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Mathematica Lab Tutorials: Lab 6

Lab Questions

Create a procedure whose inputs are the vector of functions f=\{f_1,f_2,\cdots,f_n \}, the vector of the names of the unknowns x=\{x_1,x_2,\cdots,x_n \} and the initial guesses x^{(0)}=\{x_1^{(0)},x_2^{(0)},\cdots,x_n^{(0)} \}. The procedure should do the following:

  • Find the length of the vectors f or x (These should be equal, a more robust code should stop the computations if they are not but this is not required)
  • The procedure should then form the matrix

        \[ K_{ij}=\frac{\partial f_i}{\partial x_j} \]

  • The procedure should then output \Delta x=-K^{-1} f evaluated at the values of the initial guesses.

Use the above procedure along with an appropriate while loop to find two different solutions (using two different initial guesses) for the following system of nonlinear equations. Use ε_s=0.00001. Compare with the Newton Raphson method implemented in Mathematica using (FindRoot).

    \[\begin{split} x_1^2+x_2^2&=100\\ x_1 x_2+x_2 x_3&=-102\\ x_2^2+x_3^2&=117 \end{split} \]

Video Tutorials

FindRoot and using rules for evaluations

Programing the Newton Raphson method

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