1
LECTURE 3.1B
Dr. Vyas
Overview:
•
Thomas Algorithm for solving system with a TriDiagonal Matrix
•
Optimizing the Matlab program for faster computation
•
Examples/Demos
I. Tridiagonal Matrix
1.
The system of linear algebraic equations resulting in a tridiagonal matrix
commonly occurs in scientific and engineering computation. For example, the
numerical solution to a second order linear ODE
1
( )
( )
( )
y
p t y
q t y
r t
′′
′
=
+
+
with
the boundary conditions,
( )
, ( )
y a
y b
α
β
=
=
is obtained by solving the following
system of linear equations, as represented in the matrix form:
2
1
1
1
2
1
2
1
1
2
2
2
2
2
2
2
1
1
2
2
2
2
2
2
2
1
1
1
2
2
1
0
0
1
2
1
0
0
1
2
1
0
0
1 2
N
N
N
N
N
N
N
h q
h p
y
h p
h q
h p
y
y
h p
h q
h p
y
h p
h q







+



+



+



+
M
M
2
1
0
2
2
2
2
2
1
1
N
N
N
h r
e
h r
h r
h r
e



+

=


+
M
2.
In order to solve a tridiagonal system of linear equations without having to resort
to a full fledged GaussElimination method or inversion of the coefficient matrix,
we need to come up with a procedure that eliminates the lower diagonal
(aka sub
diagonal) and converts the tridiagonal matrix to an upper triangular matrix
. This
procedure is well known as the Thomas algorithm
.
3.
Instead of following an outlined algorithm, the algorithm will be deduced from a
4 4
×
size matrix with usual symbolic notation for elements. Thereafter, a
tridiagonal system with numbers will be used to illustrate the implementation, and
finally a Matlab code will be constructed and verified.
4.
Consider the following tridiagonal system of equations.
11
12
1
1
21
22
23
2
2
32
33
34
3
3
43
44
4
4
0
0
0
0
0
0
a
a
x
b
a
a
a
x
b
a
a
a
x
b
a
a
x
b
⇔
=
AX = B
The vector
21
32
43
[
,
,
]
a
a
a
is a vector of
subdiagonal
elements, whereas the vector
12
23
34
[
,
,
]
a
a
a
is a vector of
superdiagonal
elements and
11
22
33
44
[
,
,
,
]
a
a
a
a
forms a vector
of
diagonal
(or main diagonal) elements.
1
The details of derivation and further discussion of numerical solution will be covered as part of the
section 9.9 in the textbook. Students will be required to recapitulate this algorithm at that time.
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5.
It helps to write the tridiagonal matrix into the equation form for a thorough
understanding of the algorithm derivation. To that end, let us write:
11 1
12
2
3
4
1
0
0
a x
a x
x
x
b
+
+
+
=
…………………………….(1)
21 1
22
2
23 3
4
2
0
a x
a x
a x
x
b
+
+
+
=
………………………….
.(2)
1
32
2
33 3
34
4
3
0
x
a x
a x
a x
b
+
+
+
=
………………………….
..(3)
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 Spring '08
 Vyas
 Math, Linear Algebra, Algebra, matlab, Equations, Matrices, ........., Diagonal matrix, Chess opening, Howard Staunton, Thomas Algorithm

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