This git repository is an implementation of polynomials and their use for interpolation, in the context of a work for undergraduate courses in France (TIPE in Prepa). It is meant to be use with Pycharm (community version, pure Python), but you are free to use the IDE you prefer (Pyzo, etc.).

The final report (in french) is accessible here, I might do a latex version on day.

I introduced polynomials as lists of coefficients, and defined on them in polynomials.py. A few utils are also introduced there : polynomial sum, polynomial constant, polynomial product, polynomial image, polynomial integral and derivate function, polynomial display with matplotlib.pyplot.

In the file Lagrange.py, we introduce the Lagrange polynomials, and their representation. Here are the function's specs :

• lagrange_polynomial_i(xcoords, i) : calculate Lagrange's ith polynomial in the Lagrange's formula for xcoords
• lagrange_polynomial(xcoords, ycoords) : calculate Lagrange's polynomial for xcoords, ycoords
• lagrange_graph(xcoords, ycoords) : calculate and then display Lagrange's polynomial on xcoords
• lagrange_graph_function(xcoords, f) : do the preveious operation with ycoords = map f xcoords
• lagrange_tchebychev(f, a, b, n) : calculate Tchebychev's coordinates (for degree n), and calculate Lagrange's polynomial for these xcoords (they should limit the polynomial's sup between a and b for all n degree polynomials)
• lagrange_tchebychev_graph(f, a, b, n) : display the previous polynomial on generated xcoords
• newton_cotes(f, a, b, n) : calculate the n-th Newton-Côtes formula thanks to Lagrange's polynomials (integral approximation)
• tchebychev_integral(f, a, b, n) : calculate the function's integral approximation thanks to its n-th degree tchebychev polynomial calculated between a and b

In the file hermite.py, it is quite similar to Lagrange's functions, except this time it is done for Hermite's interpolation

In the file scatter_interpolation.py, I defined a simple function lagrange_tchebychev_pts(xcoords, ycoords, n) to calculate the nearest points to n degree Tchebychev points between min and max of xcoords, and interpolate those points with a Lagrange polynomial, to then display it.

The goal is to define a good polynomial interpolation for a set of points.

In the file lagrange2D.py, I tried to simply interpolation according to two directions, x and y, and then applied it to a picture where I set some colors at the extremal points of the picture, and then interpolated the colors between them, which gave the following result.

I could try it with l'Hermite's interpolation one day.

In the file cubic_splines.py, I defined cubic_splines(f, a, b, n), which displays the n subdivision cubic splines interpolation on function f between a and b (and an alternative definition cubic_splines2, as well as in another file, but it is not that important).

I also defined a function interp_splines(f, a, b, n) to approximate the integral of f between a and b with the sum of the splines integrals.

I compared all those methods, plus truncated power series, in the file integral_comparison.py, on well known functions, such as the exponential, cosh, sinh, cos and sin.

You just need to run the file (after having run the other files to access their definitions), and then choose the degree n, and the interval [a , b].

The associated report is in french, which does not help to understand. I might work at it when I have time, but for the moment it will stay like that.

If you have any recommandation, do not hesitate to message me or to submit a pull request.