Numerical modeling of waves for a tsunami early warning system

Appendix C
Thomas algorithm
A tridiagonal system of equations is characterzied by a matrix which has
non zero elements only in the central diagonal and in its upper and lower
diagonal, an example of such a system is as follows
⎡
⎢
⎣
b1 c1 0 ... ... ... 0
a2 b2 c2 0 ... ... 0
0 a3 b3 c3 0 ... 0
... ... ... ... ... ... ...
0 ... ... 0 an−1 bn−1 cn−1
0 ... ... ... 0 an bn
⎤ ⎡
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎦ ⎣
x1
x2
x3
...
xn−1
xn
⎤ ⎡
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ = ⎢
⎥ ⎢
⎥ ⎢
⎦ ⎣
d1
d2
d3
...
dn−1
dn
⎤
⎥
⎦
(C.1)
where ai, bi and ci are the elements of the lower central and upper
diagonals of the matrix respectively; xi are the unknowns and di are the
elements of the vector known. The Gaussian elimination is an algorithm
which solves a system of linear equation finding the rank of a matrix and
calculating the inverse of an invertible square matrix. The simplified case
of tridiagonal system can be solved by means of the Thomas algorithm.
The solution of a system of n equations in n unknowns is obtained in O (n)
operations by using the Thomas algorithm instead of O (n 3 ).
The algorithm first eliminates the ai elements of the matrix by modifying
the coefficients as follows, denoting the modified coefficients with primes
and
c ′ ⎧
⎨
i =
⎩
c1
b1
ci
bi−c ′ i−1 ai
; i =1
; i =2, 3, ..., n − 1
121
(C.2)