ODELIX — M2 internship proposal |
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Title: Sparse interpolation of rational functions
Topics: symbolic computation, high performance computing
Address
Research team: MAX, Algebraic modeling and symbolic computation
Contacts
Context
The MAX team is searching for PhD candidates on the themes of the “ODELIX” ERC Advanced Grant. The present M2 internship proposal allows applicants to familiarize themselves with these themes. Upon successful completion of the internship, there will be an opportunity to pursue with a PhD.
Applying
Applications can be sent by email to the above contact persons; they
should include a CV, a transcript of records, a letter of motivation,
and optional recommendation letters. Note that we have no hard
deadlines; new applications will be considered at any time of the year,
as long as the
Description |
Large-scale symbolic computations lead to the manipulation of very large
mathematical expressions. Often, such expressions can be regarded as
so-called sparse polynomials or rational functions in suitable
variables. For instance, a sparse polynomial of degree
in
,
, and
is a polynomial
, for which the number of non-zero
terms
is significantly lower than
.
One typical motivating example of a sparse polynomial is the determinant
of the symbolic matrix
Indeed, is a polynomial of degree
in
variables, with
terms. The entries of the matrix inverse
of
are typical examples of sparse rational
functions.
One notorious enemy in symbolic computation is intermediate
expression swell. For instance, when computing
symbolically, using Gaussian elimination, the intermediate expressions
tend to become much larger than the final result. A magic way to defeat
this enemy is sparse interpolation. The idea is to evaluate the
determinant
for a sufficient number of well
chosen values of the variables
and then to
reconstruct
from these evaluations.
The idea of sparse interpolation for polynomials goes back a long while [6, 1, 7]. Various methods have been proposed to generalize it to rational functions [5, 2]. Recently, particularly efficient methods for these tasks have been proposed [3, 4], but not yet implemented. The purpose of this internship is to learn and implement these new ideas and further improve them, if possible. If successful, then this may lead to significantly faster future implementations of many basic tasks in computer algebra.
References |
M. Ben-Or and P. Tiwari. A deterministic algorithm for sparse multivariate polynomial interpolation. In Proc. ACM STOC '88, pages 301–309. New York, NY, USA, 1988.
A. Cuyt and W.-S. Lee. Sparse interpolation of multivariate rational functions. TCS, 412:1445–1456, 2011.
J. van der Hoeven and G. Lecerf. Sparse polynomial interpolation. Exploring fast heuristic algorithms over finite fields. Technical Report, HAL, 2019. https://hal.archives-ouvertes.fr/hal-02382117.
J. van der Hoeven and G. Lecerf. On sparse interpolation of rational functions and gcds. ACM SIGSAM Commun. Comput. Algebra, 55(1):1–12, 2021.
E. Kaltofen and B. M. Trager. Computing with polynomials given by black boxes for their evaluations: greatest common divisors, factorization, separation of numerators and denominators. JSC, 9(3):301–320, 1990.
R. Prony. Essai expérimental et analytique sur les lois de la dilatabilité des fluides élastiques et sur celles de la force expansive de la vapeur de l'eau et de la vapeur de l'alkool, à différentes températures. J. de l'École Polytechnique Floréal et Plairial, an III, 1:24–76, 1795. Cahier 22.
D. S. Roche. What can (and can't) we do with sparse polynomials? In Proc. ISSAC '18, pages 25–30. New York, NY, USA, 2018. ACM.