# one needs to define $f$ as a function of a vector
# f(v)=1-(v[0]-x_0)^2-0.5*(v[1]-y_0)^2$ Methods
Optimization and root finding
Exercise 1 Consider the function \(f(x,y)=1-(x-x_0)^2-0.5(y-y_0)^2\) with \(x_0=0.5\) and \(y_0=1.0\). Check the documentatin for scipy.optimize. Use it to maximze function \(f\).
Consider the function \(g(x)=0.1+exp(-x)x(1-x)\) over [0,2]. Choose the scipy function and find the root of \(g\).
#Consider the function \(h(x,y)=0.1+exp(-x)x(1-x)\) over [0,2]. Choose the scipy function and find the root of \(g\). to find the root of \(g\)?
#Interpolation
We consider the function \(f(x) = sinc(\lambda x) = \frac{sin(\lambda x)}{x}\). Let \(I=(x_i)_{i=[1,10]}\) be a regularly spaced interval between -2 and +2, containing 10 points. Call \(Y=(y_i)=f(x_i)\) the values of \(f\) on this interval. Let \(T\) be a test set with 1000 regularly spaced points between -2.5 and 2.5.
The goal is to compare several ways interpolate function f on \(T\).
Exercise 2 Define f, I, Y, T with numpy.
# Exercise 3 Construct a stepwise approximation using numpy indexing
# Exercise 4 Plot it
# Exercise 5 Construct a linear approximation using numpy
#Exercise 6 Use scipy.interpolate to interpolate the data linearly. Compare the various extrapolation options.
#Exercise 7 Use scipy.interpolate to interolate the data with cubic splines. Compare the various extrapolation options.
#Exercise 8 Plot the results
#Discretization
Exercise 9 Consider the AR1 process \(y_t = \rho y_{t-1} + \epsilon_t\) where \(\rho=0.9\) and \(\epsilon_t=0.01\). Use the quantecon library to discretize \((y_t)\) as a discrete markov chain.
#Exercise 10 Suppose \(\epsilon\) follows a normal law with standard deviation \(σ = 0.05\). Take γ = 40 and define \(U(x)=(x^{-γ})/(-γ)\) We want to compute \(C(\epsilon) = \mathbb{E} [U(exp(\epsilon)) ]\).
- Choose \(N>0\) and construct a 1d vector with \(N\) realizations of \(\epsilon\). Use it to compute the expectation.
- Estimate the standard deviation of this expectation.
- Use gauss-hermite method from numpy to compute the same expectation.
- Compare both methods.