Mathematics Exam > Mathematics Notes > Mathematics for IIT JAM, GATE, CSIR NET, UGC NET > Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths
Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download
| Download, print and study this document offline |
Please wait while the PDF view is loading
Page 1
Chapter 04.08
Gauss-Seidel Method
After reading this chapter, you should be able to:
1. solve a set of equations using the Gauss-Seidel method,
2. recognize the advantages and pitfalls of the Gauss-Seidel method, and
3. determine under what conditions the Gauss-Seidel method always converges.
Why do we need another method to solve a set of simultaneous linear equations?
In certain cases, such as when a system of equations is large, iterative methods of solving
equations are more advantageous. Elimination methods, such as Gaussian elimination, are
prone to large round-off errors for a large set of equations. Iterative methods, such as the
Gauss-Seidel method, give the user control of the round-off error. Also, if the physics of the
problem are well known, initial guesses needed in iterative methods can be made more
judiciously leading to faster convergence.
What is the algorithm for the Gauss-Seidel method? Given a general set of n equations and
n unknowns, we have
1 1 3 13 2 12 1 11
... c x a x a x a x a
n n
= + + + +
2 2 3 23 2 22 1 21
... c x a x a x a x a
n n
= + + + +
. .
. .
. .
n n nn n n n
c x a x a x a x a = + + + + ...
3 3 2 2 1 1
If the diagonal elements are non-zero, each equation is rewritten for the corresponding
unknown, that is, the first equation is rewritten with
1
x on the left hand side, the second
equation is rewritten with
2
x on the left hand side and so on as follows
Page 2
Chapter 04.08
Gauss-Seidel Method
After reading this chapter, you should be able to:
1. solve a set of equations using the Gauss-Seidel method,
2. recognize the advantages and pitfalls of the Gauss-Seidel method, and
3. determine under what conditions the Gauss-Seidel method always converges.
Why do we need another method to solve a set of simultaneous linear equations?
In certain cases, such as when a system of equations is large, iterative methods of solving
equations are more advantageous. Elimination methods, such as Gaussian elimination, are
prone to large round-off errors for a large set of equations. Iterative methods, such as the
Gauss-Seidel method, give the user control of the round-off error. Also, if the physics of the
problem are well known, initial guesses needed in iterative methods can be made more
judiciously leading to faster convergence.
What is the algorithm for the Gauss-Seidel method? Given a general set of n equations and
n unknowns, we have
1 1 3 13 2 12 1 11
... c x a x a x a x a
n n
= + + + +
2 2 3 23 2 22 1 21
... c x a x a x a x a
n n
= + + + +
. .
. .
. .
n n nn n n n
c x a x a x a x a = + + + + ...
3 3 2 2 1 1
If the diagonal elements are non-zero, each equation is rewritten for the corresponding
unknown, that is, the first equation is rewritten with
1
x on the left hand side, the second
equation is rewritten with
2
x on the left hand side and so on as follows
nn
n n n n n n
n
n n
n n n n n n n n n
n
n n
n n
a
x a x a x a c
x
a
x a x a x a x a c
x
a
x a x a x a c
x
a
x a x a x a c
x
1 1 , 2 2 1 1
1 , 1
, 1 2 2 , 1 2 2 , 1 1 1 , 1 1
1
22
2 3 23 1 21 2
2
11
1 3 13 2 12 1
1
- -
- -
- - - - - - -
-
- - - -
=
- - - -
=
- - -
=
- - -
=
? ?
? ?
?
?
? ?
? ?
These equations can be rewritten in a summation form as
11
1
1
1 1
1
a
x a c
x
n
j
j
j j ?
?
=
-
=
22
2
1
2 2
2
a
x a c
x
j
n
j
j
j ?
?
=
-
=
.
.
.
1 , 1
1
1
, 1 1
1
- -
- ?
=
- -
-
?
-
=
n n
n
n j
j
j j n n
n
a
x a c
x
nn
n
n j
j
j nj n
n
a
x a c
x
?
?
=
-
=
1
Hence for any row i ,
. , , 2 , 1 ,
1
n i
a
x a c
x
ii
n
i j
j
j ij i
i
? =
-
=
?
