Question

Solve the following systems of nonlinear equations by the Newton-Raphson method.

given the following system of nonlinear equations, obtain the values ​​of xi+1, yi+1. (perform 2 interactions).
A. u=x²+2xy-10
B. v=y² +7xy²-57
C. xi=1.5
D. yi=3.5

Answer 1

The Newton-Raphson method is used to solve a system of nonlinear equations by iterating on initial guesses xi=1.5 and yi=3.5. The process involves calculating a Jacobian matrix, evaluating the function and its derivatives, and updating the guesses with found increments for two iterations.

Step-by-step explanation:

The question asks for the solution to a system of nonlinear equations using the Newton-Raphson method. With the initial guesses of xi=1.5 and yi=3.5, we perform two iterations to update these values. The system given is:

• u=x²+2xy-10
• v=y²+7xy²-57

To apply the Newton-Raphson method, we first compute the Jacobian matrix of the system, which consists of the partial derivatives of both equations with respect to x and y. Then, we evaluate the function and its Jacobian at the initial point (xi, yi), and solve the linear system to find the increments (∆x, ∆y). These increments are used to update the guesses for x and y.

For two iterations, this process is repeated with the updated values, resulting in new guesses for xi+1 and yi+1 after each iteration. The detailed steps are generally complex and require calculus and linear algebra knowledge.