Technology & Engineering
Approximation Error
Approximation error is the difference between the exact value and the approximate value of a quantity. It is a measure of how much an approximation deviates from the true value. In engineering and technology, approximation error is an important consideration in numerical analysis and modeling.
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3 Key excerpts on "Approximation Error"
- G. Miller(Author)
- 2014(Publication Date)
- Cambridge University Press(Publisher)
1 Numerical error 1.1 Types of error The term “error” is going to appear throughout this book in different contexts. The varieties of error we will be concerned with are: • Experimental error. We may wish to calculate some function y (x 1 , ..., x n ), where the quantities x i are measured. Any such measurement has associated errors, and they will affect the accuracy of the calculated y . • Roundoff error. Even if x were measured exactly, odds are it cannot be represented exactly in a digital computer. Consider π , which cannot be represented exactly in decimal form. We can write π ≈ 3.1416, by rounding the exact number to fit 5 decimal figures. Some roundoff error occurs in almost every calculation with real numbers, and controlling how strongly it impacts the final result of a calculation is always an important numerical consideration. • Approximation Error. Sometimes we want one thing but calculate another, intention- ally, because the other is easier or has more favorable properties. For example, one might choose to represent a complicated function by its Taylor series. When substi- tuting expressions that are not mathematically identical we introduce Approximation Error. Experimental error is largely outside the scope of numerical treatment, and we’ll assume here, with few exceptions, that it’s just something we have to live with. Experi- mental error plays an important role in data fitting, which will be described at length in Chapter 8. Sampling error in statistical processes can be thought of as a type of experi- mental error, and this will be discussed in Chapter 11. Controlling roundoff error, sometimes by accepting some Approximation Error, is the main point of this chapter, and it will be a recurring theme throughout this book. J. H. Wilkinson [242] describes error analysis generally, with special emphasis on ma- trix methods. An excellent and comprehensive modern text is N. J. Higham’s [104].
eBook - PDFNumerical Methods
Fundamentals and Applications
- Rajesh Kumar Gupta(Author)
- 2019(Publication Date)
- Cambridge University Press(Publisher)
Numerical methods use arithmetic operations to solve complex mathematical problems. The numerical processes are algorithmic, so these methods can be applied easily with the advent of high-speed computers. In fact, the development of more efficient computers has played a vital role in a veritable explosion in the usage of numerical techniques for engineering and scientific problems. The common characteristic of numerical techniques is that all these involve cumbersome arithmetic operations. During the implementation of the numerical techniques on a computer, we often come across various types of errors. The precisions (number of digits in the representation of a number) of a numerical solution can be diminished by these several possible errors. This chapter deals with various types of errors, and some standard remedies to trace and reduce these errors. In Section 2.1, measurement of the error will be discussed. Section 2.2 presents the various sources of errors in mathematical modeling of a real world problem. The study of errors during the implementation of numerical methods for the solution of a mathematical model is the primary objective of Section 2.3. The last section is about some interesting discussion on error. 2.1 Absolute, Relative and Percentage Errors The difference between the exact value and an approximate value of a quantity is called error in the measurement. Its absolute value is called absolute error. Let x be the exact value and x be an approximate value of a given quantity; then the absolute error is given by Error Analysis Chapter 2 I claim to be a simple individual liable to err like any other fellow mortal. I own, however, that I have humility enough to confess my errors and to retrace my steps. Mohandas Karamchand Gandhi (Mahatma Gandhi) (October 2, 1869–January 30, 1948) He embraced non-violent civil disobedience and led India to independence from British rule.
eBook - ePub- William J. Kennedy, WIlliam J. Kennedy, James E. Gentle, Kennedy(Authors)
- 2021(Publication Date)
- CRC Press(Publisher)
Chapter 2 (except in base 10 here), gives the computer resultThe absolute error with respect to the true value is0.162 × 10− 0.161 × 10¯0.100 ×10− 10.011 − 0.010 = 0.001and the absolute error with respect to the c-true value is zero.3.4 Analyzing Error in A Finite Process
The magnitude of error due to approximations imposed by the computer and the way in which it is generated are of particular interest in scientific applications. Often a bound for the magnitude of the total error in an arithmetic process can be described mathematically in terms of original data and intermediate errors that can be bounded. The separation of total error into component parts is called an error analysis . An error analysis often alerts the prospective user to problem areas in his proposed numeric process, and it may suggest ways in which error can be reduced.There are two different general techniques for a priori analysis of error which are used in practice. These are called forward or direct error analysis and backward or inverse error analysis. It will become obvious as we discuss these techniques that other different, but somewhat similar, techniques can be defined. We will, however, concern ourselves only with the two techniques mentioned above because they are the most commonly used.A direct error analysis is carried out by obtaining an expression for error at each intermediate step of the computation in terms of true values and of errors made in previous steps. Often, errors considered in a direct analysis are relative errors defined with respect to the true value. Thus at each stage of the direct error analysis we are required to compare our computer result with the true value. This type of error analysis is usually difficult to carry out.
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