# PolSys seminar

# Numerical Linear System Solving With Parametric Entries By Error Correction

Friday, September 19, 2014Speaker(s) : Brice Boyer (PolSys team, CNRS/INRIA/UPMC)

We consider the problem of solving a full rank consistent linear system A(u) x = b(u) where A n K[u]^{m×n} and b in K[u]^m. Our algorithm computes the unique solution x = f(u)/g(u) (a vector of rational functions) by evaluating the parameter u at distinct points. Those points where the matrix A evaluates to A( xi_lambda) of lower rank, or in the numeric setting to an ill-conditioned matrix, are not identified but accounted for by error-correcting code techniques. We also correct true errors where the evaluation at some u = x_lambda results in an erroneous, possibly full rank consistent and well-conditioned scalar linear system. We have implemented our algorithm in Maple with floating point arithmetic. For the determination of the exact numerator and denominator degrees and number of errors we use SVD based numeric rank computations. The arising linear systems for the error-corrected parametric solution are shown to be well-conditioned even when the input scalars have noise.

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Elias.Tsigaridas (at) nulllip6.fr