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CHAPTER 4 NUMERICAL METHODS Ray C. Johnson, Ph.D. Higgins Professor of Mechanical Engineering Emeritus Worcester Polytechnic Institute Worcester, Massachusetts 4.1 NUMBERS/4.1 4.2 FUNCTIONS / 4.3 4.3 SERIES / 4.6 4.4 APPROXIMATIONS AND ERROR / 4.7 4.5 FINITE-DIFFERENCE APPROXIMATIONS /4.16 4.6 NUMERICAL INTEGRATION / 4.18 4.7 CURVE FITTING FOR PRECISION POINTS / 4.20 4.8 CURVE FITTING BY LEAST SQUARES / 4.22 4.9 CURVE FITTING FOR SEVERAL VARIABLES / 4.25 4.10 INTERPOLATION / 4.26 4.11 ROOT FINDING / 4.28 4.12 SYSTEMS OF EQUATIONS / 4.34 4.13 OPTIMIZATION TECHNIQUES / 4.37 REFERENCES / 4.38 In this chapter some numerical techniques particularly useful in the field of machine design are briefly summarized. The presentations are directed toward automated calculation applications using electronic calculators and digital computers. The sequence of presentation is logically organized in accordance with the preceding table of contents, and emphasis is placed on useful equations and methods