CP04_roots - Examples of nonlinear equations...

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1 1 Roots of Nonlinear Equations 2 Examples of nonlinear equations one-dimensional equations 0 ) cos( ) ln( ) exp( 0 ) cos( 0 9 6 2 2 = = = + x x x x x x x x two-dimensional equation = + = 1 ) 1 ( 2 2 3 2 y x x x y 3 Part 1. Real Roots of a Single Variable Function 0 ) ( = x f 4 1.1 Introduction . and between ip relationsh nonlinear any # equation al differenti a of solution a # equation ntal transcende a # equation algebraic an # be may , equation, nonlinear The f(x) x f(x) 0 = . 0 ) ( = = c f c x f(x) that such value the - find function nonlinear continuous a - Given 5 Prelude for root finding You should try to get some idea of what your function looks like before trying to find its roots. 0 ) ( = x f 6 Prelude for root finding (more) There are two phases to finding the roots of a nonlinear equation: 1. bounding the root 2. refining the root to the desired accuracy. Two general types of root-finding methods exist: 1. closed domain (bracketing) methods 2. open domain (nonbracketing) methods . There are numerous pitfalls in finding the roots of nonlinear equations.
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2 7 1.2 Bounding the solution Bounding the solution involves finding a rough estimate of the solution that can be used as the initial approximation, or the starting point, in a systematic procedure that refines the solution to a specified tolerance in an efficient manner. If possible, the root should be bracketed between two points at which the value of the nonlinear function has opposite signs. "The hardest thing of all is to find a black cat in a dark room, especially if there is no cat.“ Confucius (551-479 AD) 8 Possible bounding procedures 1. Graphing the function 2. Incremental search 3. Past experience with the problem or a similar problem 4. Solution of a simplified approximate model 5. Previous solution in a sequence of solutions 9 1.3 Refining the Solution Refining the solution involves determining the solution to a specified tolerance by a systematic procedure. Methods for refining the solution are: 1. Trial and error ( very inefficient method ) 2. Closed domain (bracketing) methods 3. Open domain methods 10 Closed domain (bracketing) methods Methods start with two values of x which bracket the root, x = α , and systematically reduce the interval while keeping the root trapped within the interval. Most common closed domain methods: 1. Interval halving (bisection) 2. False position (regula falsi) Bracketing methods are robust (they are guaranteed to obtain a solution since the root is trapped in the closed interval). They can be slow to converge. 11 Open domain methods Methods do not restrict the root to remain trapped in a closed interval. Consequently, they are not as robust as bracketing methods and can actually diverge. However, they use information about the nonlinear function itself to refine the estimates of the root. Thus, they are considerably more efficient than bracketing ones.
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CP04_roots - Examples of nonlinear equations...

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