1. Find root using Modified Newton Raphson method (Multivariate Newton Raphson method)
`f(x) = x^2+y^2-5=0,x^3+y^3-2=0` and Initial guesses `= 2,-1`

Solution:
`f_1=x^2+y^2-5=0 ->(1)`

`f_2=x^3+y^3-2=0 ->(2)`

In order to use Multivariate Newton Raphson method, we must first determine the functional form of the partial derivatives
`J_(1,1) = (del f_1)/(del x)=2x`

`J_(1,2) = (del f_1)/(del y)=2y`

`J_(2,1) = (del f_2)/(del x)=3x^2`

`J_(2,2) = (del f_2)/(del y)=3y^2`

`J = [[2x,2y],[3x^2,3y^2]]`

with `x_0=[2,-1]^T`, we get

Formula
`[[x_(n+1)],[y_(n+1)]] = [[x_(n)],[y_(n)]] - [[(del f_1)/(del x),(del f_1)/(del y)],[(del f_2)/(del x),(del f_2)/(del y)]]^(-1) xx f([[x_(n)],[y_(n)]])`

---

`n=0,x=2,y=-1`

`[[2],[-1]] - [[4,-2],[12,3]]^(-1) xx f([[2],[-1]])`

`=[[2],[-1]] - [[4,-2],[12,3]]^(-1) xx [[0],[5]]`

`=[[1.72222],[-1.55556]]`

---

`n=1,x=1.72222,y=-1.55556`

`[[1.72222],[-1.55556]] - [[3.44444,-3.11111],[8.89815,7.25926]]^(-1) xx f([[1.72222],[-1.55556]])`

`=[[1.72222],[-1.55556]] - [[3.44444,-3.11111],[8.89815,7.25926]]^(-1) xx [[0.3858],[-0.65586]]`

`=[[1.70779],[-1.44752]]`

---

`n=2,x=1.70779,y=-1.44752`

`[[1.70779],[-1.44752]] - [[3.41559,-2.89504],[8.74968,6.28596]]^(-1) xx f([[1.70779],[-1.44752]])`

`=[[1.70779],[-1.44752]] - [[3.41559,-2.89504],[8.74968,6.28596]]^(-1) xx [[0.01188],[-0.05213]]`

`=[[1.70942],[-1.4415]]`

---

`n=3,x=1.70942,y=-1.4415`

`[[1.70942],[-1.4415]] - [[3.41885,-2.88299],[8.76639,6.23373]]^(-1) xx f([[1.70942],[-1.4415]])`

`=[[1.70942],[-1.4415]] - [[3.41885,-2.88299],[8.76639,6.23373]]^(-1) xx [[0.00004],[-0.00014]]`

`=[[1.70943],[-1.44148]]`


Approximate root using Newton Raphson method is
`x=1.70943,y=-1.44148`