ch19

Advanced Engineering Mathematics

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301 PART E. Numeric Analysis The subdivision into three chapters has been retained. All three chapters have been updated in the light of computer requirements and developments. A list of suppliers of software (with addresses etc.) can be found at the beginning of Part E of the book and another list at the beginning of Part G. CHAPTER 19 Numerics in General Major Changes Updating of this chapter consists of the inclusion of ideas, such as error estimation by halving, changes in Sec. 19.4 on splines, the presentation of adaptive integration and Romberg integration, and further error estimation techniques in integration. SECTION 19.1. Introduction, page 780 Purpose. To familiarize the student with some facts of numerical work in general, regardless of the kind of problem or the choice of method. Main Content, Important Concepts Floating-point representation of numbers, overflow, underflow, Roundoff Concept of algorithm Stability Errors in numerics, their propagation, error estimates Loss of significant digits Short Courses. Mention the roundoff rule and the definitions of error and relative error. SOLUTIONS TO PROBLEM SET 19.1, page 786 2. 0.286403 10 1 , 0.112584 10 2 , 0.316816 10 5 6. 19.9499, 0.0501; 19.9499, 0.0501256 8. 99.980, 0.020; 99.980, 0.020004 10. Small terms first. (0.0004 0.0004) 1.000 1.001, but (1.000 0.0004) 0.0004 1 (4S) 14. The proof is practically the same as that in the text. With the same notation we get x y ( x y ) ( x x ) ( y y ) 1 2 1 2 1 2 . 16. Since x 2 2/ x 1 and 2 is exact, r ( x 2 ) r ( x 1 ) by Theorem 1b. Since x 1 is rounded to 4S, we have ( x 1 ) 0.005, hence r ( x 1 ) 0.005/39.95. im19.qxd 9/21/05 1:10 PM Page 301
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302 Instructor’s Manual This implies ( x 2 ) r ( x 2 ) x 2 r ( x 1 ) x 2 (0.005/39.95) 0.0506 0.00001. 18. 61.2 7.5 15.5 11.2 3.94 2.80 61.2 116 44.1 2.80 7.90 (( x 7.5) x 11.2) x 2.80 ( 3.56 3.94 11.2)3.94 2.80 ( 14.0 11.2)3.94 2.80 11.0 2.80 8.20 Exact: 8.336016 SECTION 19.2. Solution of Equations by Iteration, page 787 Purpose. Discussion of the most important methods for solving equations ƒ( x ) 0, a very important task in practice. Main Content, Important Concepts Solution of ƒ( x ) 0 by iteration (3) x n 1 g ( x n ) Condition sufficient for convergence (Theorem 1) Newton (–Raphson) method (5) Speed of convergence, order Secant, bisection, false position methods Comments on Content Fixed-point iteration gives the opportunity to discuss the idea of a fixed point, which is also of basic significance in modern theoretical work (existence and uniqueness of solutions of differential, integral, and other functional equations). The less important method of bisection and method of false position are included in the problem set. SOLUTIONS TO PROBLEM SET 19.2, page 796 2. x 0 1, x 1 0, x 2 1, x 3 0, • • • x 0 0.5, x 1 0.875, x 2 0.330, • • • x 0 2, x 1 7, x 2 344, x 3 40 707 583, • • • 4. g 4 x 0.2, 1, 0.9457, 0.9293, 0.9241, 0.9225, 0.9220, 0.9218, 0.9217, 0.9217 6. x x /( e 0.5 x sin x ), 1, 0.7208, 0.7617, 0.7541, 0.7555, 0.7553, 0.7553, 0.7553 8. CAS Project. (a) This follows from the intermediate value theorem of calculus. (b) Roots r 1 1.56155 (6S-value), r 2 1 (exact), r 3 2.56155 (6S-value). (1) r 1 , about 12 steps, (2) r 1 , about 25 steps, (3) convergent to r 2 , divergent, (4) convergent to 0, divergent, (5) r 3 , about 7 steps, (6) r 2 , divergent, (7) r 1 , 4 steps; this is Newton.
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