?
=
Now to find
i
x ’s, one assumes an initial guess for the
i
x ’s and then uses the rewritten
equations to calculate the new estimates. Remember, one always uses the most recent
estimates to calculate the next estimates,
i
x . At the end of each iteration, one calculates the
absolute relative approximate error for each
i
x as
Page 3
Chapter 04.08
Gauss-Seidel Method
After reading this chapter, you should be able to:
1. solve a set of equations using the Gauss-Seidel method,
2. recognize the advantages and pitfalls of the Gauss-Seidel method, and
3. determine under what conditions the Gauss-Seidel method always converges.
Why do we need another method to solve a set of simultaneous linear equations?
In certain cases, such as when a system of equations is large, iterative methods of solving
equations are more advantageous. Elimination methods, such as Gaussian elimination, are
prone to large round-off errors for a large set of equations. Iterative methods, such as the
Gauss-Seidel method, give the user control of the round-off error. Also, if the physics of the
problem are well known, initial guesses needed in iterative methods can be made more
judiciously leading to faster convergence.
What is the algorithm for the Gauss-Seidel method? Given a general set of n equations and
n unknowns, we have
1 1 3 13 2 12 1 11
... c x a x a x a x a
n n
= + + + +
2 2 3 23 2 22 1 21
... c x a x a x a x a
n n
= + + + +
. .
. .
. .
n n nn n n n
c x a x a x a x a = + + + + ...
3 3 2 2 1 1
If the diagonal elements are non-zero, each equation is rewritten for the corresponding
unknown, that is, the first equation is rewritten with
1
x on the left hand side, the second
equation is rewritten with
2
x on the left hand side and so on as follows
nn
n n n n n n
n
n n
n n n n n n n n n
n
n n
n n
a
x a x a x a c
x
a
x a x a x a x a c
x
a
x a x a x a c
x
a
x a x a x a c
x
1 1 , 2 2 1 1
1 , 1
, 1 2 2 , 1 2 2 , 1 1 1 , 1 1
1
22
2 3 23 1 21 2
2
11
1 3 13 2 12 1
1
- -
- -
- - - - - - -
-
- - - -
=
- - - -
=
- - -
=
- - -
=
? ?
? ?
?
?
? ?
? ?
These equations can be rewritten in a summation form as
11
1
1
1 1
1
a
x a c
x
n
j
j
j j ?
?
=
-
=
22
2
1
2 2
2
a
x a c
x
j
n
j
j
j ?
?
=
-
=
.
.
.
1 , 1
1
1
, 1 1
1
- -
- ?
=
- -
-
?
-
=
n n
n
n j
j
j j n n
n
a
x a c
x
nn
n
n j
j
j nj n
n
a
x a c
x
?
?
=
-
=
1
Hence for any row i ,
. , , 2 , 1 ,
1
n i
a
x a c
x
ii
n
i j
j
j ij i
i
? =
-
=
?
?
=
Now to find
i
x ’s, one assumes an initial guess for the
i
x ’s and then uses the rewritten
equations to calculate the new estimates. Remember, one always uses the most recent
estimates to calculate the next estimates,
i
x . At the end of each iteration, one calculates the
absolute relative approximate error for each
i
x as
100
new
old new
×
-
= ?
i
i i
i
a
x
x x
where
new
i
x is the recently obtained value of
i
x , and
old
i
x is the previous value of
i
x .
When the absolute relative approximate error for each xi is less than the pre-specified
tolerance, the iterations are stopped.
Example 1
The upward velocity of a rocket is given at three different times in the following table
Table 1 Velocity vs. time data.
Time, t (s) Velocity, v (m/s)
5 106.8
8 177.2
12 279.2
The velocity data is approximated by a polynomial as
( ) 12 5 ,
3 2
2
1
= = + + = t a t a t a t v
Find the values of
3 2 1
and , , a a a using the Gauss-Seidel method. Assume an initial guess of
the solution as
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
5
2
1
3
2
1
a
a
a
and conduct two iterations.
