The　long-term　optimal　trajectory　to　compute　a　minimum　point　consists　of　a　sequence　of　greatest　descent　steps　followed　by　the　Newton　step　in　the　last　iteration.　The　greatest　descent　direction　can　be　approximated　by　a　Levenberg-Marquardt　like　formula.　There　is　a　simple　way　to　prescribe　the　relative　steplengths　so　that　the　approximate　greatest　descent　(AGD)　direction　merges　into　the　Newton　direction　near　a　minimum　point.　This　enables　fast　local　convergence　of　the　AGD　method　near　a　minimum　point.　Here　we　examine　the　use　of　the　Bk　matrices,　defined　by　a　quasi-Newton　update　formula,　as　a　way　to　model　and　approximate　the　Hessian　matrix　of　a　nonlinear　function.　These　Bk　matrices　are　used　in　the　AGD　iteration　rather　than　the　Newton　iteration.　Used　in　this　manner　numerical　errors　in　the　Bk　matrices　can　be　tolerated　when　the　point　is　at　a　large　distance　from　the　minimum　point.　Furthermore,　Bk　is　not　required　to　be　positive　definite　or　nonsingular.　Instead　we　require　a　weaker　condition,　namely　the　function　is　monotonic　decreasing.　This　can　always　be　achieved　by　using　a　small　steplength.　Computational　errors　can　make　the　Bk　matrix　singular　and　not　positive　definite.　From　numerical　experiments　the　main　advantage　of　using　the　B　k　matrix　rather　than　the　Hessian　matrix　in　the　AGD　method　is　that　it　is　faster　when　the　number　of　variables　is　large.　Copyright　©　2011　Watam　Press.