optimize the integral of x from -1 to 1 using chebyshev polynomials in casadi in python

To optimize the integral of x from -1 to 1 using Chebyshev polynomials in casadi, we can use the casadi.cheb function to generate the Chebyshev basis polynomials, and then use these polynomials to approximate the integrand. We can then use casadi.integral to compute the integral of the approximation over the domain [-1,1].

Here's some example code that demonstrates how to do this:

main.py
import casadi as cs
import numpy as np

# Define the degree of the Chebychev approximation
degree = 5

# Define the domain of integration
a = -1.0
b = 1.0

# Create the Chebyshev basis polynomials
basis = cs.cheb(degree+1)
n = basis.shape[1]

# Create the weighting function
w = cs.sum2(cs.mtimes(cs.T(basis),basis))

# Create the integration function for our test function
x = cs.MX.sym('x')
f = x
g = cs.Function('g',[x],[f])
integrand = cs.mtimes(g(basis),(b-a)/2)*w
integral_func = cs.Function('integral_func',[basis],[integrand])

# Create the optimization problem
x = cs.MX.sym('x')
obj = integral_func(x)
nlp = {"x": x, "f": obj}
solver = cs.nlpsol('solver', 'ipopt', nlp)

# Solve the optimization problem
sol = solver(x0=np.zeros(n))

# Extract the optimal coefficients
coeffs = sol['x']

# Create the final Chebyshev approximation function
final_func = cs.Function('final_func',[x],[mtimes(basis,coeffs)])

# Compute the integral of the approximation
result = cs.integral(integrand, a, b)

print(f"The approximated integral is: {result}")
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In this example, we first define the degree of the Chebyshev approximation (the number of polynomials to use), and the domain of integration. We then create the Chebyshev basis polynomials using casadi.cheb, and the weighting function using cs.sum2(cs.mtimes(cs.T(basis),basis)).

We then define our test function (f = x) and create an integration function integrand using the Chebyshev basis polynomials and the weighting function. We also create a cs.Function object for this function so we can evaluate it later.

Next, we define the optimization problem using the integrand function as the objective function. We use the ipopt solver to solve the problem.

We extract the optimal coefficients from the solution and create a final Chebyshev approximation function (final_func) using these coefficients and the Chebyshev basis polynomials.

Finally, we compute the integral of the approximation using cs.integral.