5. Hollow dies

5.1 Large problems in linear algebra

Problems obtained by discretizing partial differential equations easily generate gigantic matrices. For example, discretization of the heat equation in dimension 3 on a parallelepiped [0, 1] ³ requires the domain to be cut along three directions; if we take as unknowns the temperature at each point of the discretized cube, we have N ³ unknowns if each side of the cube is discretized into N points. We'll also have as many (linear) equations by writing the discrete approximations of the Laplacian and partial derivatives,...