Solving equation numerical using lipschitz

[Larsson]

Homework Statement

This problem is about solving an equation system numerical using lipschitz method or whatever the name is.

x_1 = sqrt(1-x^2)
x_2 = sqrt((9-5x_1^2)/21)

and we create
(x_1,x_2) = x
g(x) = (x_1,x_2) (fixed point equation)

lipschitz states:
||x^(k+1) - x^*|| <= L*||x^k - x^*||
0<L<1

where x^* is the fix-point, which means x^* = g(x^*)

(k is not an exponent, it's indexation)

Homework Equations

Se above

The Attempt at a Solution

The problem now is to really get started and doing the iterations, and that's where I'm kind of stuck. I suppose I'll have to start with a guess on x_1 and x_2, so lets.

(x_1)^(k =0) = 1
(x_2)^(k =0) = 1

I try using the lipschitz for x_1
||x_1^(k =2) - sqrt(1-(x_2)^2) || <= L* ||1 - sqrt(1-(x_2)^2) ||
But this just looks like crap to me. From this I want to solve x_1^(k = 2), but I don't see how that's done. And I'm not even sure this is the right setup with lipschitz. I don't see how the iteration is supposed to be done, and all I know about L is that it's between 0 and 1, what value should L have?. Could someone please show me the first the of the iteration so I see how it works?

 

[mighty2000]

I understand your confusion with using the Lipschitz method to solve this equation system. The Lipschitz method is a numerical method used to solve fixed-point equations, which is what we have in this problem. Let me walk you through the steps of using this method to solve the given equation system.

First, let's define the fixed-point equation, as you have already done:

g(x) = (x_1,x_2) = (sqrt(1-x^2), sqrt((9-5x_1^2)/21))

The goal of the Lipschitz method is to find a fixed point of this equation, denoted by x^*. This means that x^* = g(x^*). Now, we will use the Lipschitz condition to find this fixed point.

The Lipschitz condition states that for a given function g, there exists a constant L such that for any two points x and y, the distance between g(x) and g(y) is less than or equal to L times the distance between x and y. This can be written as:

||g(x) - g(y)|| <= L*||x - y||

Now, in our case, we want to find a fixed point x^* = (x_1^*, x_2^*) such that x^* = g(x^*), or in other words, g(x^*) - x^* = 0. So, we can rewrite the Lipschitz condition as:

||g(x^*) - x^*|| <= L*||x^* - x^*||

Since the distance between a point and itself is always 0, this simplifies to:

||g(x^*) - x^*|| <= 0

This means that the fixed point x^* must be a point where g(x^*) = x^*. So, our goal now is to find this point.

The Lipschitz method uses iterations to approximate the fixed point x^*. We start with an initial guess for x^*, denoted by x^(0). In this case, you have chosen x^(0) = (1,1). Then, we use the fixed-point equation g(x) to calculate a new point, denoted by x^(1):

x^(1) = g(x^(0)) = (sqrt(1-(x^(0))^2), sqrt((9-5(x