ME 261: Numerical Analysis

Transcription

1 ME 6: Numerical Analysis Dr. A.B.M. Toufique Hasan Associate Professor Department of Mechanical Engineering, BUET Term: July September 06 Lecture-: Approimation and Errors ME 6: Numerical Analysis Error Loss of Significant Figures (Subtractive Cancellation) Problem: Use four-digit rounding arithmetic to evaluate the roots of the quadratic equation: True answer: and Solution: b + a and b a ( 6.0) ( 4.000)(.000)(.000) b a S.F. (loss of significance) Subtraction of nearly equal numbers A poor approimation to -0.06, with the large relative error % 4.5% 0.06!!!!!!!!!!!!! And, b with the small relative error a Addition of nearly equal numbers % 0.0% 6.08 ME 6: Numerical Analysis

2 Error To obtain a more accurate approimation of Reformulation For, change the form of the quadratic formula by rationalize the numerator b + b a b b + c This avoids the subtraction of nearly equal number To avoid the subtraction of nearly equal numbers Then, No loss of significance with the small relative error in approimation of % % 0.06 <<< 4.5% (without manipulation) Reformulation results in good approimation with no loss of significance. ME 6: Numerical Analysis Error Problem: Evaluate f( at 4.7 using three-digit arithmetic. 6.. Eact (chopping) (rounding) Eact: f(4.7) (chopping): f(4.7) -.5!!!!! High relative error of 5.5% (rounding): f(4.7) -.4!!!!! High relative error of 6.06% ME 6: Numerical Analysis 4

3 Error To obtain a more accurate evaluation of f( f( can be written in a nested manner f( ((-6.)+.) (-6.) (-6.)+. ((-6.)+.) (chopping) (rounding) Eact: f(4.7) (chopping): f(4.7) -4.!!!!! Low relative error of 0.447% (rounding): f(4.7) -4.!!!!! Low relative error of 0.5% Polynomials should always be epressed in nested form before performing an evaluation to deal with finite digit arithmetic and for better approimation. ME 6: Numerical Analysis 5 Sources of error in numerical computations. Errors in Mathematical Modeling A mathematical model is the basic input to the numerical process to computationally represent a physical system. In many situations, it is impossible to include all the realistic phenomena acting on the physical problem. And thus, simplified approimations or assumptions in mathematical modeling are required to represent a physical system. For eample: (a) steady state, (b) frictionless flow, (c) D consideration are the frequent approimations which are used in mathematical models of many mechanical engineering problems. These approimations result in error in modeling of a physical system. These errors are not relevant to the numerical approimation during mathematical operations.. Errors in Numerical Input These errors are those that are present in the data supplied to the numerical model. These are sometimes called data error- the input errors occur due to the uncertainties associated with eperimental measurements since each instrument (such as anemometer, thermocouple, pressure gauge, UTM, ruler, etc.) has certain degree of accuracy and precision. However, good precision instrument can reduce this source of error. ME 6: Numerical Analysis 6

4 Sources of error in numerical computations contd... Machine Error The floating point representation of numbers involves rounding and chopping errors since each computing machine (computer, calculator etc) can holds a finite number of digits. This error is introduced at each arithmetic operation during the computation. This depends on the bit of computer OS and associated software packages-6bit/bit/64bit. According to IEEE (for 64 bit PC)- numbers occurring in calculations that have a magnitude less than 0-08 result in underflow and generally set to zero. Numbers greater than 0 08 results in overflow and typically cause computation to halt. Certain mathematical operations are undefined such as 0/0, sqrt(-); when a program tries to do any of these, the IEEE standard substitutes another special bit pattern that is displayed as NaN (Not a Number). Inherent Error Errors in Mathematical Model + Errors in numerical input+ Machine Error ME 6: Numerical Analysis 7 Sources of error in numerical computations contd.. 4. Blunders Blunders are errors that are caused due to human imperfection. Such errors may cause a serious disaster in the result. Some common sources of such errors are- (i) Selecting a wrong numerical method for solving the mathematical model. (ii) Selecting a wrong algorithm for implementing the numerical method. (iii) Making mistakes in the computer programming (coding). All these mistakes can be avoided through a sound understanding of the numerical methods and use of good programming techniques and tools. 5. Computational Errors (Numerical Error) Computational errors are introduced during the process of implementation of a numerical method. They come in two forms- roundoff error and truncation error. Roundoff error- occurs when a fied number of digits are used to represent eact numbers. Such error is introduced at every mathematical operation. Consequently, even though an individual roundoff error could be very small, the cumulative effect of a series of computations can be very significant. ME 6: Numerical Analysis 8 4

5 Sources of error in numerical computations contd.. Rounding a number can be done in two ways- Chopping- Etra digits are dropped. This is called truncating a number. For eample, an eact number can be approimated to 4.9 ( digit chopping), 4.96 ( digit chopping) or 4.96 (4 digit chopping). Symmetric Roundoff- The last retained significant digit is rounded up by if the first discarded digit is larger or equal 5; otherwise, the last retained digit is unchanged. For eample can be approimated to 4.79 (Correct to 4 significant digits/ decimal places) and can be approimated to Truncation Error- Occurs when some unwanted (less significant) terms for a eact formula are discarded. In mechanical engineering problem, infinite series in numerical operation are often appeared. Consider the Taylor series- f ( ) f ( h! f (! Can be approimated to (/truncated to) f ( 4! n 0 n f ( h n! n f ( ) f ( h! f ( ) f ( h! ME 6: Numerical Analysis f (! O ( h ) O ( h ) Order of error 9 5