Chapter 11
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Special Matrices and Gauss-Seidel
Chapter 11
• Certain matrices have particular structures that can be
exploited to develop efficient solution schemes.
– A banded matrix is a square matrix that has all elements equal to zero,
with the exception of a band centered on the main diagonal. These
matrices typically occur in solution of differential equations.
– The dimensions of a banded system can be quantified by two
parameters: the band width BW and half-bandwidth HBW. These two
values are related by BW=2HBW+1.
• Gauss elimination or conventional LU decomposition methods
are inefficient in solving banded equations because pivoting
becomes unnecessary.
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Figure 11.1
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Tridiagonal Systems
• A tridiagonal system has a bandwidth of 3:
 f1
e
 2



g1
f2
g2
e3
f3
e4
  x1   r1 
  x  r 
  2    2 
g 3   x3  r3 
   
f 4   x4  r4 
• An efficient LU decomposition method, called Thomas
algorithm, can be used to solve such an equation. The
algorithm consists of three steps: decomposition, forward
and back substitution, and has all the advantages of LU
decomposition.
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Gauss-Seidel
• Iterative or approximate methods provide an
alternative to the elimination methods. The
Gauss-Seidel method is the most commonly
used iterative method.
• The system [A]{X}={B} is reshaped by solving
the first equation for x1, the second equation
for x2, and the third for x3, …and nth equation
for xn. For conciseness, we will limit ourselves
to a 3x3 set of equations.
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b1  a12 x2  a13 x3
x1 
a11
b2  a21 x1  a23 x3
x2 
a22
b3  a31 x1  a32 x2
x1 
a33
•Now we can start the solution process by choosing
guesses for the x’s. A simple way to obtain initial
guesses is to assume that they are zero. These zeros
can be substituted into x1equation to calculate a new
x1=b1/a11.
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• New x1 is substituted to calculate x2 and x3. The
procedure is repeated until the convergence criterion
is satisfied:
 a ,i
xij  xij 1

100%   s
j
xi
For all i, where j and j-1 are the present and previous
iterations.
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Fig. 11.4
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Convergence Criterion for GaussSeidel Method
• The Gauss-Seidel method has two fundamental
problems as any iterative method:
– It is sometimes nonconvergent, and
– If it converges, converges very slowly.
• Recalling that sufficient conditions for convergence
of two linear equations, u(x,y) and v(x,y) are
u u

1
x y
v v

1
x y
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• Similarly, in case of two simultaneous equations,
the Gauss-Seidel algorithm can be expressed as
b1 a12
u ( x1 , x2 ) 

x2
a11 a11
b2 a21
v( x1 , x2 ) 

x1
a22 a22
u
0
x1
u
a12

x2
a11
v
a
  21
x1
a22
v
0
x2
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• Substitution into convergence criterion of two linear
equations yield:
a12
1
a11
a21
1
a22
• In other words, the absolute values of the slopes
must be less than unity for convergence:
a11  a12
a22  a21
For n equations :
n
aii   ai , j
j 1
j i
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Figure 11.5
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