A system of simultaneous algebraic equations with nonzero coefficients only on the main diagonal, the lower diagonal, and the upper diagonal is called a tridiagonal system of equations. Consider a tridiagonal system of N equations with N unknowns, u1, u2, u3,… uN as follows:

A standard method for solving a system of linear, algebraic equations is Gaussian elimination. Thomas’ algorithm, also called the tridiagonal matrix algorithm (TDMA) is essentially a shortened variant of the Gaussian elimination method to solve the tridiagonal system of equations.

The ith equation in the system may be written as

where a1 = 0 and bN = 0. Looking at the system of equations, we see that the ith unknown can be expressed in terms of (i+1)th unknowns. That is,

where Pi and Qi are constants. Note that if all the equations in the system are expressed in this fashion, the coefficient matrix of the system would transform to upper triangular matrix.

To determine the constants Pi and Qi, we plug Equation (A.4) in (A.2) to yield

These are the recurring relations for the constants P and Q. It shows that Pi can be calculated if Pi-1 is known. To start the computation, we use the fact that a1 = 0. Now, P1 and Q1 can be easily calculated because terms involving P0 and Q0 vanish. Therefore,

Once the values of P1 and Q1 are known, we can use the recurring expressions for Pi and Qi for all values of i.

To start the back substitution, we use the fact that bN = 0. As a consequence, from Equation (A.6), we have PN = 0. Therefore,

Once the value of uN is known, we use Equation (A.3) to obtain uN-1, uN-2,…u1.