ON IP-APPROXIMATE SOLUTIONS OF LINEAR-EQUATIONS

For 1 less-than-or-equal-to p less-than-or-equal-to infinity, the l(p) -approximate solutions of Ax = b are the minimizers of \\Ax - b\\p, wh ere \\.\\p is the l(p)-norm. We consider the special case where the nu ll space of A(T) is one-dimensional. Sample results: (a) If 1 less-tha n-or-equal-to p less-than-or-equal-to and A is m X (m - 1) of rank m - 1, then there is a matrix A{p} (depending on A and p) such that, for every b is-an-element-of R(m), the vector A{p}b is an l(p)-approximate solution of Ax = b, which is unique if 1 < p < infinity. (b) If 1 < p < infinity and A is m x n of rank m - 1, then there is a matrix A{2}{ p} (depending on A and p) such that for every b is-an-element-of R(m) the vector A{2}{p}b is the l(p)-approximate solution of minimal euclid ean norm. (c) Let 1 less-than-or-equal-to p less-than-or-equal-to infi nity, and let A be m X n of rank m - 1. Then there is a vector r{p} (c omputed from any least-squares solution of Ax = b) such that any ordin ary solution of the auxiliary equation Ax = b + r{p} is an l(p)-approx imate solution of Ax = b.