Useful Notes and Equations

For a comprehensive and thorough summary of the theory, check MuChenSun’s wonderful note here.

My own notes:

Useful Equations:

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Textbook Exercises Attempts

Exercise 6.8 Use Newton–Raphson iterative numerical root finding to perform two steps in finding the root of

\[g(x,y) = \begin{bmatrix} x^2 - 4\\ y^2 - 9\\ \end{bmatrix}\]

when your initial guess is \((x^{0}, y^{0})=(1,1)\). Write the general form of the gradient (for any guess \((x,y)\)) and compute the results of the first two iterations. You can do this by hand or write a program. Also, give all the correct roots, not just the one that would be found from your initial guess. How many are there?

From \(g(x,y)\) we know that \(x_{d} = (4,9)^{T}\), \(f(\theta)=(x^{2}, y^{2})^{T}\).

\[\frac{\delta f}{\delta \theta} = J(\theta) = \begin{bmatrix} 2x & 0 \\ 0 & 2y \\ \end{bmatrix}\] \[J^{-1}(\theta) = \begin{bmatrix} \frac{1}{2x} & 0 \\ 0 & \frac{1}{2y} \\ \end{bmatrix}\] \[\begin{align} \begin{split} \theta^{1} &= \begin{bmatrix} \frac{1}{2} & 0 \\ 0 & \frac{1}{2} \\ \end{bmatrix} \begin{bmatrix} 4-1^{2} \\ 9-1^{2} \\ \end{bmatrix} + \begin{bmatrix} 1 \\ 1 \\ \end{bmatrix} \\ &= \begin{bmatrix} 2.5 \\ 5 \end{bmatrix} \\ \theta_{2} &= \begin{bmatrix} \frac{1}{5} & 0 \\ 0 & \frac{1}{10} \\ \end{bmatrix} \begin{bmatrix} 4 - 2.5^{2} \\ 9 - 5^{2} \\ \end{bmatrix} + \begin{bmatrix} 2.5 \\ 5 \\ \end{bmatrix} \\ &= \begin{bmatrix} 2.05 \\ 3.4 \\ \end{bmatrix} \end{split} \end{align}\]

There should be a total of 4 solutions, since x could be -2 or 2, and y could be -3 or 3.

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