Tridiagonal Matrix and the Thomas Algorithm

The tridiagonal matrix is a square matrix in which all elements not on the diagonal line and the two lines parallel to the diagonal line are zero. Elements on these three lines may or may not be zero. For example, a 7 x 7 tridiagonal 

With this special form of the tridiagonal matrix, the Thomas algorithm (Von Rosenberg 1969) can be used to effect a solution. The algorithm for a square N XN tridiagonal matrix is simple and is given as 

Step 2. The values of the dependent variables are calculated from 

coming back to the problem at hand in the last section, let us choose five intervals that is, V = 5 and Ax = 1/5 = 0.2. The five linear equations are (written in matrix form) 

The first equation comes from the boundary condition at x = 0 (Eq. 12.123), the next four come from Eq. 12.1206, and the last comes from the boundary condition at x = 1 (Eq. 12.124). Solving this set of equations using the Thomas algorithm just presented will yield solutions for discrete values y, to y. Table 12.1 shows these approximate values and the exact solution, which is 

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