Appendix: Tridiagonal Matrix Algorithm

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,… u N 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 i th equation in the system may be written as where a 1 = 0 and b N = 0. Looking at the system of equations, we see that the i th unknown can be expressed in terms of (i+1) th unknowns. That is, where P i and Q i 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 P i and Q i , we plug Equation (A.4) in (A.2) to yield These are the recurring relations for the constants P and Q. It shows that P i can be calculated if P i-1 is known. To start the computation, we use the fact that a 1 = 0. Now, P 1 and Q 1 can be easily calculated because terms involving P 0 and Q 0 vanish. Therefore, Once the values of P 1 and Q 1 are known, we can use the recurring expressions for P i and Q i for all values of i . To start the back substitution, we use the fact that b N = 0. As a consequence, from Equation (A.6), we have P N = 0. Therefore, Once the value of uN is known, we use Equation (A.3) to obtain u N-1 , u N-2 ,…u1 .