Thomas algorithm

In Crank Nicolson solution procedure, we get a system of algebraic equations which assumes the form of a tridiagonal matrix problem. Here we shall discuss a very well known solution procedure known as Thomas algorithm (1949) which utilizes efficiently the advantage of the tridiagonal form. A tridiagonal system is:

The Thomas Algorithm is a modified Gaussian matrix-solver applied to a tridingonal system.

The idea is to transform the coefficient matrix into a upper triangular form. The intermediate steps that solve for x₁, x₂, ...x_N .

Change d_i and c_i arrays as

i = 2,3,....N

and

Similarly

i = 2,3,....N

and

At this stage the matrix is in upper triangular form. The solution is then obtained by back substitution as

and