The Thomas method can be used to solve effectively the tridiagonal matrix equation.
Let us consider the matrix equation
- $${\bf{AY}} = {\bf{Z}}$$
with tridiagonal matrix A
- $${\bf{A}} = \left( {\matrix{ {{C_1}} & {{B_1}} & 0 & 0 & {...} & 0 \cr {{A_2}} & {{C_2}} & {{B_2}} & 0 & {...} & 0 \cr 0 & {{A_3}} & {{C_3}} & {{B_3}} & {...} & 0 \cr {...} & {...} & {...} & {...} & {...} & {{B_{N - 1}}} \cr 0 & 0 & 0 & 0 & {{A_N}} & {{C_N}} \cr } } \right)$$
$${\bf{Y}} = \left( \matrix{
{y_1} \hfill \cr
{y_2} \hfill \cr
... \hfill \cr
{y_N} \hfill \cr} \right)$$
$${\bf{Z}} = \left( \matrix{
{Z_1} \hfill \cr
{Z_2} \hfill \cr
... \hfill \cr
{Z_N} \hfill \cr} \right)$$
We can look for a solution in the form
- $${y_i} = {\alpha _{i + 1}}{y_{i + 1}} + {\beta _{i + 1}}$$
At first, from the boundary condition
$$\eqalign{
& {y_1} = {y_a} \cr
& {y_N} = {y_b} \cr} $$
we can calculate $\alpha_2$ and $\beta_2$ using Eq. (3). Then we can calculate $\alpha_{i+1}$ and $\beta_{i+1}$ recursively for $i=2,...,N-1$ (forward sweep) using the following recurrent formulas
- $$\eqalign{
& {\alpha _{i + 1}} = {{ - {B_i}} \over {{A_i}{\alpha _i} + {C_i}}} \cr
& {\beta _{i + 1}} = {{{Z_i} - {A_i}{\beta _i}} \over {{A_i}{\alpha _i} + {C_i}}} } $$
Since we know all coefficients $\alpha_i$ and $\beta_i$ for $i=2,...,N$ and we know $y_N$, we can calculate the solution $y_i$ from Eq. (3) for $i=N-1,...,2$ (backward sweep).