Description
Prof. Boris Houska
1. Let us apply Newton type method for solving the unconstained optimization problem
min F(x) for. x
• What is the gradient of the function F at a given point x?
• What is the Hessian matrix of F at a given point x? • We implement a Newton type method method of the form
xk+1 = xk− M−1∇F(xk)
with the constant Hessian approximation
.
What is the local convergence rate of this method?
2. Find the explicit solution x(t) of the following scalar differential equations:
• x˙(t) = 2 with x(0) = −1,
• x˙(t) = −x(t) with x(0) = 1,
• x˙(t) = −x(t) + 1 with x(0) = 0,
• with x(0) = 1,
• x˙(t) = −sin(t) with x(0) = 1.
3. Implement the standard Runge-Kutta method of order 4 with constant step size h for the differential equation
∀t ∈ [0,10], with x1(0) = x2(0) = 1.
for h = 0.1, h = 0.01 and h = 0.001. Plot your results. Can you estimate the numerical approximation error, maxk∥yk − x(tk)∥∞ for 0 ≤ tk = 0.01 ∗ k ≤ 10, of the approximate solution (y0,y1,…,y1000) for h = 0.01 by using your result for the step size h = 0.001? Justify your arguments.
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