Pivoting for structured matrices and rational tangential interpolation

摘要: Gaussian elimination is a standard tool for computing triangular factorizations general matrices, and thereby solving associated linear systems of equations. As well-known, when this classical method implemented in finite-precision-arithmetic, it often fails to compute the solution accurately because accumulation small roundoffs accompanying each elementary floating point operation. This problem motivated number interesting important studies modern numerical algebra; our purposes paper we only mention that starting with breakthrough work Wilkinson, several pivoting techniques have been proposed stabilize behavior elimination.Interestingly, matrix interpretations many known new algorithms various applied problems can be seen as way structured where different patterns structure arise context physical problems. The special such matrices [e.g., Toeplitz, Hankel, Cauchy, Vandermonde, etc.] allows one speed-up computation its factorization, i.e., efficiently obtain fast implementations procedure. There vast literature about methods which are under names, e.g., Cholesky, elimination, generalized Schur, or Schur-type algorithms. However, without further improvements they efficient numerically inaccurate [for indefinite matrices] method.In survey recent results on implementation allowed us improve accuracy variety approach led formulate more accurate factorization J-unitary rational functions, tangential interpolation problems, Toeplitz-like Toeplitz-plus-Hankel-like solvers, divided differences schemes. We beleive similar used design algorithm other colleagues supports anticipation.