Update README.md

Author: lorenzotomada

README.md

@@ -22,13 +22,19 @@ We implemented various GitHub workflows, which include unit testing, documentati
 3. After each push, the code is automatically formatted using the `black` formatter.
 
 ## Where to find important files
-All the important files are in the `src/pyclassify folder`. In the root directory, the only interesting files are the `CMakeLists.txt` and the `setup.py`. While it is going to be deprecated, the `setup.py` file made it easier to automatically compile the library during installation and to deal with external dependencies, e.g. `Eigen`.
+All the important files are in the `src/pyclassify folder`. In the root directory, the only interesting files are the `CMakeLists.txt` and the `setup.py`. Notice that the `setup.py` was added to the `pyproject.toml` file as it made it easier to automatically compile the library during installation and to deal with external dependencies, e.g. `Eigen`.
 
 In the `src/pyclassify` folder, the file `utils.py` contains some helper functions, e.g. the ones need to check that a matrix is of the correct shape.
 The `cxx_utils.cpp` file contains the implementation in `C++` of some functions that are needed in the divide and conquer algorithm (e.g. the implementations of deflation, QR method and secular solvers).
 In addition, the `parallel_tridiag_eigen.py` contains the actual implementation of the divide and conquer method, while `eigenvalues.py` contains the implementation of the Lanczos algorithm.
 The `zero_finder.py` just consists of a first implementation of some of the functions in `cxx_utils.cpp` and has not been removed since it is used in tests to ensure that the `C++` implementation is correct.
 
+## What did we implement?
+In order to solve an eigenvalue problem, we considered multiple strategies.
+1. The most trivial one was to implement the power method in order to be able to compute (at least) the biggest eigenvalue. We then used `numba` to try and optimize it, but in this case just-in-time compilation was not extremely beneficial.The implementation of the power method is contained in `eigenvalues.py`.
+2. Lanczos + QR: this is an approach (tailored to the case of symmetric matrices) to compute *all* the eigenvalues and eigenvectors. Notice that, also in the case of the QR method,`numba` was not very beneficial in terms of speed-up, resulting in a pretty slow methodology. For this reason, we implemented the QR method in `C++` and used `pybind11` to expose it to `Python`. All the code written in `C++` can be found in `cxx_utils.cpp`.
+3. `CuPy` implementation of all of the above: we implemented all the above methodologies using `CuPy` to see whether using GPU could speed up computations. Since this was not the case, we commented all the lines of code involving `CuPy`, so that installation of the package is no longer required and we can use our code also on machines that do not have GPU.
+4. The core of the project is the implementation (as well as a generalization of the simplified case in which $\rho=\$ considered in our reference) of the _divide et implera_ method for the computation of eigenvalues of a symmetric matrix. Some helpers were originally written in `Python` and then translated to `C++` for efficiency reasons: their original implementation is in `zero_finder.py` and is still present in the project for testing purposes. The translated version can be found in `cxx_utils.cpp`. Instead, the implementation of the actual method to compute the eigenvalues starting from a tridiagonal matrix is contained in `parallel_tridiag_eigen.py` and makes use of `mpi4py`.
 
 # Results
 The results of the profiling (runtime vs matrix size, memory consumption, scalability, and so on) are discussed in detail in `Documentation.ipynb`.