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LECTURE 04

# LECTURE 04 - Lecture 4 MEEN 357 Engineering Analysis for...

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Lecture 4 MEEN 357 Engineering Analysis for Mechanical Engineers 1 Lecture-04-MEEN357-2009.doc RMB Lecture 4 Roundoff and Truncation Errors Chapter 4 of the textbook. Purpose: Understand the concept of error in the context of Numerical Analysis. Error Analysis : An attempt to understand the source of the error and to estimate its size. Types of Errors: Round Off Error : Approximating π and other irrational numbers by a finite number of digits. Computers can only store a finite number of digits. Truncation Error : Arises when series representations of functions are approximated by a finite number of terms. Measures of Numerical Error: (For one dimensional problems) True Percent Relative Error: 1 true value - approximation 100% true value t ε (4.1) Note: We usually do not know the true value! Iterative Percent Relative Error: current approximation - previous approximation 100% current approximation a (4.2) Note: We shall see other measures of error in later chapters. Also, for multidimensional problems, i.e. when the answer is a vector or a matrix, the error measures are more complicated. Note: In many applications, we are not concerned with the sign of the error (positive or negative). We are frequently only concerned with the magnitude of the error (absolute error). For example, if you are given ( ) Error tolerance a positive number s = (4.3) 1 The error definitions in this lecture all include the absolute value calculation in the definition. This was adopted in Lecture 3 but it was not shown in the definitions.

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Lecture 4 MEEN 357 Engineering Analysis for Mechanical Engineers 2 Lecture-04-MEEN357-2009.doc RMB then in an iterative process, one would continue the iteration until the absolute value of the error is below the given tolerated error, i.e. until as ε < (4.4) Floating Point Representation: (For Positive and Negative Numbers) N N Mantissa Base of Number System , where exponent e mb e ×= (4.5) The “Mantissa” is sometimes called the “Significant” IEEE 64-bit Floating-Point Number Representation (Section 1.2.1) As explained in Lecture 3, the IEEE 64 bit floating-point number system is virtually the standard for computer number systems. It is the number system used within MATLAB. The system, as the name indicates, starts with 64 bits as the binary representation of the floating point number illustrated in the following figure. In this figure, , SE xp and M are binary numbers of size 1,11 and 52 bits respectively. The rules for converting each of these binary numbers into a base-10 number were given during Lecture 3. These rules, repeated are, The rules tell us how to take the three pieces of information: 1. 63 0 1 Sb = = (4.6) 2. 62 61 60 59 58 57 56 55 54 53 52 11 bits E x p bbbbbbbbbbb = ±²²²²²³²²²²²´ (4.7) 3.
Lecture 4 MEEN 357 Engineering Analysis for Mechanical Engineers 3 Lecture-04-MEEN357-2009.doc RMB 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 7 6 5 4 3 2 1 0 52 bits M bbbbbbbbbbbbbbbbb bbbbbbbb = ⋅⋅⋅ ±²²²²²²²²²²²²³²²²²²²²²²²²²´ (4.8) and calculate a real number which we have called υ .

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LECTURE 04 - Lecture 4 MEEN 357 Engineering Analysis for...

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