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A convergent minimization algorithm made up of repetitive line searches is considered in ℝ. It is shown that the uniform nonsingularity of the matrices consisting of n successive normalized search directions guarantees a speed of convergence which is at least n-step Q-linear. Consequences are given for multistep methods, including Powell's 1964 procedure for function minimization without calculating derivatives as well as Zangwill's modifications of this procedure. © 1977 Plenum Publishing Corporation.