Solution
The polynomial is going through three data points ( ) ( ) ( )
3 3 2 2 1 1
, and , , , , v t v t v t where from the
above table
8 . 106 , 5
1 1
= = v t
2 . 177 , 8
2 2
= = v t
2 . 279 , 12
3 3
= = v t
Requiring that ( )
3 2
2
1
a t a t a t v + + = passes through the three data points gives
( )
3 1 2
2
1 1 1 1
a t a t a v t v + + = =
( )
3 2 2
2
2 1 2 2
a t a t a v t v + + = =
( )
3 3 2
2
3 1 3 3
a t a t a v t v + + = =
Substituting the data ( ) ( ) ( )
3 3 2 2 1 1
, and , , , , v t v t v t gives
( ) ( ) 8 . 106 5 5
3 2
2
1
= + + a a a
( ) ( ) 2 . 177 8 8
3 2
2
1
= + + a a a
( ) ( ) 2 . 279 12 12
3 2
2
1
= + + a a a
Page 4
Chapter 04.08
Gauss-Seidel Method
After reading this chapter, you should be able to:
1. solve a set of equations using the Gauss-Seidel method,
2. recognize the advantages and pitfalls of the Gauss-Seidel method, and
3. determine under what conditions the Gauss-Seidel method always converges.
Why do we need another method to solve a set of simultaneous linear equations?
In certain cases, such as when a system of equations is large, iterative methods of solving
equations are more advantageous. Elimination methods, such as Gaussian elimination, are
prone to large round-off errors for a large set of equations. Iterative methods, such as the
Gauss-Seidel method, give the user control of the round-off error. Also, if the physics of the
problem are well known, initial guesses needed in iterative methods can be made more
judiciously leading to faster convergence.
What is the algorithm for the Gauss-Seidel method? Given a general set of n equations and
n unknowns, we have
1 1 3 13 2 12 1 11
... c x a x a x a x a
n n
= + + + +
2 2 3 23 2 22 1 21
... c x a x a x a x a
n n
= + + + +
. .
. .
. .
n n nn n n n
c x a x a x a x a = + + + + ...
3 3 2 2 1 1
If the diagonal elements are non-zero, each equation is rewritten for the corresponding
unknown, that is, the first equation is rewritten with
1
x on the left hand side, the second
equation is rewritten with
2
x on the left hand side and so on as follows
nn
n n n n n n
n
n n
n n n n n n n n n
n
n n
n n
a
x a x a x a c
x
a
x a x a x a x a c
x
a
x a x a x a c
x
a
x a x a x a c
x
1 1 , 2 2 1 1
1 , 1
, 1 2 2 , 1 2 2 , 1 1 1 , 1 1
1
22
2 3 23 1 21 2
2
11
1 3 13 2 12 1
1
- -
- -
- - - - - - -
-
- - - -
=
- - - -
=
- - -
=
- - -
=
? ?
? ?
?
?
? ?
? ?
These equations can be rewritten in a summation form as
11
1
1
1 1
1
a
x a c
x
n
j
j
j j ?
?
=
-
=
22
2
1
2 2
2
a
x a c
x
j
n
j
j
j ?
?
=
-
=
.
.
.
1 , 1
1
1
, 1 1
1
- -
- ?
=
- -
-
?
-
=
n n
n
n j
j
j j n n
n
a
x a c
x
nn
n
n j
j
j nj n
n
a
x a c
x
?
?
=
-
=
1
Hence for any row i ,
. , , 2 , 1 ,
1
n i
a
x a c
x
ii
n
i j
j
j ij i
i
? =
-
=
?
?
=
Now to find
i
x ’s, one assumes an initial guess for the
i
x ’s and then uses the rewritten
equations to calculate the new estimates. Remember, one always uses the most recent
estimates to calculate the next estimates,
i
x . At the end of each iteration, one calculates the
absolute relative approximate error for each
i
x as
100
new
old new
×
-
= ?
i
i i
i
a
x
x x
where
new
i
x is the recently obtained value of
i
x , and
old
i
x is the previous value of
i
x .
