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The approach to solving linear systems with structured matrices by means ofthe bidiagonal factorization of the inverse of the coefficient matrix is firstconsidered, the starting point being the classical Bjorck-Pereyra algorithmsfor Vandermonde systems, published in 1970 and carefully analyzed by Higham in1987. The work of Higham showed the crucial role of total positivity forobtaining accurate results, which led to the generalization of this approach tototally positive Cauchy, Cauchy-Vandermonde and generalized Vandermondematrices. Then, the solution of other linear algebra problems (eigenvalue andsingular value computation, least squares problems) is addressed, a fundamentaltool being the bidiagonal decomposition of the corresponding matrices. Thisbidiagonal decomposition is related to the theory of Neville elimination,although for achieving high relative accuracy the algorithm of Nevilleelimination is not used. Numerical experiments showing the good behaviour ofthese algorithms when compared with algorithms which ignore the matrixstructure are also included.

Bjorck-Pereyra-type methods and total positivity