Qfyilyi

I have been given a project, I need to show the use of a version of Newton's method to solve these non-linear equations. The version of Newton's method I am required to use is: $ X_{n+1} = X_n - J^{(-1)} F(X_n) $. I have all the values required here, and this works to find the point $ X_1 $. However I need to find a code that inputs the following points $ X_2, X_3, ..., X_n $ automatically. Here are the given values:

f[x_, y_] := x^2 + y^2 - 5;

g[x_, y_] := x^3 - y^3 - 7;



x0 = 2.1;

y0 = 0.9;



f[x0, y0]

g[x0, y0]

0.22

1.532

M = {{2*x0, 2*y0}, {3*x0^2, -3*y0^2}}

{{4.2, 1.8}, {13.23, -2.43}}

J = Inverse[M]

{{0.0714286, 0.0529101}, {0.388889, -0.123457}}

F0 = {{f[x0, y0]}, {g[x0, y0]}}

X0 = {{x0}, {y0}}

X1 = X0 - J.F0

{{0.22}, {1.532}}

{{2.1}, {0.9}}

{{2.00323}, {1.00358}}

Thank you in Advance!

jac = D[{f[x, y], g[x, y]}, {{x, y}, 1}];

Xlist = NestList[# - Inverse[jac /. Thread[{x, y} -> #]].{f @@ #, g @@ #} &, {x0, y0}, 5]

{{2.1, 0.9}, {2.0032275, 1.0035802}, {2.0000033, 1.0000051}, {2., 1.}, {2., 1.}, {2., 1.}}

You can get the same from FindRoot:

{res, {steps}} = Reap[FindRoot[{f[x, y], g[x, y]}, {{x, x0}, {y, y0}}, 

                       Method -> "Newton", StepMonitor :> Sow[{x, y}]]]

{{x -> 2., y -> 1.}, {{{2.0032275, 1.0035802}, {2.0000033, 1.0000051}, {2., 1.}, {2., 1.}}}}

I'll show it with FixedPointList

f[x_, y_] = {x^2 + y^2 - 5, x^3 - y^3 - 7};

j[x_, y_] = Grad[f[x, y], {x, y}];

with LinearSolve:

FixedPointList[(# - LinearSolve[j[Sequence @@ #], f[Sequence @@ #]]) &, {2.1, 0.9}, 3]

{{2.1, 0.9}, {2.00323, 1.00358}, {2., 1.00001}, {2., 1.}}

with Inverse:

FixedPointList[(# - Inverse[j[Sequence @@ #]].f[Sequence @@ #]) &, {2.1, 0.9}, 3]

{{2.1, 0.9}, {2.00323, 1.00358}, {2., 1.00001}, {2., 1.}}