When the absolute relative approximate error for each xi is less than the pre-specified
tolerance, the iterations are stopped.
Example 1
The upward velocity of a rocket is given at three different times in the following table
Table 1 Velocity vs. time data.
Time, t (s) Velocity, v (m/s)
5 106.8
8 177.2
12 279.2
The velocity data is approximated by a polynomial as
( ) 12 5 ,
3 2
2
1
= = + + = t a t a t a t v
Find the values of
3 2 1
and , , a a a using the Gauss-Seidel method. Assume an initial guess of
the solution as
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
5
2
1
3
2
1
a
a
a
and conduct two iterations.
Solution
The polynomial is going through three data points ( ) ( ) ( )
3 3 2 2 1 1
, and , , , , v t v t v t where from the
above table
8 . 106 , 5
1 1
= = v t
2 . 177 , 8
2 2
= = v t
2 . 279 , 12
3 3
= = v t
Requiring that ( )
3 2
2
1
a t a t a t v + + = passes through the three data points gives
( )
3 1 2
2
1 1 1 1
a t a t a v t v + + = =
( )
3 2 2
2
2 1 2 2
a t a t a v t v + + = =
( )
3 3 2
2
3 1 3 3
a t a t a v t v + + = =
Substituting the data ( ) ( ) ( )
3 3 2 2 1 1
, and , , , , v t v t v t gives
( ) ( ) 8 . 106 5 5
3 2
2
1
= + + a a a
( ) ( ) 2 . 177 8 8
3 2
2
1
= + + a a a
( ) ( ) 2 . 279 12 12
3 2
2
1
= + + a a a
or
8 . 106 5 25
3 2 1
= + + a a a
2 . 177 8 64
3 2 1
= + + a a a
2 . 279 12 144
3 2 1
= + + a a a
The coefficients
3 2 1
and , , a a a for the above expression are given by
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
2 . 279
2 . 177
8 . 106
1 12 144
1 8 64
1 5 25
3
2
1
a
a
a
Rewriting the equations gives
25
5 8 . 106
3 2
1
a a
a
- -
=
8
64 2 . 177
3 1
2
a a
a
- -
=
1
12 144 2 . 279
2 1
3
a a
a
- -
=
Iteration #1
Given the initial guess of the solution vector as
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
5
2
1
3
2
1
a
a
a
we get
25
) 5 ( ) 2 ( 5 8 . 106
1
- -
= a
6720 . 3 =
( ) ( )
8
5 6720 . 3 64 2 . 177
2
- -
= a
8150 . 7 - =
( ) ( )
1
8510 . 7 12 6720 . 3 144 2 . 279
3
- - -
= a
36 . 155 - =
The absolute relative approximate error for each
i
x then is
100
6720 . 3
1 6720 . 3
1
×
-
= ?
a
% 76 . 72 =
100
8510 . 7
2 8510 . 7
2
×
-
- -
= ?
a
% 47 . 125 =
100
36 . 155
5 36 . 155
3
×
-
- -
= ?
a
% 22 . 103 =
Page 5
Chapter 04.08
Gauss-Seidel Method
After reading this chapter, you should be able to:
1. solve a set of equations using the Gauss-Seidel method,
2. recognize the advantages and pitfalls of the Gauss-Seidel method, and
3. determine under what conditions the Gauss-Seidel method always converges.
Why do we need another method to solve a set of simultaneous linear equations?
In certain cases, such as when a system of equations is large, iterative methods of solving
equations are more advantageous. Elimination methods, such as Gaussian elimination, are
prone to large round-off errors for a large set of equations. Iterative methods, such as the
Gauss-Seidel method, give the user control of the round-off error. Also, if the physics of the
problem are well known, initial guesses needed in iterative methods can be made more
judiciously leading to faster convergence.
What is the algorithm for the Gauss-Seidel method? Given a general set of n equations and
n unknowns, we have
1 1 3 13 2 12 1 11
... c x a x a x a x a
n n
= + + + +
2 2 3 23 2 22 1 21
... c x a x a x a x a
n n
= + + + +
. .
. .
