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I've got a complicated expression, which looks like: e(4ik+3iω+ip)−e(4ik+2iω+2ip)−e(4ik+2iω)+e(4ik+iω+ip)+2e(2ik+4iω+ip)−e(2ik+3iω+2ip)−e(2ik+3iω) + ...

How would you do to factor the exponentials with the same power of k, i.e. e(4ik)(...) + e(2ik)(...) and so on?

Let $f=f(x_1(t),x_2(t))$ be defined as follows:

sage: var('t mu')
sage: x=list()   
sage: x.append( function('x1')(t) )
sage: x.append( function('x2')(t) )
sage: f = x[0] - mu*x[0]^2*x[1]
sage: f

$f = x_1(t) - \mu x_1(t)^2 x_2(t)$.

Now, I need to to substitute the term containing $x_1^2(t) x_2(t)$ in $f$ by some new auxiliary variable $x_3(t)$, that is, to obtain $f = x_1(t) - \mu x_3(t) $. However, this code doesn't work:

sage: x.append( function('x3')(t) )
sage: f.subs({x[0]^2*x[1] : x[2]})

Some ideas? Thanks.

This code is motivated by symbolic manipulation with ODEs (think of $f$ as being the right-hand side term in the autonomous ODE system $\dot{x}(t) = f(x(t))$ ).

For the OP: Sage ships with the command oeis to play with directly in the REPL, eg.

sage: oeis([1, 4, 36, 288, 7200], max_results=2)
0: A277174: a(n) = Product_{i=1..n} i*rad(i) where rad(n) = A007947(n).
1: A316297: a(n) = n! times the denominator of the n-th harmonic number H(n).

minimize_constrained expects [x,0.5,0] to be the initial point for finding the minimum, but the 1st entry, x, is a symbolic expression. Perhaps you can let min_fx = lambda x: minimize_constrained(f, [c_1,c_2,c_3,c_4], [x, 0.5, 0], algorithm='l-bfgs-b') and then operate on min_fx?

I couldn't resist to comment that the vector calculus example notebooks are delightful!

I want to run the following code (I'm using SAGE v6.10 release version 2015-12-18):

P = MixedIntegerLinearProgram()
x = P.new_variable()
A = random_matrix(RR, 3, 2); 
P.add_constraint(A*x <= [2.1,1.5,0.4]) 
P.polyhedron()

However, it outputs the error message:

AttributeError: type object 'float' has no attribute 'fraction_field'

I noticed that if instead of using real variables I use only integer variables, then it works fine. For instance:

P = MixedIntegerLinearProgram()
x = P.new_variable()
A = random_matrix(ZZ, 3, 2); 
P.add_constraint(A*x <= [2,1,0]) 
P.polyhedron()

outputs:

A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 1 vertex and 2 rays

What do you suggest to overcome this problem?

Thanks

Thanks for finding and fixing the bug. Feel free to CC me when you create the ticket.

For numerical integrals in Sage, use numerical_integral. The function integrate is more appropriate for symbolic integration.

With your example:

sage: import scipy.stats as st
sage: numerical_integral(lambda x : st.norm.pdf(x), (-oo, 0.5))
(0.6914624612740132, 1.580169726891987e-09)

The first component is the value of the integral, and the second component is an error estimate.

As far as i'm aware, Sage does not expose a multi-dimensional numerical integration function. This can be worked around by a nested call to numerical_integral, but it will not be as efficient as it could be for some applications. There are more examples in Mathematical Computation with SageMath, Ch. 14, page 315.

what optimisation algorithms are available.

Check those provided by SciPy's optimization and root finding library. All of them are available through the Sage interpreter by calling the appropriate scipy.optimize function. For example, the basin-hopping algorithm finds the global minimum of a scalar function:

sage: import numpy as np
sage: from scipy.optimize import basinhopping
sage: func = lambda x: np.cos(14.5 * x - 0.3) + (x + 0.2) * x
sage: x0=[1.]
sage: minimizer_kwargs = {"method": "BFGS"}
sage: ret = basinhopping(func, x0, minimizer_kwargs=minimizer_kwargs, niter=200)

                                              fun: -1.0008761844426548
          lowest_optimization_result:       fun: -1.0008761844426548
...

In some (but not all) cases, Sage provides an interface to these algorithms, so that you can directly use Sage symbolic functions as arguments.

It is worth mentioning that "out there" you'll find dozens of Python libraries for mathematical optimization (*). Notice that standard Python packages can be installed with the terminal command

$ sage --pip install --user package_name

and if the installation goes well, it is available the next time you start Sage via import package_name.

(*) See eg.:

• pymeigo, docs