in addition to { }we will make use of the vectors { defined via the recursionr : - > - ( at the final iteration tthe bfgs updating formula ( can be rewritten in the formct > - ct- at- ut dt > by iterating the recursion ( backwards to we can writect > > > = that iswe can express ct in terms of the initial and the entire history of all bfgs values { }as claimed furtherwith the { computed via ( and ( )we can writect > > > = > > > = > > > = hencefrom the definition of the { in ( )we obtain ct > > = > > = > rt rt given and the history of all recent bfgs values { }hj= the computation of the quasinewton search direction -ch can be accomplished via the recursions ( and ( as summarized in algorithm note that if is diagonal matrixsay the identity matrixthen is cheap to compute and the cost of running algorithm is ( dthusfor fixed length of the bfgs historythe cost of the limited-memory bfgs updating grows linearly in dmaking it viable optimization algorithm in large-scale applications