. .
n n nn n n n
c x a x a x a x a = + + + + ...
3 3 2 2 1 1
If the diagonal elements are non-zero, each equation is rewritten for the corresponding
unknown, that is, the first equation is rewritten with
1
x on the left hand side, the second
equation is rewritten with
2
x on the left hand side and so on as follows
nn
n n n n n n
n
n n
n n n n n n n n n
n
n n
n n
a
x a x a x a c
x
a
x a x a x a x a c
x
a
x a x a x a c
x
a
x a x a x a c
x
1 1 , 2 2 1 1
1 , 1
, 1 2 2 , 1 2 2 , 1 1 1 , 1 1
1
22
2 3 23 1 21 2
2
11
1 3 13 2 12 1
1
- -
- -
- - - - - - -
-
- - - -
=
- - - -
=
- - -
=
- - -
=
? ?
? ?
?
?
? ?
? ?
These equations can be rewritten in a summation form as
11
1
1
1 1
1
a
x a c
x
n
j
j
j j ?
?
=
-
=
22
2
1
2 2
2
a
x a c
x
j
n
j
j
j ?
?
=
-
=
.
.
.
1 , 1
1
1
, 1 1
1
- -
- ?
=
- -
-
?
-
=
n n
n
n j
j
j j n n
n
a
x a c
x
nn
n
n j
j
j nj n
n
a
x a c
x
?
?
=
-
=
1
Hence for any row i ,
. , , 2 , 1 ,
1
n i
a
x a c
x
ii
n
i j
j
j ij i
i
? =
-
=
?
?
=
Now to find
i
x ’s, one assumes an initial guess for the
i
x ’s and then uses the rewritten
equations to calculate the new estimates. Remember, one always uses the most recent
estimates to calculate the next estimates,
i
x . At the end of each iteration, one calculates the
absolute relative approximate error for each
i
x as
100
new
old new
×
-
= ?
i
i i
i
a
x
x x
where
new
i
x is the recently obtained value of
i
x , and
old
i
x is the previous value of
i
x .
When the absolute relative approximate error for each xi is less than the pre-specified
tolerance, the iterations are stopped.
Example 1
The upward velocity of a rocket is given at three different times in the following table
Table 1 Velocity vs. time data.
Time, t (s) Velocity, v (m/s)
5 106.8
8 177.2
12 279.2
The velocity data is approximated by a polynomial as
( ) 12 5 ,
3 2
2
1
= = + + = t a t a t a t v
Find the values of
3 2 1
and , , a a a using the Gauss-Seidel method. Assume an initial guess of
the solution as
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
5
2
1
3
2
1
a
a
a
and conduct two iterations.
Solution
The polynomial is going through three data points ( ) ( ) ( )
3 3 2 2 1 1
, and , , , , v t v t v t where from the
above table
8 . 106 , 5
1 1
= = v t
2 . 177 , 8
2 2
= = v t
2 . 279 , 12
3 3
= = v t
Requiring that ( )
3 2
2
1
a t a t a t v + + = passes through the three data points gives
( )
3 1 2
2
1 1 1 1
a t a t a v t v + + = =
( )
3 2 2
2
2 1 2 2
a t a t a v t v + + = =
( )
3 3 2
2
3 1 3 3
a t a t a v t v + + = =
Substituting the data ( ) ( ) ( )
3 3 2 2 1 1
, and , , , , v t v t v t gives
( ) ( ) 8 . 106 5 5
3 2
2
1
= + + a a a
( ) ( ) 2 . 177 8 8
3 2
2
1
= + + a a a
( ) ( ) 2 . 279 12 12
3 2
2
1
= + + a a a
or
8 . 106 5 25
3 2 1
= + + a a a
2 . 177 8 64
3 2 1
= + + a a a
2 . 279 12 144
3 2 1
= + + a a a
The coefficients
3 2 1
and , , a a a for the above expression are given by
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
2 . 279
2 . 177
8 . 106
1 12 144
1 8 64
1 5 25
3
2
1
a
a
a
Rewriting the equations gives
25
5 8 . 106
3 2
1
a a
a
- -
=
8
64 2 . 177
3 1
2
a a
a
- -
=
1
12 144 2 . 279
2 1
3
a a
a
- -
=
Iteration #1
Given the initial guess of the solution vector as
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
5
2
1
3
2
1
a
a
a
we get
25
) 5 ( ) 2 ( 5 8 . 106
1
- -
= a
6720 . 3 =
( ) ( )
8
5 6720 . 3 64 2 . 177
2
- -
= a
8150 . 7 - =
( ) ( )
1
8510 . 7 12 6720 . 3 144 2 . 279
3
- - -
= a
36 . 155 - =
The absolute relative approximate error for each
i
x then is
100
6720 . 3
1 6720 . 3
1
×
-
= ?
