139_Thomas Algorithm

A tridiagonal matrix is a square matrix in which only the main diagonal and the diagonals immediately above and below the main diagonal contain non-zero elements, with all other elements being zero.

 

 

In this post, I'll discuss the Thomas algorithm, a method for finding solutions to systems involving such tridiagonal matrices without having to compute the inverse of matrix A. This algorithm can conveniently work with multiple right-hand side vectors and compute solutions without the need to calculate the inverse matrix.

 

 

First, the decomposition can be compute as follows.

 

 

Therefore, 

 

 

Therefore, the matrix can be represented as follows.

 

 

If the same matrix is represented using LU decomposition, it would be as follows.

 

 

The values can be seen to match those of the LU decomposition. Forward substitution can be represented as follows.

 

 

Then, the solution can be obtained through back substitution using the $U$ matrix.