A new algorithm for linear programming, based on nested decomposition principles, is presented and illustrated. The method, labelled the primal Decomplex method (from DECOMposition and simPLEX), is distinct from other well known linear programming algorithms in that a sequence of extreme points is generated which is neither all primally nor all dually feasible. Rather, primal feasibility to only pars of the constraint set is enforced at each instant. A distinguishing feature of the method is its relative simplicity : an m-row linear program is decomposed into m sub-problems each consisting of a single (transformed) constraint and a convexity row.