Matrix calculation

A large number of mathematical and applied problems involve studying (and solving) systems of linear equations. The system (S) :

(where the unknowns are the numbers and where the numbers a _ij and b _i are given in ) is also noted, in matrix form :

or more compactly:

AX = B

(this equality means nothing more than the n equalities of system (S)).

The following example illustrates this situation.

Consider the differential equation y'' + q (x) y = f (x), found in strength of materials with boundary conditions of the form y (a) = A, y (b) = B, the functions q and f being continuous on [a, b]. This boundary problem admits a unique solution y, whose approximations can be sought as follows. We establish a mesh of the segment [a, b] by subdividing it into n + 1 equal sub-segments whose extremities are denoted a = x ₀ , x ₁ , ..., x _n , x _{n + 1} = b. We know...