Newton’s Method for Functions of Several Variables

1. Newton’s Method for Functions of Several Variables By: Kelly Martin

2. One Variable: Converges to the root of the function. Several Variables: Find solution for systems of equations of nonlinear and “more than one equation.” Will find where the equations intersect.

3. f1(u,v)=0 Two unknowns: u,v f2(u,v)=0 Vector valued function: F(u,v,)=(f1, f2) -> F(x)=0, where x= (u,v) Jacobian Matrix: Represents derivative f’. First order partial derivatives of the vector valued function. DF(x)=

4. Jacobian MatrixExample: Given: f1(x,y)= 3x+y3 f2(x,y)= x3+2y JacobianMatrix: DF(x,y)=

5. One Variable: Several Variables: -1 Do not divide by matrix, multiply by the inverse

6. Inverse Matrices:

7. Example: Use Newton’s Method with starting guess (1,2), for 2 steps, to find a solution to the system: ƒ1(x, y) = y – x3 ƒ2(x, y) = x2 + y2 -1 y – x3 = 0 x2 + y2 -1 = 0 -1 -1

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9. -1 -1