a
% 76 . 72 =
100
8510 . 7
2 8510 . 7
2
×
-
- -
= ?
a
% 47 . 125 =
100
36 . 155
5 36 . 155
3
×
-
- -
= ?
a
% 22 . 103 =
At the end of the first iteration, the estimate of the solution vector is
?
?
?
?
?
?
?
?
?
?
-
- =
?
?
?
?
?
?
?
?
?
?
36 . 155
8510 . 7
6720 . 3
3
2
1
a
a
a
and the maximum absolute relative approximate error is 125.47%.
Iteration #2
The estimate of the solution vector at the end of Iteration #1 is
?
?
?
?
?
?
?
?
?
?
-
- =
?
?
?
?
?
?
?
?
?
?
36 . 155
8510 . 7
6720 . 3
3
2
1
a
a
a
Now we get
( )
25
) 36 . 155 ( 8510 . 7 5 8 . 106
1
- - - -
= a
056 . 12 =
( )
8
) 36 . 155 ( 056 . 12 64 2 . 177
2
- - -
= a
882 . 54 - =
( ) ( )
1
882 . 54 12 056 . 12 144 2 . 279
3
- - -
= a
= 34 . 798 -
The absolute relative approximate error for each
i
x then is
100
056 . 12
6720 . 3 056 . 12
1
×
-
= ?
a
% 543 . 69 =
( )
100
882 . 54
8510 . 7 882 . 54
2
×
-
- - -
= ?
a
% 695 . 85 =
( )
100
34 . 798
36 . 155 34 . 798
3
×
-
- - -
= ?
a
% 540 . 80 =
At the end of the second iteration the estimate of the solution vector is
?
?
?
?
?
?
?
?
?
?
-
- =
?
?
?
?
?
?
?
?
?
?
54 . 798
882 . 54
056 . 12
3
2
1
a
a
a
and the maximum absolute relative approximate error is 85.695%.
Conducting more iterations gives the following values for the solution vector and the
corresponding absolute relative approximate errors.
Read More
|
556 videos|198 docs
|
FAQs on Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths - Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
| 1. What is the Gauss-Seidel method for solving systems of linear algebraic equations? |  |
Ans. The Gauss-Seidel method is an iterative technique used to solve systems of linear algebraic equations. It starts with an initial guess for the solution and then updates the values iteratively until a desired level of accuracy is achieved. In each iteration, the method uses the updated values to compute new approximations for the unknowns, improving the solution at each step.
| 2. How does the Gauss-Seidel method differ from the Gauss elimination method? | |
Ans. The Gauss-Seidel method differs from the Gauss elimination method in the way it updates the values of the unknowns. In the Gauss elimination method, all the unknowns are updated simultaneously using the values from the previous iteration. On the other hand, the Gauss-Seidel method updates each unknown as soon as its updated value is available, using the most recent estimates of the other unknowns. This iterative process continues until the desired level of accuracy is achieved.
| 3. What are the advantages of using the Gauss-Seidel method? | |
Ans. The Gauss-Seidel method has several advantages:
- It is relatively easy to implement and understand.
- It converges faster for certain types of matrices, especially those that are diagonally dominant.
- It can handle large systems of equations efficiently, as it updates the unknowns one at a time.
| 4. Are there any limitations to using the Gauss-Seidel method? | |
Ans. Yes, there are limitations to using the Gauss-Seidel method:
- It may not converge for certain types of matrices, particularly those that are not diagonally dominant.
