ME 261: Numerical Analysis Lecture-2: Approximation & Error
1ME 261: Numerical AnalysisLecture-2: Approximation &ErrorMd. Tanver HossainDepartment of Mechanical Engineering,BUEThttp://tantusher.buet.ac.bd
2Floating Point Arithmetic fractional quantities are typically represented in computers usingfloating point format this approach is very much similar to scientific notation for example, fixed point number 17.542 is the same as the floatingpoint number .17542*102 which is often displayed as .17542e2 another example, -.004428 is same as -.4428*10-2 General form of floating-point number in Computers:
3Floating Point Arithmetic
4Floating Point Arithmetic
5Floating Point Arithmetic
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9SIGNIFICANT FIGURES Whenever we employ a number in acomputation, we must have assurance that itcan be used with confidence. The concept of a significant figure, or digit,has been developed to formally designatethe reliability of a numerical value. The significant digits of a number are thosethat can be used with confidence. If we say we know the value is 48 with twosignificant digits. The value is 48.5 withthree significant digits and vice versa. Although quantities such as π, e, or 7represent specific quantities, they cannot beexpressed exactly by a limited number ofdigits.
10SIGNIFICANT FIGURES Although it is usually a straightforward procedure to ascertain thesignificant figures of a number, some cases can lead to confusion. For example, zeros are not always significant figures because theymay be necessary just to locate a decimal point. The numbers 0.00001845, 0.0001845, and 0.001845 all have foursignificant figures. Similarly, when trailing zeros are used in large numbers, it is notclear how many, if any, of the zeros are significant. For example, at face value the number 45,300 may have three, four,or five significant digits, depending on whether the zeros are knownwith confidence. Such uncertainty can be resolved by using scientificnotation, where 4.53 104, 4.530 104, 4.5300 104 designate thatthe number is known to three, four, and five significant figures,respectively.
11ERROR DEFINITIONS True value approximation error Et true value approximationEt is used to designate the exact value. For example, we want to measure the length of bridge and a rivet. We measured the length of bridge is 9999 cm and the length of the rivet is 9 cm. The exact values are 10000 and 10 cm, respectively. The absolute error for both cases is 1 cm. The percentage relative errors are 0.01% and 10%.
12Sources of Error in Numerical Computations1. Errors in Mathematical Modeling2. Errors in Numerical Input3. Machine ErrorThe floating point representation (in binary digits) of numbers involves roundingand chopping errors since each computing machine (computer, calculator etc.) canhold a finite number of digits. These errors are introduced at each arithmeticoperation during the computations. It Depends on the bit of computer operatingsystem and associated software packages-16 Bit/32 Bit/64 Bit.Inherent Error Errors in Mathematical Model Errors in numericalinput Machine Error
134. BlundersBlunders are errors that are caused due to human imperfection. Such errors maycause a very serious disaster in the result. Some common sources of such errorsare(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). Thus when a large program is written,it is a good practice to divide it into smaller sub-programs and test each sub-program separatelyfor accuracy.5. Computational ErrorsComputational errors are introduced during the process of implementation of anumerical method. They come in two formsRounding a number can be done in two waysChopping- Extra digits are dropped. This is called truncating the number.Example- a exact number 43.92638 can be approximated to 43.92 (2 digit cho
43.926 (3 digit chopping), 43.9263 (4 digit chopping) Symmetric Roundoff- The last retained significant digit is “rounded up” by 1 if the first discarded digit is larger or equal to 5; otherwise, the last retained digit
Introduction of Chemical Reaction Engineering Introduction about Chemical Engineering 0:31:15 0:31:09. Lecture 14 Lecture 15 Lecture 16 Lecture 17 Lecture 18 Lecture 19 Lecture 20 Lecture 21 Lecture 22 Lecture 23 Lecture 24 Lecture 25 Lecture 26 Lecture 27 Lecture 28 Lecture
Lecture 1: A Beginner's Guide Lecture 2: Introduction to Programming Lecture 3: Introduction to C, structure of C programming Lecture 4: Elements of C Lecture 5: Variables, Statements, Expressions Lecture 6: Input-Output in C Lecture 7: Formatted Input-Output Lecture 8: Operators Lecture 9: Operators continued
“numerical analysis” title in a later edition [171]. The origins of the part of mathematics we now call analysis were all numerical, so for millennia the name “numerical analysis” would have been redundant. But analysis later developed conceptual (non-numerical) paradigms, and it became useful to specify the different areas by names.
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Partial Di erential Equations MSO-203-B T. Muthukumar tmk@iitk.ac.in November 14, 2019 T. Muthukumar tmk@iitk.ac.in Partial Di erential EquationsMSO-203-B November 14, 2019 1/193 1 First Week Lecture One Lecture Two Lecture Three Lecture Four 2 Second Week Lecture Five Lecture Six 3 Third Week Lecture Seven Lecture Eight 4 Fourth Week Lecture .
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methods are the only option for the majority of problems in numerical analysis, and may actually be quicker even when a direct method exists. I„e word “iterative” derives from the latin iterare, meaning “to repeat”. Numerical Analysis II - ARY 4 2017-18 Lecture Notes
