Thomas algorithm proof and theory

1. THOMAS ALGORITHM
Tri-Diagonal Matrix

2. Matrix
Add a subheadingMatrix, a set of numbers arranged in rows and
columns so as to form a rectangular array. The numbers are
called the elements, or entries, of the matrix.

3. Banded Matrix
A band matrix is a type of sparse matrix where its non-zero values are concentrated within a diagonal band, which
includes the central diagonal as well as adjacent diagonals on both sides.

4. The matrix may exhibit symmetry, where it possesses an
equal count of sub-diagonals and super-diagonals. In the
case of a matrix containing just one sub-diagonal and one
super-diagonal, it qualifies as a tridiagonal matrix. The
quantity of super-diagonals is termed the upper bandwidth
(illustrated as two in the example), while the number of sub-
diagonals is referred to as the lower bandwidth (depicted as
three in the example). The combined total of diagonals,
which amounts to six in this instance, is known as the
bandwidth.
Banded matrix

5. TriDigonal Matrix
A tridiagonal matrix is a band matrix that has nonzero elements only on the main
diagonal, the first diagonal below this, and the first diagonal above the main
diagonal.

6. TriDigonal Matrix Algorithim
• Thomas’s algorithm also called TriDiagonal Matrix Algorithm (TDMA) is
essentially the result of applying Gaussian elimination to the tridiagonal
system of equations
• 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

7. Generalising TriDigonal Matrix
Consider a tridiagonal system of N equations with N
unknowns, u1, U2, U3, " Uvas given below:

8. Generalising TriDigonal Matrix
Note: a1 = 0 & CN = 0
The matrix can be written as
Looking at the system of equations, we see that unknown can
be
expressed as a function of (i + 1)th unknown. That is

9. Let us consider the system of
equations
Lets solve a problem for clear
understanding..!!

10. Stage 1: Converting Mx = r into Ux=p

15. Solution
X1 = 0.952
X2 = 3.857
X3 = 3.619