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Newton's Method for Functions of  Several Variables

Newton's Method for Functions of  Several Variables. Joe Castle & Megan Grywalski. Recall, to find x n+1 using Newton’s Method, the formula was:

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Newton's Method for Functions of  Several Variables

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  1. Newton's Method for Functions of  Several Variables Joe Castle& Megan Grywalski

  2. Recall, to find xn+1 using Newton’s Method, the formula was: • The formula of Newton’s method for nonlinear systems is similar to this previous formula except the derivative expression is not in the denominator but in the numerator as the inverse of the matrix (F) defined: The idea here is to use Xn to generate a new value (Xn+1) by using the tangent line to estimate the point(s) of intersection.

  3. Xn: initial guess (vector) • F: the original system of nonlinear equations displayed as a vector (set equal to zero). • J: (Jacobian Matrix) partial derivatives of F evaluated at Xn.

  4. Finding the Inverse To find A-1, multiply this new matrix by 1/the determinate of A. The determinate is: ad-bc

  5. Steps to Solve • Given a set of nonlinear equations, set each equation equal to 0 and represent these equations in vector notation: F. • Take the partial derivatives of F to find the Jacobian matrix: J. • Find the Jacobian matrix’s inverse: J-1. • Use the initial guess (Xn) to evaluate and find Xn+1.

  6. Example 1. Set equations equal to 0 and represent in vector notation. 2. Take the partial derivatives of F to find the Jacobian Matrix. 3. Find the inverse Jacobian Matrix. 4.Use initial guess (X0) to solve the equation for X1.

  7. ADVANTAGES • Rapid convergence • If there is more than one intersection, the initial guess will find the closest point of intersection. DISADVANTAGES • F must be differentiable • The Jacobian Matrix must be nonsingular-its inverse must exist. • Cannot compute the number of iterations needed • Some initial guesses won’t give you a point of intersection (if the guess is exactly between the intersections)

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