5_System of Nonlinear Equations

5_System of Nonlinear Equations - System of Nonlinear...

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System of Nonlinear Equations
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Consider the n - dimensional version of solving a single equation f(x) = 0 Statement of the problem n simultaneous nonlinear equations to be solved
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The simplest and easiest method is Newton – Raphson Method Need a good starting point Variants of this method have good convergence properties
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Taylor series expansion of about point x By dropping the higher order terms J(x) is the Jacobian matrix (n×n) made up of the partial derivatives Linear approximation of the vector valued function f in the vicinity of x
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Assume x as the current approximate solution of f(x) = 0 Let ° + ∆° be an improved solution To find the correction ∆± put ² ± + ∆± = 0 Result a set of linear equations for ∆± ³ ° ∆° = −´(°) Solve for the increments and improve the solution
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Steps in Newton – Raphson Method Estimate the solution vector x Evaluate f(x) Compute the Jacobian matrix J(x)
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