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# class_7 - CS205 Class 7 Readings Heath 5.6 Systems of...

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CS205 - Class 7 Readings : Heath 5.6 Systems of Nonlinear Equations 1. Let’s turn our attention back to systems of nonlinear equations, i.e. A(x)=b or F(x)=0. a. Here the Jacobian matrix, J(x), is rather useful as a linearization of the nonlinear problem. b. Here we define / ij i j J F x   where each equation of F(x)=0 is written individually as ( ) 0 i F x , and each j x is the j-th component of the x vector. c. For example, consider 1 1 2 ( ) sin 4 0 F x x x and 2 2 1 2 ( ) 0 F x x x which can be written in matrix form as 1 2 2 1 2 sin 4 ( ) 0 x x F x x x . Then the Jacobian matrix is 2 1 1 cos ( ) 2 1 x J x x . d. Note that J(x), that is J, generally depends on the x vector. e. In general, we write J(x)=F’(x) and note that the Jacobian is the multidimensional generalization of f’(x). f. Thus conditioning in multiple dimensions depends on the Jacobian matrix just as conditioning for scalars depends on f’(x).
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