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Unformatted text preview: Nonlinear Equations Numerical Methods in One Dimension Methods for Systems of Nonlinear Equations Scientic Computing: An Introductory Survey Chapter 5 Nonlinear Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at UrbanaChampaign Copyright c 2002. Reproduction permitted for noncommercial, educational use only. Michael T. Heath Scientic Computing 1 / 55 Nonlinear Equations Numerical Methods in One Dimension Methods for Systems of Nonlinear Equations Outline 1 Nonlinear Equations 2 Numerical Methods in One Dimension 3 Methods for Systems of Nonlinear Equations Michael T. Heath Scientic Computing 2 / 55 Nonlinear Equations Numerical Methods in One Dimension Methods for Systems of Nonlinear Equations Nonlinear Equations Solutions and Sensitivity Convergence Nonlinear Equations Given function f , we seek value x for which f ( x ) = 0 Solution x is root of equation, or zero of function f So problem is known as root nding or zero nding Michael T. Heath Scientic Computing 3 / 55 Nonlinear Equations Numerical Methods in One Dimension Methods for Systems of Nonlinear Equations Nonlinear Equations Solutions and Sensitivity Convergence Nonlinear Equations Two important cases Single nonlinear equation in one unknown, where f : R R Solution is scalar x for which f ( x ) = 0 System of n coupled nonlinear equations in n unknowns, where f : R n R n Solution is vector x for which all components of f are zero simultaneously , f ( x ) = Michael T. Heath Scientic Computing 4 / 55 Nonlinear Equations Numerical Methods in One Dimension Methods for Systems of Nonlinear Equations Nonlinear Equations Solutions and Sensitivity Convergence Examples: Nonlinear Equations Example of nonlinear equation in one dimension x 24sin( x ) = 0 for which x = 1 . 9 is one approximate solution Example of system of nonlinear equations in two dimensions x 2 1x 2 + 0 . 25 = 0x 1 + x 2 2 + 0 . 25 = 0 for which x = . 5 0 . 5 T is solution vector Michael T. Heath Scientic Computing 5 / 55 Nonlinear Equations Numerical Methods in One Dimension Methods for Systems of Nonlinear Equations Nonlinear Equations Solutions and Sensitivity Convergence Existence and Uniqueness Existence and uniqueness of solutions are more complicated for nonlinear equations than for linear equations For function f : R R , bracket is interval [ a,b ] for which sign of f differs at endpoints If f is continuous and sign( f ( a )) 6 = sign( f ( b )) , then Intermediate Value Theorem implies there is x * [ a,b ] such that f ( x * ) = 0 There is no simple analog for n dimensions Michael T. Heath Scientic Computing 6 / 55 Nonlinear Equations Numerical Methods in One Dimension Methods for Systems of Nonlinear Equations Nonlinear Equations Solutions and Sensitivity Convergence Examples: One Dimension Nonlinear equations can have any number of solutions exp( x ) + 1 = 0 has no solution exp(x )x = 0 has one solution x 24sin( x ) = 0 has two solutions...
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 Fall '08
 Layton,A
 Linear Equations, Equations

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