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26 problems ADVANCED COMPUTING RESEARCH akdk analysis of quasi-Newton BFGS METHOD Ai-Ping BFGS method corrects BFGS method performs BFOS mBFOS BFOS BFOS(IO Bksk Byrd and Nocedal circumstances this modified corrects large eigenvalues corrects the eigenvalues cos2 0k denotes due to Byrd eigenvalues better eigenvalues more efficiently equality holds f satisfies Assumption f(xk FORTRAN function evaluations Garbow and Hillstrom global convergence Hessian approximation inequality of 25 inexact line search largest real root least pk Lemma Liao line search satisfying matrix mBFOS BFOS mBFOS mBFOS(M modified BFGS method modified BFGS update Nocedal and Yuan number of function numbers of iterations Numerical results obtain the condition positive definite presents the numbers quasi-Newton equation quasi-Newton methods rank-one update self-correcting mechanism self-correcting property sign of equality small eigenvalues study the global superlinearly convergent Suppose that f term in 31 Theorem 2.6 unconstrained optimization problem variable metric methods Wolfe condition xTBkx yjsk