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Unformatted text preview: Part E has three chapters. Chapter 19 familiarizes you with general concepts needed throughout numerics (floating point, roundoff, stability, algorithm, errors, etc.) and with general tasks (solution of equations, interpolation, numeric integration and differentiation). Chapter 20 concerns algebra, in particular, numerics for linear systems of equations and eigenvalue problems for matrices. Finally, Chapter 21 deals with differential equations (ODEs and PDEs). Chap . 19 Numerics in General This chapter has five sections, the first on general concepts needed throughout numerics, and the other four on three basic areas, namely, solution of equations (Sec. 19.2), interpolation (Secs. 19.3 and 19.4) and numeric integration and differentiation (Sec. 19.5). In this chapter you should also obtain a feel for the spirit, viewpoint, and nature of numerics. You will notice that this field has a flavor distinct from that of calculus. Sec . 19 . 1 Introduction Problem Set 19 . 1 . Page 786 7 . Quadratic equation . The equation to be solved is x 2 u 20 x u 1 U 0. The 4Scalculation is x 1,2 U 10.00 ¡ 100.0 u 1.000 U 10.00 ¡ 99.00 U 10.00 ¡ 9.950, hence 19.95 and 0.05. Now use (7). Then x 1 remains as before and x 2 becomes x 2 U 1 x 1 U 1 19.95 U 0.05013. The 2Scalculation gives x 1,2 U 10 ¡ 100 u 1.0 U 10 ¡ 99 U 10 ¡ 99 U 10 ¡ 9.9, hence 10 u 9.9 U 10 u 10 U 20 (2S) and 10 u 9.9 U 10 u 10 U 0. By (7) you obtain x 1 as before and x 2 U 1 x 1 U 1 20 U 0.050. 13 . Rounding and adding . For instance, in rounding to, say, 1D, the given numbers a 1 U 1.03 and a 2 U 0.24 you get ã 1 U 1.0 and ã 2 U 0.2, hence the sum 1.2. But if you add first, you obtain 1.27. 168 Numeric Analysis Part E Rounded to 1D this gives 1.3, which is a more accurate approximation of the true value 1.27 than the approximation 1.2 obtained before. In terms of general formulas you have ã 1 u a 1 u U 1 ã 2 u a 2 u U 2 , where U 1 and U 2 are the errors due to rounding, hence they are less than or equal to 1/2 unit of the last decimal in absolute value. If you round first and add then, you add the rounded numbers ã 1 and ã 2 , that is, ã 1 ¡ ã 2 u a 1 ¡ a 2 u u U 1 ¡ U 2 U . You see that in this case the error U 1 ¡ U 2 is a number between 0 and 1 unit of the last decimal in absolute value. But if you add first, the sum is a 1 ¡ a 2 , and in rounding it you make an error between 0 and 1/2 unit of the last decimal in absolute value. Similarly for n numbers, where the sum of the rounded numbers is a number with an error between 0 and n /2 units of the last decimal in absolute value, whereas in adding and then rounding the error is between 0 and 1/2 unit of the last decimal in absolute value, as before in the case of two numbers....
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This note was uploaded on 04/08/2012 for the course MT 423 taught by Professor M.lee during the Spring '12 term at National Taiwan University.
 Spring '12
 M.lee

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