CS205 - Class 7Readings: Heath 5.6Systems of Nonlinear Equations1.Let’s turn our attention back to systems of nonlinear equations, i.e. A(x)=b or F(x)=0.a.Here the Jacobianmatrix, J(x), is rather useful as a linearization of the nonlinear problem.b.Here we define /ijijJFx where each equation of F(x)=0 is written individually as ( )0iF x, and each jxis the j-th component of the x vector.c.For example, consider 112( )sin40F xxxand 2212( )0Fxxxwhich can be written in matrix form as 12212sin4( )0xxF xxx. Then the Jacobian matrix is 211cos( )21xJ xx.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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Nonlinear system, xk, Systems of Nonlinear Equations, Jacobian matrix, xk yk