F[x_,y_] = ((((x-11))^2)/64) - ((((y-7))^2)/100) - 1;

G[x_,y_] = (((x-3))^2)+(((y-1))^2)-400;

FGPlot = ContourPlot[{F[x,y]==0,G[x,y]==0},{x,-25,25}, {y, -25, 25}, ContourStyle-> {{Red, Thickness[0.01]},{Blue,Thickness[0.01]}}, PlotRange-> {{-25,25},{-25,25}}, Axes->True, Frame->False, AspectRatio-> 1]

(FGMat = {{F[x,y]},{G[x,y]}})//MatrixForm

Flatten[N[FGMat/.{{x->-2,y-> 20}}]]

(DFGMat= {{D[F[x,y],x],D[F[x,y],y]},{D[G[x,y],x],D[G[x,y],y]}})//MatrixForm

(NewtIter= (Factor[Simplify[{{x},{y}}-Inverse[DFGMat].FGMat]]))//MatrixForm

NewtIter/.{x->-0.25+0.433 I, y-> 0.433 - 0.75 I}

(Subscript[x, 0]) = -2;(Subscript[y, 0]) = 20;

(Seq1 = N[NestList[Flatten[NewtIter]/.{x->#[[1]] ,y-> #[[2]]}&,{(Subscript[x, 0]),(Subscript[y, 0])},5],15])//MatrixForm

root1 =Seq1[[5]]

FindRoot[{F[x,y]==0,G[x,y]==0},{{x,-2},{y,20}}, WorkingPrecision->15]

(Subscript[x, 0]) = 20;(Subscript[y, 0]) = 13;

(Seq2 = N[NestList[Flatten[NewtIter]/.{x->#[[1]] ,y-> #[[2]]}&,{(Subscript[x, 0]),(Subscript[y, 0])},5],15])//MatrixForm

root2 = Seq2[[5]]

FindRoot[{F[x,y]==0,G[x,y]==0},{{x,20},{y,13}}, WorkingPrecision->15]

(Subscript[x, 0]) = 20;(Subscript[y, 0]) =-4;

(Seq3 = N[NestList[Flatten[NewtIter]/.{x->#[[1]] ,y-> #[[2]]}&,{(Subscript[x, 0]),(Subscript[y, 0])},5],15])//MatrixForm

root3 = Seq3[[5]]

FindRoot[{F[x,y]==0,G[x,y]==0},{{x,20},{y,-4}}, WorkingPrecision->15]

(Subscript[x, 0]) = -10;(Subscript[y, 0]) =-16;

(Seq4 = N[NestList[Flatten[NewtIter]/.{x->#[[1]] ,y-> #[[2]]}&,{(Subscript[x, 0]),(Subscript[y, 0])},5],15])//MatrixForm

root4 = Seq4[[5]]

FindRoot[{F[x,y]==0,G[x,y]==0},{{x,-10},{y,-16}}, WorkingPrecision->15]

RootPlots = Graphics[{PointSize[0.03],Black,Point[{root1,root2,root3,root4}]}];

Show[FGPlot,RootPlots]

F[x_,y_] = 3(x^2)y - (y^3)+5 x -8;

G[x_,y_] = 3 x (y^2)-(x^3)-4 y + 2;

FGPlot = ContourPlot[{F[x,y]==0,G[x,y]==0},{x,-10,10}, {y, -10, 10}, ContourStyle-> {{Red, Thickness[0.01]},{Blue,Thickness[0.01]}}, PlotRange-> {{-10,10},{-10,10}}, Axes->True, Frame->False, AspectRatio-> 1]

(FGMat = {{F[x,y]},{G[x,y]}})//MatrixForm

Flatten[N[FGMat/.{{x->-2+I,y-> 5-3 I}}]]

(DFGMat= {{D[F[x,y],x],D[F[x,y],y]},{D[G[x,y],x],D[G[x,y],y]}})//MatrixForm

(NewtIter= (Factor[Simplify[{{x},{y}}-Inverse[DFGMat].FGMat]]))//MatrixForm

NewtIter/.{x->-0.25+0.433 I, y-> 0.433 - 0.75 I}

(Subscript[x, 0]) = 7.-10. I;(Subscript[y, 0]) = -5. + 3. I;

