CHAPTER 2 Numerical Derivatives and Roundoff Error_082709

CHAPTER 2 Numerical Derivatives and Roundoff Error_082709 -...

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1 CHAPTER 2: Numerical Derivatives and Round-off Error It is very easy to compute the numerical derivative of a function. Numerical derivatives also provide excellent examples of the impact of round-off error on numerical approximations—which will be major focus in this chapter. We will develop simple expressions for the most common derivatives used in science and engineering and then use these expressions to compute the numerical derivatives of elementary functions. By applying our expressions to elementary functions we will make it easy to compare the numerical values of the derivatives to the true value of the derivatives. This will give us a better feel for the difference between the true value of the solution to a mathematical expression and the value computed with a numerical method. It is also very easy to evaluate and to reduce the truncation error in a numerical derivative. We will look at several expressions for approximating the numerical value of the first derivative of a function, with truncation error varying from first order to fourth order. By computing the number of significant figures based on both the true error and the computational error we will then be able to see how truncation error affects the significant figures in a numerical approximation of a solution. In particular we will see how to vary systematically the convergence of a numerical approximation to maximize our ability to determine the number of significant figures in the approximation. Additionally, the simplest expressions for numerical derivatives are surprisingly sensitive to round-off error. By looking at round-off error in this chapter, we will be able to see how the round-off error that is inherent in numerical computation using a digital computer places a fundamental limitation on the accuracy of our approximations to the solution of mathematical expressions. Round-off error and truncation error interact to reduce the accuracy of our numerical solutions. By the time you finish this chapter you should have a feel for how truncation error and round-off error interact in limiting the accuracy of our numerical estimates. One of the most important conclusions you should reach by the end of this chapter is that the use of expressions with smaller truncation error is the most effective way to overcome the limitations on accuracy imposed by round-off error. Importance While the expressions we will use for numerical derivatives are quite simple, this does not mean that they are not very powerful. The most important application of numerical derivatives is in the solution to ordinary and partial differential equations models of the real world. The most powerful and sophisticated software used to simulate physical and engineering systems is based on the use of numerical derivatives in what are called finite difference solutions to differential equation models. These types of software are used for simulations of
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This note was uploaded on 01/26/2011 for the course CH E 2112 taught by Professor Dr.harwell during the Spring '10 term at The University of Oklahoma.

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CHAPTER 2 Numerical Derivatives and Roundoff Error_082709 -...

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