iterative methods for solving linear equation system
TRANSCRIPT
Johann Carl Friedrich Gauss
30 April 1777 Brunswick – 23 February
1855 Brunswick
Philipp Ludwig von Seidel
23 October 1821, Zweibrücken, Germany
– 13 August 1896, Munich
SPECIAL MATRICES
BANDED MATRIX
Banded matrix is a square matrix that
has all elements equal to zero, with
the exception of a band centered on
the main diagonal. Banded system is
frequently encountered in engineering
and scientific practice because of they
typically occurred in the solution of
differential equation.
The dimensions of a banded system
can be quantified by two parameters:
• The banded width BW
• The half bandwidth HBW
These two values are related by
BW=2HBW+1, 𝑎𝑖𝑗 = 𝑖𝑓 𝑖 − 𝑗 >HBW
SPECIAL MATRICES
TRIDIAGONAL BANDED
A tridiagonal matrix is a
matrix that is "almost" a
diagonal matrix. To be
exact: a tridiagonal matrix
has nonzero elements only
in the main diagonal, the
first diagonal below this,
and the first diagonal above
the main diagonal
JACOBI METHOD
The Jacobi method is an algorithm
for determining the solutions of a
system of linear equations with
largest absolute values in each row
and column dominated by the
diagonal element. Each diagonal
element is solved for, and an
approximate value plugged in. The
process is then iterated until it
converges. The method is named
after German mathematician Carl
Gustav Jakob Jacobi
JACOBI METHOD
This method makes two assumptions:
1. That the system given by the next system equation has a unique solution
2. That the coefficient matrix A has no zeros on its main diagonal. If any
of the diagonal entries are zero, then rows or columns must be
interchanged to obtain a coefficient matrix that has nonzero entries on
the main diagonal.
JACOBI METHOD
To begin the Jacobi method, solve the first equation for the second
equation for and so on, as follows
Then make an initial approximation of the solution 𝑥1, 𝑥2, 𝑥3,…𝑥𝑛 ,
initial aproximation, and substitute xi these values of into the right-hand
side of the rewritten equations to obtain the first approximation. After this
procedure has been completed, one iteration has been performed. In the
same way, the second approximation is formed by substituting the first
approximation’s x-values into the right-hand side of the rewritten
equations. By repeated iterations, you will form a sequence of
approximations that often converges to the actual solution.
JACOBI METHOD
Example
Use the Jacobi method to approximate the solution of the following system
of linear equations.
Continue the iterations until two successive approximations are identical
when rounded to three significant digits.
Solution To begin, write the system in the form :
Because you do not know the actual solution, choose
Initial approximation
JACOBI METHOD
As a convenient initial approximation. So, the first approximation is
Continuing this procedure, you obtain the sequence of approximations shown in
Table
n 0 1 2 3 4 5 6 7
𝑥1 0,000 -0,200 0,146 0,192 0,181 0,185 0,186 0,186
𝑥2 0,000 0,222 0,203 0,328 0,332 0,329 0,331 0,331
𝑥3 0,000 -0,429 -0,517 -0,416 -0,421 -0,424 -0,423 0,423
JACOBI METHOD
For the system of linear equations given in Example 1, the Jacobi method
is said to converge. That is, repeated iterations succeed in producing an
approximation that is correct to three significant digits. As is generally true
for iterative methods, greater accuracy would require more iterations.
Because the last two columns in Table are identical, you can conclude that to
three significant digits the solution is
THE GAUSS-SEIDEL METHOD
You will now look at a modification of the
Jacobi method called the Gauss-Seidel
method, named after Carl Friedrich Gauss
(1777–1855) and Philipp L. Seidel (1821–
1896). This modification is no more
difficult to use than the Jacobi method,
and it often requires fewer iterations to
produce the same degree of accuracy
THE GAUSS-SEIDEL METHOD
With the Jacobi method, the values of 𝑥𝑖obtained in the nth approximation remain
unchanged until the entire(𝑛 + 1) th approximation has been calculated. With the
Gauss- Seidel method, on the other hand, you use the new values of each 𝑥𝑖 as
soon as they are known. That is, once you have determined 𝑥1 from the first
equation, its value is then used in the second equation to obtain the new 𝑥2
Similarly, the new 𝑥1 and 𝑥2 are used in the third equation to obtain the new 𝑥3
and so on.
Example
Use the Gauss-Seidel iteration method to approximate the solution to the system
of equations given by
THE GAUSS-SEIDEL METHOD
Solution To begin, write the system in the form :
Using 𝑥1, 𝑥2, 𝑥3 = 0,0,0 as the initial approximation, you obtain the following
new value for 𝑥1
Now that you have a new value for 𝑥1, however, use it to compute a new
value for 𝑥2 That is,
THE GAUSS-SEIDEL METHOD
Similarly, use 𝑥1 = −0,200 𝑎𝑛𝑑 𝑥2 = 0,156 to compute a new value for 𝑥3 that is
So the first approximation is 𝑥1 = −0,200 𝑎𝑛𝑑 𝑥2 = 0,156 and 𝑥3 = −0,508
Continued iterations produce the sequence of approximations shown in Table
n 0 1 2 3 4 5
𝑥1 0,000 -0,200 0,167 0,191 0,186 0,186
𝑥2 0,000 0,156 0,334 0,333 0,331 0,331
𝑥3 0,000 -0,508 -0,429 -0,422 -0,423 -0,423
Note that after only five iterations of the Gauss-Seidel method, you
achieved the same accuracy as was obtained with seven iterations of the
Jacobi method
THE GAUSS-SEIDEL METHOD
An Example of Divergence
Apply the Jacobi method to the system
Repeated iterations produce the sequence of approximations shown in Table
using the initial approximation 𝑥1, 𝑥2 = 0,0 and show that the method
diverges.
n 0 1 2 3 4 5 6 7
𝑥1, 0 -4 -34 -174 -1244 -6124 -42,874 -214,374
𝑥2 0 -6 -34 -244 -1244 -8574 -42874 -300,124
For this particular system of linear equations you can determine that the actual
solution is x1=1 and x2=1. So you can see from Table, that the approximations
given by the Jacobi method become progressively worse instead of better, and you
can conclude that the method diverges.
THE GAUSS-SEIDEL METHOD
The problem of divergence in Example 3 is not resolved by using the Gauss-
Seidel method rather than the Jacobi method. In fact, for this particular
system the Gauss-Seidel method diverges more rapidly, as shown in Table
n 0 1 2 3 4 5
𝑥1 0 -4 -174 -6124 -214,374 -7,503,124
𝑥2 0 -34 -1224 -42,874 -1,500624 -52,521,874
With an initial approximation of 𝑥1, 𝑥2 = 0,0 neither the Jacobi method
nor the Gauss-Seidel method converges to the solution of the system of
linear equations given in Example
BIBLIOGRAPHY
• http://college.cengage.com/mathematics/larson/elementary_linear/5e
/students/ch08-10/chap_10_2.pdf
• Numerical Methods for Engineers, Fifth Edition, Steven C.
Chapra and Raymond P. Canale.