- It may converge slowly for matrices with highly non-diagonally dominant elements.
- It may oscillate or diverge if the initial guess is far from the actual solution.
| 5. How can the convergence of the Gauss-Seidel method be improved? | |
Ans. The convergence of the Gauss-Seidel method can be improved by using the following techniques:
- Reordering the equations or unknowns to achieve a more diagonally dominant matrix.
- Applying a relaxation factor to control the rate of convergence. This involves multiplying the updated value of each unknown by a factor between 0 and 1. A smaller factor results in slower convergence but may improve stability.
- Using an appropriate initial guess that is close to the actual solution. A good initial guess can significantly reduce the number of iterations required to reach the desired accuracy.
Related Exams
About this Document
|
4.74/5 Rating |
|
Dec 23, 2024 Last updated |
Document Description: Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths for Mathematics 2024 is part of Mathematics for IIT JAM, GATE, CSIR NET, UGC NET preparation.
The notes and questions for Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths have been prepared according to the Mathematics exam syllabus. Information about Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths covers topics
like and Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths Example, for Mathematics 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths.
Introduction of Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths in English is available as part of our Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
for Mathematics & Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths in Hindi for Mathematics for IIT JAM, GATE, CSIR NET, UGC NET course.
Download more important topics related with notes, lectures and mock test series for Mathematics
Exam by signing up for free. Mathematics: Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
Description
Full syllabus notes, lecture & questions for Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET - Mathematics | Plus excerises question with solution to help you revise complete syllabus for Mathematics for IIT JAM, GATE, CSIR NET, UGC NET | Best notes, free PDF download
Information about Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths
In this doc you can find the meaning of Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths defined & explained in the simplest way possible. Besides explaining types of
Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths theory, EduRev gives you an ample number of questions to practice Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths tests, examples and also practice Mathematics
tests
|
556 videos|198 docs
|
Download as PDF
|
Explore Courses for Mathematics exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
Related Searches
Viva Questions
,Previous Year Questions with Solutions
,Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths | Mathematics for IIT JAM
,Summary
,ppt
,CSIR NET
,GATE
,Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths | Mathematics for IIT JAM
,mock tests for examination
,Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths | Mathematics for IIT JAM
,UGC NET
,Sample Paper
,UGC NET
,past year papers
,UGC NET
,GATE
,CSIR NET
,Exam
,CSIR NET
,Extra Questions
,GATE
,Objective type Questions
,Semester Notes
,video lectures
,practice quizzes
,shortcuts and tricks
,study material
,Free
,Important questions
,MCQs
;
Additional Information about Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths for Mathematics Preparation
Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths Free PDF Download
The Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths is an invaluable resource that delves deep into the core of the Mathematics exam.
These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective.
With the help of these notes, you can grasp complex subjects quickly, revise important points easily,
and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner,
allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material,
or toppers' notes, this PDF has got you covered. Download the Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths now and kickstart your journey towards success in the Mathematics exam.
Importance of Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths
The importance of Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths cannot be overstated, especially for Mathematics aspirants.
This document holds the key to success in the Mathematics exam.
It offers a detailed understanding of the concept, providing invaluable insights into the topic.
By knowing the concepts well in advance, students can plan their preparation effectively.
Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.
Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths Notes
Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths.
It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation.
Practice papers and question papers enable you to assess your progress effectively.
Additionally, the paper analysis provides valuable tips for tackling the exam strategically.
Access to Toppers' notes gives you an edge in understanding complex concepts.
Whether you're a beginner or aiming for advanced proficiency, Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths Notes on EduRev are your ultimate resource for success.
Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths Mathematics Questions
The "Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths Mathematics Questions" guide is a valuable resource for all aspiring students preparing for the
Mathematics exam. It focuses on providing a wide range of practice questions to help students gauge
their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths on the App
Students of Mathematics can study Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths,
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of Solution of Systems of Linear Algebraic Equations Using Gauss-Seidel Method - CSIR-NET Maths is prepared as per the latest Mathematics syllabus.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup