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Modified Newton Raphson method (Multivariate Newton Raphson method) example
( Enter your problem )
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- Algorithm & Example `f(x) = x^2+y^2-5=0,x^3+y^3-2=0` and Initial guesses `= 2,-1`
- 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`
- Example `f(x) = 3x-cos(yz)-3/2=0,4x^2-625y^2+2z-1=0,20z+e^(-xy)+9=0` and Initial guesses `= 1,1,1`
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1. Algorithm & Example `f(x) = x^2+y^2-5=0,x^3+y^3-2=0` and Initial guesses `= 2,-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`
This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then
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