#### Date of Award

5-1-1998

#### Degree Type

Thesis

#### University or Center

Clark Atlanta University(CAU)

#### School

School of Arts and Sciences

#### Degree Name

M.S.

#### Department

Mathematical Sciences

#### First Advisor

J. Ernest Wilkins, Jr.

#### Abstract

As a general rule, it is not possible to express the integral, **∫**_{a}^{b}*f*(x)*dx, *of a real valued function* f** *of a single real variable *x, *in terms of relatively elementary functions. Even when this can be done, numerical calculations based on such expressions may be lengthy and tedious. For these reasons, mathematicians have devised several numerical quadrature schemes to provide approximate values for the integral (1). The error in any such scheme is defined as the difference between the integral and the numerical quadrature scheme. In this paper, we studied the errors in four specific numerical quadrature schemes: trapezoidal rule, midpoint rule, Simpson’s rule, and Bode’s rule. Our interest was to find estimates for these errors. The approach was first to find an explicit formula for the error. This formula involved an integral, one factor of whose integrand was a derivative of some order of the function *f*(x). Both upper and lower bounds were found for the error in terms of the maximum and minimum values of that derivative. Numerical results were finally calculated for two specific functions *f(*x). We found that Bode’s rule gave the most accurate answer while the trapezoidal rule was least accurate.

#### Recommended Citation

Wilson, Joycelyn N., "Errors in Numerical Quadrature Schemes" (1998). *ETD Collection for AUC Robert W. Woodruff Library*. 507.

http://digitalcommons.auctr.edu/dissertations/507

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