2. Example `f(x) = x^2-2x+y^2-z+1=0,xy^2-x-3y+yz+2=0,xz^2-3z+yz^2+xy=0` and Initial guesses `= 1,2,3`
Find root using Modified Newton Raphson method (Multivariate Newton Raphson method)
`f(x) = x^2-2x+y^2-z+1=0,xy^2-x-3y+yz+2=0,xz^2-3z+yz^2+xy=0` and Initial guesses `= 1,2,3`
Solution:
`f_1=x^2+y^2-2x-z+1=0 ->(1)`
`f_2=xy^2-x-3y+yz+2=0 ->(2)`
`f_3=xz^2-3z+yz^2+xy=0 ->(3)`
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-2`
`J_(1,2) = (del f_1)/(del y)=2y`
`J_(1,3) = (del f_1)/(del z)=-1`
`J_(2,1) = (del f_2)/(del x)=y^2-1`
`J_(2,2) = (del f_2)/(del y)=2xy-3+z`
`J_(2,3) = (del f_2)/(del z)=y`
`J_(3,1) = (del f_3)/(del x)=z^2+y`
`J_(3,2) = (del f_3)/(del y)=z^2+x`
`J_(3,3) = (del f_3)/(del z)=2xz-3+2yz`
`J = [[2x-2,2y,-1],[y^2-1,2xy-3+z,y],[z^2+y,z^2+x,2xz-3+2yz]]`
with `x_0=[1,2,3]^T`, we get
Formula
`[[x_(n+1)],[y_(n+1)],[z_(n+1)]] = [[x_(n)],[y_(n)],[z_(n)]] - [[(del f_1)/(del x),(del f_1)/(del y),(del f_1)/(del z)],[(del f_2)/(del x),(del f_2)/(del y),(del f_2)/(del z)],[(del f_3)/(del x),(del f_3)/(del y),(del f_3)/(del z)]]^(-1) xx f([[x_(n)],[y_(n)],[z_(n)]])`
`n=0,x=1,y=2,z=3`
`[[1],[2],[3]] - [[0,4,-1],[3,4,2],[11,10,15]]^(-1) xx f([[1],[2],[3]])`
`=[[1],[2],[3]] - [[0,4,-1],[3,4,2],[11,10,15]]^(-1) xx [[1],[5],[20]]`
`=[[0.10256],[1.64103],[2.5641]]`
`n=1,x=0.10256,y=1.64103,z=2.5641`
`[[0.10256],[1.64103],[2.5641]] - [[-1.79487,3.28205,-1],[1.69297,-0.09928,1.64103],[8.21565,6.67719,5.94149]]^(-1) xx f([[0.10256],[1.64103],[2.5641]])`
`=[[0.10256],[1.64103],[2.5641]] - [[-1.79487,3.28205,-1],[1.69297,-0.09928,1.64103],[8.21565,6.67719,5.94149]]^(-1) xx [[0.93425],[1.45832],[3.93945]]`
`=[[1.52062],[1.41113],[0.19859]]`
`n=2,x=1.52062,y=1.41113,z=0.19859`
`[[1.52062],[1.41113],[0.19859]] - [[1.04125,2.82226,-1],[0.99129,1.49018,1.41113],[1.45057,1.56006,-1.83557]]^(-1) xx f([[1.52062],[1.41113],[0.19859]])`
`=[[1.52062],[1.41113],[0.19859]] - [[1.04125,2.82226,-1],[0.99129,1.49018,1.41113],[1.45057,1.56006,-1.83557]]^(-1) xx [[2.06374],[-0.44578],[1.66565]]`
`=[[1.94123],[0.77134],[0.89465]]`
`n=3,x=1.94123,y=0.77134,z=0.89465`
`[[1.94123],[0.77134],[0.89465]] - [[1.88247,1.54269,-1],[-0.40503,0.88936,0.77134],[1.57174,2.74163,1.85362]]^(-1) xx f([[1.94123],[0.77134],[0.89465]])`
`=[[1.94123],[0.77134],[0.89465]] - [[1.88247,1.54269,-1],[-0.40503,0.88936,0.77134],[1.57174,2.74163,1.85362]]^(-1) xx [[0.58624],[-0.4102],[0.98455]]`
`=[[1.06737],[1.19117],[0.48353]]`
`n=4,x=1.06737,y=1.19117,z=0.48353`
`[[1.06737],[1.19117],[0.48353]] - [[0.13473,2.38234,-1],[0.41889,0.02636,1.19117],[1.42497,1.30116,-0.81588]]^(-1) xx f([[1.06737],[1.19117],[0.48353]])`
`=[[1.06737],[1.19117],[0.48353]] - [[0.13473,2.38234,-1],[0.41889,0.02636,1.19117],[1.42497,1.30116,-0.81588]]^(-1) xx [[0.9399],[-0.55044],[0.34888]]`
`=[[1.26825],[0.95182],[0.88028]]`
`n=5,x=1.26825,y=0.95182,z=0.88028`
`[[1.26825],[0.95182],[0.88028]] - [[0.53651,1.90364,-1],[-0.09404,0.29458,0.95182],[1.72672,2.04315,0.90858]]^(-1) xx f([[1.26825],[0.95182],[0.88028]])`
`=[[1.26825],[0.95182],[0.88028]] - [[0.53651,1.90364,-1],[-0.09404,0.29458,0.95182],[1.72672,2.04315,0.90858]]^(-1) xx [[0.09764],[-0.13686],[0.28663]]`
`=[[0.95899],[1.03384],[0.96813]]`
`n=6,x=0.95899,y=1.03384,z=0.96813`
`[[0.95899],[1.03384],[0.96813]] - [[-0.08202,2.06767,-1],[0.06882,-0.049,1.03384],[1.9711,1.89626,0.85861]]^(-1) xx f([[0.95899],[1.03384],[0.96813]])`
`=[[0.95899],[1.03384],[0.96813]] - [[-0.08202,2.06767,-1],[0.06882,-0.049,1.03384],[1.9711,1.89626,0.85861]]^(-1) xx [[0.10237],[-0.03463],[-0.04513]]`
`=[[1.00171],[1.00007],[0.99718]]`
`n=7,x=1.00171,y=1.00007,z=0.99718`
`[[1.00171],[1.00007],[0.99718]] - [[0.00343,2.00014,-1],[0.00014,0.00075,1.00007],[1.99443,1.99608,0.99227]]^(-1) xx f([[1.00171],[1.00007],[0.99718]])`
`=[[1.00171],[1.00007],[0.99718]] - [[0.00343,2.00014,-1],[0.00014,0.00075,1.00007],[1.99443,1.99608,0.99227]]^(-1) xx [[0.00297],[-0.00282],[0.00075]]`
`=[[1],[1],[1]]`
`n=8,x=1,y=1,z=1`
`[[1],[1],[1]] - [[0,2,-1],[0,0.00001,1],[2,2,1.00001]]^(-1) xx f([[1],[1],[1]])`
`=[[1],[1],[1]] - [[0,2,-1],[0,0.00001,1],[2,2,1.00001]]^(-1) xx [[0],[0],[0.00001]]`
`=[[1],[1],[1]]`
Approximate root using Newton Raphson method is
`x=1,y=1,z=1`
This material is intended as a summary. Use your textbook for detail explanation.
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