Ray C. Johnson, Ph.D.
Higgins Professor of Mechanical Engineering Emeritus
Worcester Polytechnic Institute
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 rather
than on the derivation of theory.
In the design and analysis of machines it is necessary to obtain quantities for various
items of interest, such as dimensions, material properties, area, volume, weight,
stress, and deflection. Quantities for such items are expressed by numbers accompa-
nied by the units of measure for a meaningful perspective. Also, numbers always
have an algebraic sign, which is assumed to be positive unless clearly designated as
negative by a minus sign preceding the number. The various kinds of numbers are
defined in Sec. 2-7, which see.
4.1.1 Real Numbers, Precision, and Rounding
Any numerical quantity is expressed by a real number which may be classified as an
integer, a rational number, or an irrational number. For practical purposes of calcu-
lation or manufacturing, it is often necessary to approximate a real number by a
specified number of digits. For some cases, significant numbers may be useful, and
the following relates to the obtainable degree of precision.
Degree of Precision.