On a new iterative algorithm for solving linear equations

AbstractLet V and W be two real or complex spaces which, by means of the choice of an inner product, we can identify with their duals. Let A:V→W be a linear map. A′:W→V be its dual. There is a very large number of numerical algorithms (see [2] or [3] for solving the linear equation A·x=b.Some of these algorithms, such as Gaussian elimination, produce the entire k-dimensional affine solution space in V, others, such as gradient methods [3], produce a particular solution. It is not always clear which solution they produce, or indeed by what criterion one ought to single out a particular solution in the k-parameter family. Here we are interested in a particular solution, which we call the canonical solution of (1). We shall write R(A) for the range of A and r(A) for the rank