nonl_eq - Chapter 5 Nonlinear Equations Given a function...

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Chapter 5 Nonlinear Equations Given a function f(x) find x such that f(x) =0 Solution x* is called a root or a zero Problem is called root-finding • Case 1: single nonlinear equation • Case 2: system of n nonlinear equations in n unknowns
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Examples x 1 = cos( x 1 ) 81 + x 2 2 9 + sin( x 3 ) 3 x 2 = sin( x 1 ) 3 + cos( x 3 ) 3 x 3 = cos( x 1 ) 9 + x 2 3 + sin( x 3 ) 6
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Examples • Single equation x 2 – 4sin(x)=0 x* is approx 1.9 • Nonlinear system in 2 unknowns x 1 x 2
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Existence and uniqueness For linear systems Ax=b : unique solution, no solution, infinite no. of solutions • For nonlinear system f(x)=0 can have any possibilities: no solution, 1 soln, 2 solns,… x 1 x 1
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Multiplicity If m=1 then x* is a simple root
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Bisection Method Given a > b such that f(a) and f(b) have different signs While (b-a) > tol m = a + (b-a)/2 if sign(f(a))=sign(f(m)) then a=m else b=m end a m b
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f(x) = x 2 4 sin(x) = 0 a f(a) b f(b) 1.000000 2.365884 3.000000 8.435520 1.000000 2.365884 2.000000 0.362810 1.500000 1.739980 2.000000 0.362810 1.750000 0.873444 2.000000 0.362810 1.875000 0.300718 2.000000 0.362810 1.875000 0.300718 1.937500 0.019849 1.906250 0.143255 1.937500 0.019849 1.921875 0.062406 1.937500 0.019849 1.929688 0.021454 1.937500 0.019849 1.933594 0.000846 1.937500 0.019849 1.933594 0.000846 1.935547 0.009491 1.933594 0.000846 1.934570 0.004320 1.933594 0.000846 1.934082 0.001736 16 / 55
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Bisection method is slow (linear convergence) But it cannot fail The interval is reduced by half at every iteration One bit of accuracy is gained at each iterations Given an interval [a,b] and a tolerance tol, how many iterations will it take to achieve the tolerance? Regardless of how easy or hard
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This note was uploaded on 01/15/2012 for the course ECON 101 taught by Professor N/a during the Fall '11 term at Middlesex CC.

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nonl_eq - Chapter 5 Nonlinear Equations Given a function...

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