(a) (x1, y1)=(2√2,0)
(b) (x1,y1) = (1 - √2/2,0)
Applying the Gauss-Newton Method to Minimize Distances to Circles
The Gauss-Newton method is a powerful technique for solving non-linear least squares problems. In this case, we're using it to find the point (x¯, y¯) that minimizes the sum of squared distances to three circles. Here's how we can carry out the first step for both scenarios:
a) Centers (0,1), (1,1), (0,-1) and all radii 1:
1. Define the objective function: We want to minimize the sum of squared distances from (x, y) to each circle. This can be written as:
f(x, y) = (x - 0)² + (y - 1)² + (x - 1)² + (y - 1)² + (x - 0)² + (y + 1)²
2. Calculate the Jacobian matrix: This matrix contains the partial derivatives of the objective function with respect to x and y.
J(x, y) = [2(x - 0) + 2(x - 1), 2(y - 1) + 2(y - 1) + 2(y + 1)]
3. Evaluate the residual vector: This vector represents the difference between the actual distances and the desired distance of 1.
r(x, y) = [sqrt((x - 0)^2 + (y - 1)^2) - 1, sqrt((x - 1)² + (y - 1)²) - 1, sqrt((x - 0)² + (y + 1)²) - 1]
4. Solve the linear system: We use the Jacobian and residual vector to solve for the update vector (δx, δy) that minimizes the objective function in the first iteration. This involves multiplying the Jacobian transpose by the residual vector and then inverting the resulting matrix:
δxδy = -J^T * r
5. Update the initial guess: Add the update vector to the initial guess to obtain the new approximation for (x, y):
(x1, y1) = (x0, y0) + (δx, δy)
b) Centers (-1,0), (1,1), (1,-1) and all radii 1:
Repeat steps 1-5 with the following modifications:
Objective function:
f(x, y) = (x + 1)² + (y - 0)² + (x - 1)² + (y - 1)² + (x - 1)² + (y + 1)²
Jacobian matrix:
J(x, y) = [2(x + 1) + 2(x - 1), 2(y - 0) + 2(y - 1) + 2(y + 1)]
Residual vector:
r(x, y) = [sqrt((x + 1)² + (y - 0)²) - 1, sqrt((x - 1)² + (y - 1)²) - 1, sqrt((x - 1)² + (y + 1)²) - 1]
These steps will provide you with the new points (x1, y1) in both scenarios for the first iteration of the Gauss-Newton method. Remember that this is just the first step, and the process needs to be iterated until the objective function converges to a minimum.