Root Finding and Nonlinear Sets of Equations part 2

Tham khảo tài liệu 'root finding and nonlinear sets of equations part 2', công nghệ thông tin, kỹ thuật lập trình phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 350 Chapter 9. Root Finding and Nonlinear Sets of Equations for i 1 i ISCR i printf c scr i 1 printf n printf 8s 44s n x1 x2 CITED REFERENCES AND FURTHER READING Stoer J. and Bulirsch R. 1980 Introduction to Numerical Analysis New York Springer-Verlag Chapter 5. Acton . 1970 Numerical Methods That Work 1990 corrected edition Washington Mathematical Association of America Chapters 2 7 and 14. Ralston A. and Rabinowitz P. 1978 A First Course in Numerical Analysis 2nd ed. New York McGraw-Hill Chapter 8. Householder . 1970 The Numerical Treatment of a Single Nonlinear Equation New York McGraw-Hill . Bracketing and Bisection We will say that a root is bracketed in the interval a b if f a and f b have opposite signs. If the function is continuous then at least one root must lie in that interval the intermediate value theorem . If the function is discontinuous but bounded then instead of a root there might be a step discontinuity which crosses zero see Figure . For numerical purposes that might as well be a root since the behavior is indistinguishable from the case of a continuous function whose zero crossing occurs in between two adjacent floating-point numbers in a machine s finite-precision representation. Only for functions with singularities is there the possibility that a bracketed root is not really there as for example f x x c Some root-finding algorithms . bisection in this section will readily converge to c in . Luckily there is not much possibility of your mistaking c or any number x close to it for a root since mere evaluation of f x will give a very large rather than a very small result. If you are given a function in a black box there is no sure way of bracketing its roots or of even determining that it has roots. If you like pathological examples think about the problem of locating the two real roots of equation which dips below zero only in the ridiculously small interval of about x k 10-667. In the next .

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