RE: 2 dimension Newton Raphson

• To: mathgroup at smc.vnet.net
• Subject: [mg71245] RE: [mg71218] 2 dimension Newton Raphson
• From: "David Park" <djmp at earthlink.net>
• Date: Sat, 11 Nov 2006 03:39:17 -0500 (EST)

```This is a pretty simple problem. In the first place the equations can be
directly solved.

eqns = {(x - 4)^2 + (y - 4)^2 == 5, x^2 + y^2 == 16};

Solve[eqns]
{{x -> (1/16)*(43 - Sqrt[199]), y -> (1/16)*(43 + Sqrt[199])},
{x -> (1/16)*(43 + Sqrt[199]), y -> (1/16)*(43 - Sqrt[199])}}

You could also turn it into a 1-dimensional root search. I use Ted Ersek's
RootSearch package from MathSource because it is quite convenient here. We
parametrize the circle in terms of t and then travel around it looking for
roots of the first equation.

Needs["Ersek`RootSearch`"]

f[t_] = Simplify[(x - 4)^2 + (y - 4)^2 - 5 /.
{x -> 4*Cos[t], y -> 4*Sin[t]}]
43 - 32 Cos[t] - 32 Sin[t]

Plot[f[t], {t, 0, 2Pi}];

tsols = RootSearch[f[t] == 0, {t, 0, 2Pi}]
xysols = {x -> 4Cos[t], y -> 4Sin[t]} /. tsols
{{t -> 0.468398}, {t -> 1.1024}}
{{x -> 3.56917, y -> 1.80583}, {x -> 1.80583, y -> 3.56917}}

eqns /. xysols
{{True, True}, {True, True}}

David Park

From: ms z [mailto:ms-z- at hotmail.com]
To: mathgroup at smc.vnet.net

I have tried to solve the roots of the simultaneous nonlinear equations
(x-4)^2 + (y-4)^2 = 5
x^2 + y^2 = 16

by writing this function:

nr2method[xl1_, xl2_, es1_] :=
Block[{x1, x2, ea, es, x1new, u, v},
u = (x1 - 4)^2 + (x2 - 4)^2 - 5;
v = x1^2 + x2^2 - 16;
ea = 100; es = es1;
For[i = 1, ea > es, i++,
(x1new[x1_, x2_] = x1 - (u*D[
v, x2] - v*D[u, x2])/(D[u, x1]*D[v, x2] - D[u, x2]*D[v, x1]);
If[i == 1, x1 = xl1, x1 = b];
x2 = xl2;
b = x1new[x1, x2];
ea = Abs[(b - x1)/b 100];
Clear[x1, x2, x1new];)];
ea = 100; es = es1;
For[i = 1, ea > es, i++,
(x2new[x1_, x2_] = x2 - (v*D[u, x1] - u*D[v, x1])/(D[u, x1]*D[v, x2] -
D[u, x2]*D[v, x1]);
If[i == 1, x2 = xl2, x2 = c];
x1 = xl1;
c = x2new[x1, x2];
ea = Abs[(c - x2)/c 100];
Clear[x1, x2, x2new];)];
Print["The value of x1 is ", b];
Print["The value of x2 is ", c];]

Is this function a good one? Is there a way to make this function simpler?

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```

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