Lecture 1& 2 NUMERICAL METHODS USING MATLAB Why use Numerical Methods? To solve problems that cannot be solved exactly To solve problems that are intractable! Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. How do we solve an engineering problem? Problem description Mathematical Model Solution of Mathematical Model Using the solution Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Mathematical Procedures Error Analysis Nonlinear Equations Simultaneous Linear Equations Curve Fitting Interpolation Differentiation Integration Ordinary Differential Equations Other Advanced Mathematical Procedures: – Partial Differential Equations – Optimization – Fast Fourier Transforms Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Er r o r AnalysIs The error of a quantity is the difference between it’s true value and approximate. True Value = Approximation + Error Et = True value – Approximation (+/-) Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Types of Error i) Absolute Error ii) Relative Error iii) Percentage Error Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Absolute Error An approximate value of π (pi)is 3.142 8571 true value is 3.1415926. Find the absolute Error πΈπ΄ = | π − π₯ | = | 3.1415926 − 3.1428571 | = | − 0 .0 0 1 2 6 4 5 | = 0 .0 0 1 2 6 4 5 Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. and Relative Error. The relative error of a quantity is the ratio of it’s absolute error to it’s true value. It is denoted by ER EA ER = -------------- π Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Percentage Error Formula PE =|π‘ππ’π π£πππ’π−ππππππ₯ππππ‘ππ π£πππ’π| x 100 ---------------------------------------------|π‘ππ’π π£πππ’π| Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Approximations and Round-Off Errors For many engineering problems, we cannot obtain analytical solutions. Numerical methods yield approximate results, results that are close to the exact analytical solution. We cannot exactly compute the errors associated with numerical methods. β¦ Only rarely given data are exact, since they originate from measurements. Therefore there is probably error in the input information. β¦ Algorithm itself usually introduces errors as well, e.g., unavoidable round-offs, etc … β¦ The output information will then contain error from both of these sources. How confident we are in our approximate result? The question is “how much error is present in our calculation and is it tolerable?” Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. For numerical methods, the true value will be known only when we deal with functions that can be solved analytically (simple systems). In real world applications, we usually not know the answer a priori. Then Approximate error ο₯a = ο΄100% Approximation Iterative approach, example Newton’s method Current approximation - Previous approximation ο₯a = ο΄100% Current approximation (+ / -) Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Sources of Errors Gross error caused by human mistakes or by the computer Rounding Error caused by rounding off numbers Truncation error caused by truncating the series Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Round-off Errors Numbers such as p, e, or cannot be expressed by a 7 fixed number of significant figures. Computers use a base-2 representation, they cannot precisely represent certain exact base-10 numbers. Fractional quantities are typically represented in computer using “floating point” form, e.g., Integer part m.be mantissa exponent Base of the number system used Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Chopping Example: p=3.14159265358 to be stored on a base-10 system carrying 7 significant digits. p=3.141592 chopping error ο₯t=0.00000065 If rounded p=3.141593 ο₯t=0.00000035 Some machines use chopping, because rounding adds to the computational overhead. Since number of significant figures is large enough, resulting chopping error is negligible. Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Truncation Error Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Accuracy. How close is a computed or measured value to the true value Precision (or reproducibility). How close is a computed or measured value to previously computed or measured values. Inaccuracy (or bias). A systematic deviation from the actual value. Imprecision (or uncertainty). Magnitude of scatter. Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 3.2 Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Significant Figures Number of significant figures indicates precision. Significant digits of a number are those that can be used with confidence, e.g., the number of certain digits plus one estimated digit. 53,800 How many significant figures? 5.38 x 104 3 5.380 x 104 4 5.3800 x 104 5 Zeros are sometimes used to locate the decimal point not significant figures. 0.00001753 4 0.0001753 4 0.001753 4 Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Cont.. Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Cont.. Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
