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LECTURE 03

LECTURE 03 - Lecture 3 MEEN 357 Engineering Analysis for...

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Lecture 3 MEEN 357 Engineering Analysis for Mechanical Engineers 1 Lecture-03-MEEN357-2009.doc RMB Lecture 3 Roundoff and Truncation Errors Chapter 4 of the textbook. Purpose: Understand the concept of error in the context of Numerical Analysis. Physical problems typically involve: a) A theoretical mathematical model. a. Involves approximations as models of real phenomena. b) A solution. a. Exact mathematical solution for the assumed model. b. Approximate mathematical solution for the assumed model. c) An analysis of the solution. a. Using numerical values that themselves might be approximations. Conclusion: Errors, whether or they occur in the model, in the solution or in the data are inevitable. Error Analysis : An attempt to understand the source of the error and to estimate its size. Issues are: Identification , Quantification and Minimization of Errors Accuracy and Precision Section 4.1 of the textbook. Characterization of Errors: Accuracy : Refers to how closely a computed or measured value agrees with the true value. Precision : Refers to how closely computed or measured values agree with each other. a) Inaccurate and imprecise b) Accurate and imprecise c) Inaccurate and precise d) Accurate and precise

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Lecture 3 MEEN 357 Engineering Analysis for Mechanical Engineers 2 Lecture-03-MEEN357-2009.doc RMB Note: The word error is used to represent both errors of inaccuracy and errors of imprecision. Error Definitions Section 4.1.2 Source of Numerical Errors: Use of approximations to represent exact mathematical operations and quantities. 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. Various Measures of Numerical Error: (For one dimensional problems) Example: True or Absolute Error true value - approximation t E = (3.1) Problem: This definition does not take into account of the size of the numerical quantity being measured. 5 5 55 true value = 1000 10 Very good approximation. true value = 1 10 Good approximation. true value = 10 10 Very bad approximation. t t t E E E −− =⇒ Example: Normalized or Relative Error (Which avoids above problem.) true value - approximation 100% true value t ε (3.2) Another Problem: We usually do not know the true value!
Lecture 3 MEEN 357 Engineering Analysis for Mechanical Engineers 3 Lecture-03-MEEN357-2009.doc RMB Iterative Approach: Certain numerical methods involve an iterative approach, i.e. present approximation is based upon previous approximation . The error at each stage is measured by current approximation - previous approximation 100% current approximation a ε (3.3) 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.

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LECTURE 03 - Lecture 3 MEEN 357 Engineering Analysis for...

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