For　unconstrained　optimization,　an　inexact　Newton　algorithm　is　proposed　recently,　in　which　the　preconditioned　conjugate　gradient　method　is　applied　to　solve　the　Newton　equations.　In　this　paper,　we　improve　this　algorithm　by　efficiently　using　automatic　differentiation　and　establish　a　new　inexact　Newton　algorithm.　Based　on　the　efficiency　coefficient　defined　by　Brent,　a　theoretical　efficiency　ratio　of　the　new　algorithm　to　the　old　algorithm　is　introduced.　It　has　been　shown　that　this　ratio　is　greater　than　1,　which　implies　that　the　new　algorithm　is　always　more　efficient　than　the　old　one.　Furthermore,　this　improvement　is　significant　at　least　for　some　cases.　This　theoretical　conclusion　is　supported　by　numerical　experiments.