Black box methods for least squares problems

We present algorithms to construct an efficient black box for the pseudoinverse, <i>A</i>^t, of a black box matrix <i>A</i>. This in also known as the Moore-Penrose inverse of <i>A</i>. For the system <i>Ax</i> = <i>b</i> over a subfield of the complex numbers: the vector <i>x</i> = <i>A</i>^t<i>b</i> is the least squares solution having the least norm. When we say that <i>A</i> is a black box matrix we mean simply that methods are given to compute the matrix-vector products <i>Au</i> and <i>u^TA</i>, for vectors <i>u</i> and vectors <i>v</i> of compatible length. No other assumptions are made about the structure or representation of the matrix. The entries may be integers or rational numbers or elements of a finite field. For the integer or rational case the method uses homomorphic images, and is based on an algorithm capable of computing the black box for the pseudoinverse, when it exists, over an arbitrary field. We discuss the asymptotic costs and some variants that are likely to be useful in practice. Over a finite field the solution, <i>x</i> = <i>A</i>^t<i>b</i>, is uniquely defined when it exists. The ability to produce this particular solution is a useful addition to the range of ways solution to singular linear systems may be offered in the black box model.