## Section: Software

Keywords : algebraic number, bezoutian, C++, effective algebraic geometry, eigenvalues, genericity, linear algebra, links symbolic-numeric, geometry, sparse matrices, structured matrices, iterative methods, polynomials, solving, resultant, stability.

### Synaps, a module for symbolic and numeric computations

Participants : Ioannis Emiris, Bernard Mourrain [ contact person ] , Jean-Pascal Pavone, Olivier Ruatta, Jean-Pierre Técourt, Philippe Trébuchet, Elias Tsigaridas, Julien Wintz.

See synaps web site: http://www-sop.inria.fr/galaad/logiciels/synaps/ .

Until recently, symbolic and numeric computations were separated domains: software for manipulating formulas is often not effective for numerical linear algebra; while the numerically stable and efficient tools in linear algebra are usually not adapted to the computations with polynomials.

We design the software synaps (SYmbolic Numeric APplicationS) for symbolic and numerical computations with polynomials. It contains tools to compute with algebraic data structures such as polynomials in one or more variables, ideals, ring quotients, ..., as well as numerical computations on vectors, matrices, iterative processes, ...Specialized tools such as lapack, gmp, superlu, rs, gb, ... are also connected and can be imported in a transparent way. A set of solvers for polynomial equations have been developped in this environment and are exploited in geometric computation on algebraic curves and surfaces (topology, intersection, singularity detection and analysis, ...).

These developments are based on C++, and attention is paid to the generic
structures so that effectiveness would be maintained.
Thanks to the parameterization of the code (*template* )
and to the control of their instantiations (*traits,
template expression* ), they offer generic programming without losing
effectiveness.
This powerful kernel contains univariate and multivariate algebraic solvers
as well as subdivision solvers and several
resultant-based methods for projection operations.

Many functionalities of the library are now also available through the computer algebra system mathemagix , or the geometric modeler axel , as dynamic binary libraries.