SSM-Ch19 - Part E has three chapters Chapter 19...

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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 u2212 20 x u002b 1 u003d 0. The 4S-calculation is x 1,2 u003d 10.00 u00b1 100.0 u2212 1.000 u003d 10.00 u00b1 99.00 u003d 10.00 u00b1 9.950, hence 19.95 and 0.05. Now use (7). Then x 1 remains as before and x 2 becomes x 2 u003d 1 x 1 u003d 1 19.95 u003d 0.05013. The 2S-calculation gives x 1,2 u003d 10 u00b1 100 u2212 1.0 u003d 10 u00b1 99 u003d 10 u00b1 99 u003d 10 u00b1 9.9, hence 10 u002b 9.9 u003d 10 u002b 10 u003d 20 (2S) and 10 u2212 9.9 u003d 10 u2212 10 u003d 0. By (7) you obtain x 1 as before and x 2 u003d 1 x 1 u003d 1 20 u003d 0.050. 13 . Rounding and adding . For instance, in rounding to, say, 1D, the given numbers a 1 u003d 1.03 and a 2 u003d 0.24 you get ã 1 u003d 1.0 and ã 2 u003d 0.2, hence the sum 1.2. But if you add first, you obtain 1.27.
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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 u003d a 1 u2212 u03b5 1 ã 2 u003d a 2 u2212 u03b5 2 , where u03b5 1 and u03b5 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 u002b ã 2 u003d a 1 u002b a 2 u2212 ue0a2 u03b5 1 u002bu03b5 2 ue0a3 . You see that in this case the error u03b5 1 u002bu03b5 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 u002b 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. Sec . 19 . 2 Solution of Equations by Iteration Problem Set 19 . 2 . Page 796 3 . Nonmonotonicity (as in Example 2 in the book, Fig. 424) occurs if g ue0a2 x ue0a3 is monotone decreasing, that is, g ue0a2 x 1 ue0a3 u2264 g ue0a2 x 2 ue0a3 if x 1 u003e x 2 . (A) (Make a sketch to better understand the reasoning.) Then g ue0a2 x ue0a3 u2265 g ue0a2 s ue0a3 if and only if x u2264 s (B) where s
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