The author introduces a minimal B-error algorithm for interative techniques in solving the matrix equation Ax+b=0 following the general concepts introduced by G. H. Golub and R. S. Varga (1961). (MR0468134 )
We survey recent results on tractability of multivariate problems. We mainly restrict ourselves to linear multivariate problems studied in the worst case setting. Typical examples include multivariate integration and function approximation for weighted spaces of smooth functions.
The authors state their objectives as follows. They wish to present to the Polish audience the main aspects of a modern approach to numerical methods based on the concept of computational complexity. Instead of looking for computational methods which compare favorably with other commonly used techniques, and "gradually'' improving numerical methods for solving a given set of problems, one could formulate an optimality criterion based on the cost of performing numerical computations and regard the choice of a computational technique as a problem of optimization theory. This approach is illustrated by specific examples, such as the choice of a fast Fourier transformation, and a solution of an algebraic system of linear equations. (see MR0519667)
The authors introduce an algorithm combining the minimal error and the C̆ebyšev technique (the T-method) for solving equations of the form Ax+b=0, where x, b are vectors. A is an n×n matrix. A computer program is included as part of the article. (MR0448833)