(Seq1 = SetPrecision[NestList[Flatten[NewtIter]/.{x->#[[1]] ,y-> #[[2]]}&,{(Subscript[x, 0]),(Subscript[y, 0])},12],12])//MatrixForm

root1 =Seq1[[12]]

FGMat/.{x->root1[[1]],y->root1[[2]]}

F[x_,y_,z_] = (((x-5))^2)+(((y-9))^2)+(((z-4))^2)-49;

G[x_,y_,z_] = (((x-2))^2)+(((y-7))^2)+(((z-13))^2)-100;

H[x_,y_,z_] = (((x-6))^2)+(((y-11))^2)+(((z-10))^2)-64;

FGPlot = ContourPlot3D[{F[x,y,z]==0,G[x,y,z]==0,H[x,y,z]==0},{x,-10,25}, {y, -10, 25}, {z,-10,25},ContourStyle-> {{Red, Thickness[0.01]},{Blue,Thickness[0.01]},{Yellow,Thickness[0.01]}}, PlotRange-> {{-10,25},{-10,25},{-10,25}}, Axes->True, AspectRatio-> 1, Mesh->None, AxesLabel->{"x","y","z"}]

(FGHMat = {{F[x,y,z]},{G[x,y,z]},{H[x,y,z]}})//MatrixForm

Flatten[N[FGHMat/.{{x->-2.,y-> 14., z->6.}}]]

(DFGHMat= {{D[F[x,y,z],x],D[F[x,y,z],y],D[F[x,y,z],z]},{D[G[x,y,z],x],D[G[x,y,z],y],D[G[x,y,z],z]},{D[H[x,y,z],x],D[H[x,y,z],y],D[H[x,y,z],z]}})//MatrixForm

(NewtIter= (Factor[Simplify[{{x},{y},{z}}-Inverse[DFGHMat].FGHMat]]))//MatrixForm

NewtIter/.{x->-2.,y-> 14., z->6.}

(Subscript[x, 0]) = -2.;(Subscript[y, 0]) = 14.;(Subscript[z, 0])= 6.;

(Seq1 = SetPrecision[NestList[Flatten[NewtIter]/.{x->#[[1]] ,y-> #[[2]],z-> #[[3]]}&,{(Subscript[x, 0]),(Subscript[y, 0]),(Subscript[z, 0])},5],12])//MatrixForm

root1 =Seq1[[5]]

FGHMat/.{x->root1[[1]],y->root1[[2]], z->root1[[3]]}

FindRoot[{F[x,y,z]==0,G[x,y,z]==0,H[x,y,z]==0},{{x,-2.},{y,14.},{z,6.}}, WorkingPrecision->15]

(Subscript[x, 0]) =8.;(Subscript[y, 0]) = 5.;(Subscript[z, 0])= 7.;

(Seq2 = SetPrecision[NestList[Flatten[NewtIter]/.{x->#[[1]] ,y-> #[[2]],z-> #[[3]]}&,{(Subscript[x, 0]),(Subscript[y, 0]),(Subscript[z, 0])},5],12])//MatrixForm

root2 =Seq2[[5]]

FGHMat/.{x->root1[[1]],y->root1[[2]], z->root1[[3]]}

FindRoot[{F[x,y,z]==0,G[x,y,z]==0,H[x,y,z]==0},{{x,8.},{y,5.},{z,7.}}, WorkingPrecision->15]

(Subscript[x, 0]) =15.-20. I;(Subscript[y, 0]) = -14.+8. I;(Subscript[z, 0])= -50.-9. I;

(Chop[Seq3 = SetPrecision[NestList[Flatten[NewtIter]/.{x->#[[1]] ,y-> #[[2]],z-> #[[3]]}&,{(Subscript[x, 0]),(Subscript[y, 0]),(Subscript[z, 0])},15],12],(10^(-15))])//MatrixForm