Numerical Analysis Questions Long
The gradient descent method is an iterative optimization algorithm used to find the minimum of a function. It is particularly useful for unconstrained optimization problems where there are no constraints on the variables.
The basic idea behind the gradient descent method is to iteratively update the current solution by taking steps proportional to the negative gradient of the function at that point. The negative gradient points in the direction of steepest descent, so by moving in the opposite direction, we can approach the minimum of the function.
The algorithm starts with an initial guess for the solution and then iteratively updates it using the following update rule:
x_{k+1} = x_k - α * ∇f(x_k)
where x_k is the current solution, α is the step size (also known as the learning rate), and ∇f(x_k) is the gradient of the function at x_k.
The step size α determines the size of the steps taken in each iteration. If α is too large, the algorithm may overshoot the minimum and fail to converge. On the other hand, if α is too small, the algorithm may take a long time to converge. Choosing an appropriate step size is crucial for the success of the gradient descent method.
An example of using the gradient descent method for unconstrained optimization is minimizing the function f(x) = x^2. Let's start with an initial guess x_0 = 5 and a step size α = 0.1. We can calculate the gradient of the function as ∇f(x) = 2x.
Iteration 1:
x_1 = x_0 - α * ∇f(x_0)
= 5 - 0.1 * 2 * 5
= 5 - 1
= 4
Iteration 2:
x_2 = x_1 - α * ∇f(x_1)
= 4 - 0.1 * 2 * 4
= 4 - 0.8
= 3.2
Iteration 3:
x_3 = x_2 - α * ∇f(x_2)
= 3.2 - 0.1 * 2 * 3.2
= 3.2 - 0.64
= 2.56
We can continue this process until we reach a desired level of accuracy or convergence.
In this example, the gradient descent method will converge to the minimum of the function f(x) = x^2, which is x = 0. As the algorithm progresses, the steps taken become smaller as the gradient decreases, allowing the algorithm to converge towards the minimum.