Problem $4$

Define a function norm$\_$approx$\_$gen $(\backslash (\backslash$mathbf$\{$M$\},\backslash$mathbf$\{$x$\}\backslash \_\backslash$mathbf$\{0\},\backslash$mathbf$\{$k$\}\backslash ))$ that approximates the dominant eigenvector and eigenvalue of any square matrix. Your function should take as input a square matrix m of any size, as well as an initial guess $\backslash (\backslash$mathbf$\{$x$\}\backslash$mathbf$\{0\}0\backslash )$ and integer $\backslash (\backslash$mathbf$\{$k$\}\backslash ),$ and use the power iteration method described above to approximate the dominant eigenvector and eigenvalue of $\backslash (\backslash$mathbf$\{$m$\}\backslash )$ up to the $\backslash ($ k $\backslash )$ th iterate.

Your function should normalize at each step by dividing the eigenvector approximation by the largest absolute value of its entries. You function should return the approximation $\backslash (\backslash$mathbf$\{$x$\}\_\{$k$\}\backslash )$ of the eigenvector obtained in this way, as well as the eigenvalue approximation obtained by dividing the first entries of $\backslash (\backslash$mathbf$\{$w$\}\_\{$k$\}\backslash )$ and $\backslash (\backslash$mathbf$\{$x$\}\_\{$k$-1\}\backslash ).$

Quick Note

The command np$.$abs $($a$)$ returns a vector whose entries are the absolute value of the entries in the NumPy vector a$.$ The command np$.$ max $($a$)$ returns the maximum element of the NumPy vector a$.$ You should also check this against simple examples for which you can compute eigenvalues by hand, perhaps a $\backslash (3\backslash$times $3\backslash )$ matrix from the homework. Keep in mind that power iteration will only work for matrices that have a dominant eigenvalue.

Test Cases

You can test that your code is correct by evaluating these commands and comparing outputs:

norm$\_$approx$\_$gen$($np$.$array$([[2,4,6],[4,8,0],[1,2,9]]),$ np$.$array$([1,5,-1]),10)$

$($array$([0\mathrm{.}98994349,1.,0\mathrm{.}98491523]),12\mathrm{.}01744017467949)$