Chapter 2 the numerical solution of thenonlinear equationsSection I Import of practical problemSection II DichotomySection III Fixed point iteration methodSection IV Newton methodSection V Secant Method
Section II Dichotomy Chapter 2 the numerical solution of the nonlinear equations Section III Fixed point iteration method Section IV Newton method Section V Secant Method Section I Import of practical problem
S 1 Import of practical problemThe nonlinear phenomenon widely exists in the material worldand the social life,Many practical problems are transformed into nonlinear equationsortheproblemtosolveequations.We look at an example.In celestial mechanics, Kepler equationx-t-sinx=0,0<8<1Where t represents time, x represents the radian, a planetarymotion track x is a function of t.There is a sole x corresponding to each t in Kepler equationIn this chapter, we discuss the single-variable nonlinear equationf(x)= 0the problem of finding roots, here x E R, f(x) eC[a,b]
§1 Import of practical problem The nonlinear phenomenon widely exists in the material world and the social life, Many practical problems are transformed into nonlinear equations, or the problem to solve equations. We look at an example. In celestial mechanics, Kepler equation x t x − − = sin 0,0 1 Where t represents time, x represents the radian, a planetary motion track x is a function of t. There is a sole x corresponding to each t in Kepler equation. In this chapter, we discuss the single-variable nonlinear equation. f x( ) 0 = the problem of finding roots, here x f x C a b R, ( ) [ , ]
Seeking the root of this equation f (x) = 0Principle:Iff eC[a, bl, and f(a) : f(b)< O, then there must beone root off on (a, b)1、 The Graphic method: Give sketche of y=f(x) , determine theapproximate location of the root.2、 Test method: take some of the data appropriately to test,search out the inter-cell of sign changes, namely the inter-cell tomeet f(ak):f(bk)<0
Seeking the root of this equation f (x) = 0 Principle:If f C[a, b]，and f (a) · f (b) < 0，then there must be one root of f on (a, b) . 1、The Graphic method: Give sketche of y＝f (x) , determine the approximate location of the root. 2、Test method: take some of the data appropriately to test, search out the inter-cell of sign changes, namely the inter-cell to meet f (ak )·f (bk ) < 0
s2 Dichotomy41中弟ab[f(x)<&2Xk+1-x|<8orCan not guaranteethe accuracyof x82x
a b x1 x2 a b 1 1 x x ε k+ − k 2 f (x) ε or x* 2 x* x §2 Dichotomy Can not guarantee the accuracy of x
ErronAnalysis:a+bb-aStep 1 generatehas the error k,-x*|≤Xi22b-athe k-th step xk has the error x -x*|≤24the number k of dichotomy required can be estimated according to thgivenaccuracy:[n(b - a)- In a]b-ak<82kIn 2① Simple;② The require of f (x) is not high ( just continuous) Can not demand complex roots and even re-root②Slow convergenceNote: when finding roots by dichotomy, it is best to give thesketche of f (x) in order to determine the approximate locationof the root
Error Analysis： Step 1 generate 2 1 a b x + = has the error 2 1 b a |x x*| − − the k-th step xk has the error k k b a |x x*| 2 − − the number k of dichotomy required can be estimated according to the given accuracy ： ( ) ln 2 ln ln 2 b a ε ε k b a k − − − ① Simple; ② The require of f (x) is not high ( just continuous). ① Can not demand complex roots and even re-root ② Slow convergence Note: when finding roots by dichotomy, it is best to give the sketche of f (x) in order to determine the approximate location of the root