rational approximants with a common denominator (Type II) or one tries  ...  Multiple orthogonal polynomials are related to Hermite-Padé approximation. The type II polynomials appear in vectors that are simultaneously orthogonal to a vector of measures.  ... 
doi:10.1016/j.cam.2015.01.035 fatcat:zj3pgnd2v5gxteyfsiv3c25tzy
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