The "heat equation" problem. In the following, I address a few remarks concerning implementation.

PART 1

Most mathematical functions concerning vectors were kept outside of the Vector class, including the dot product. This happend so that I could allow using vectors of different type (ex. int and double).
I added more mathematical functions to have a more complete functionality.
In addition, I added more class functions for comparing two vectors based on a tolerance. One function for comparing only the element-wise difference and another to compare the squared norm to the aforementioned tolerance.

PART 2

In the matvec method in Matrix class, I implemented a sparse matrix-vector multiplication based on the COO format that is facilitated by the key array<int, 2> of the map.

I slightly changed the algorithm of the pseudocode to increase efficiency at the cost of memory.

1 vector of length N to store the result of A.matvec(p_k) to avoid computing it twice.
2 scalars for the squared norms. Like that, I calculate each squared norm once instead of thrice.

U renamed the template<int n> class Heat into template<int n> class HeatnD
All Heat1D, Heat2D and HeatnD classes were made. I started with the first 2 unsure of whether there would be enough time for the template class.
Almost duplicate code exists between these classes, but moving the specific functions outside of the classes seemed counter-intuitive.
Functions print_solution and save_solution do what their titles suggest in COO format. Therefore, the distance between each node is 1 arbitrary unit and no dx. In addition, the homogeneous boundary conditions are not included. The decision of how to address these issues is left to the user.

One file and Makefile for simplicity.
C ++14 standards used.
By profiling with -pg, we can see information such as that the matvec function takes up most of the computation time. This shows the importance of the sparse matrix-vector multiplication.
The result were confirmed by viewing print-outs on the screen and by saving the solutions on a file and then plotting them with Gnuplot.