Here we have included details about relaxation method and some examples . Contribution - Parinda Rajapakha, Hashan Wanniarachchi, Sameera Horawalawithana, Thilina Gamalath, Samudra Herath and Pavithri Fernando.

1. Relaxation Method

2. Introduction
• Relaxation method is an iterative approach
solution to systems of linear equations.
• Basic idea behind this method is to improve
the solution vector successively by reducing the
largest residual at a particular iteration.

3. What is a residual?
• Suppose x(i) € R is an approximation to the
solution of the linear system defined by
Ax=b
• Residual vector for x(i) with respect to this system
is
R(i) =b-A x(i) in ith iteration

4. • The error: E(I )= x-x(i)
• R(i) = b –Ax(i) = Ax –Ax(i) = A(x –x(i)) = AE(i):
• Residual equation:
– AE(i)=R(i)

5. Let x(p) =( x1
(p),x2
(p) … xn
(p))T
be the solution vector obtained after pth
iteration. If Ri
(p) denotes residual,
ai1x1 + ai2x2 + … + ainxn = bi
Define by,
Ri
(p) = bi- (ai1x1 + ai2x2 + … + ainxn)

6. Applying relaxation method
• Transfer all the terms to the right hand side of the
equation
• Reorder the equations in a way such that largest co-
efficient in the equations appear on the diagonal
• Select the largest residual and give an increment
dx=-r(i)/aii
• Change x(i) to x(i) +dx(i) to relax R(i) that is to reduce
R(i) to zero

7. Example :
6x1-3x2+x3 = 11
2x1+x2-8x3 =-15
x1-7x2+x3 = 10
0= 11- 6x1 - 3x2 - x3 R1
0= 10- x1 + 7x2 - x3  R2
0= -15- 2x1 - x2 + 8x3  R3

8. • Start with initial guesses x1=x2=x3=0
• R1=11,
• R2=10,
• R3=-15
• Largest residual is R3
• So that dx3 = - R3 /a33
• dx3= -15/-8 = 1.875

9. New guesses:
x1=0
x2=0 and
x3=1.875
Continue the process until r  0

10. Final result would be like this
Iteration
no R1 R2 R3
Max
Ri dx(i) x1 x2 x3
0 11 10 -15 1.875 0 0 0
1 -9.125 8.125 0 9.125 1.5288 0 0 1.875
2 0.0478 6.5962 -3.0576 6.5962 -0.9423 1.5288 0 1.875
3 -2.8747 0.0001 -2.1153 -2.8747 -0.4791 1.5288 -0.9423 1.875
4 -0.0031 0.4792 -1.1571 -1.1571 0.1446 1.0497 -0.9423 1.875
5 0.1447 0.3346 0.0003 0.3346 -0.0478 1.0497 -0.9423 2.0196
6 0.2881 0.0000 0.0475 0.2881 0.0480 1.0497 -0.9901 2.0196
7 -0.0001 0.048 0.1435 0.1435 -0.0179 1.0017 -0.9901 2.0196
8 0.0178 0.0659 0.0003 - - 1.0017 -0.9901 2.0017

11. • At ith iteration we can see that
R1,R2 and R3 are small enough,
• So xi values in this iteration
x1 = 1.007,
x2 = -0.9901,
x3 = 2.0017
• Which are very close to the Exact solutions
x1 = 1.0
x2 = -1.0
x3 = 2.0

12. Convergence
• Converges slowly for large systems of equations
(large n)

13. Special cases
• Simple to implement
• Not useful as a stand alone solution method
• Key ingredients to multi grid methods
– Jacobi
– Gauss seidel
– red

14. Comparison with Other
Methods

15. Methods available to find solutions
 Direct
Elimination
 Gaussian elimination
 Gauss-Jordan
elimination
Decomposition
 Court's reduction
(Cholesky's reduction)
 Iterative
 Jacobi's method
 Gauss-Seidel method
 Relaxation method

16. Advantages and Disadvantages
 Relaxation method is the core part of linear algebra.
 This method provide preconditions for new methods.
 Easily adoptable to computers.
 Can solve more than 100s of linear equations
simultaneously.
 Slower progress than the competing methods

17. Solve:
6x - 3y + z = 11
2x + y - 8z = -15
x - 7y + z = 10
Gaussian
Elimination
Gauss-
Jordan
Elimination
Courts
Reduction
Relaxation
method
X 1 1 1 1.0017
Y -1 -1 -1 -0 9901
Z 2 2 2 2.0017

18. Relaxation method is the best
method for :
 Relaxation method is highly used for image
processing .
 This method has been developed for analysis of
hydraulic structures .
 Solving linear equations relating to the radiosity
problem.
 Relaxation methods are iterative methods for solving
systems of equations, including nonlinear systems.
 Relaxation method used with other numerical
methods in mono-tropic programs.

19. Completed Researches

20. Why relaxation methods?
• Direct methods are robust.
• Direct methods are less computational costly.
But
• They require high memory access.
• Slow in convergence.

21. Evolution of relaxation methods
• Gauss Siedel Iteration
 Gauss’s letter to Gerling
 Era of electronic computing

22. •Work of David Young
 Notions - “Consistent Ordering” and “Property A”
 Convergence of the methods
• Ostrowski (1937)
 Relevant properties for M-Matrices
• Theorem of Stein – Rosenburg (1948)
 Asymptotic rates
• Concept of Irreducibility
 Grid oriented matrices

23. •Concept of Cyclic Matrices
 Convergence theory of SOR methods
•Varga’s Contribution
 Generalization of Young’s results
 Matrix Iterative Analysis (1962)
 Notions – Regular Splittings
 Theories -Stieltjes and M-Matrices
 Semi Iterative Methods
Richard Varga

24. • 1960s and 1970s
 Chaotic Relaxations
 Chazan , Miranker , Miellou , Robert
• Multigrid Methods
 Krylov subspace method
 Use of Eugene values

25. References
Rao, K.S., Year. Numerical Methods for Scientists and
Engineers. 2nd ed. Delhi: Prentice-Hall of India.
Yousef Sadd and , Henk A. van der Vorst, Iterative Solution of
Linear Systems in the 20th Century [pdf]. Available at: <www-
users.cs.umn.edu/~saad/PDF/umsi-99-152.pdf> Accessed [12th
July 2012]
Relaxation Methods for Iterative Solution to Linear Systems of
Equations Gerald Recktenwald Portland State University
Mechanical Engineering Department
Scientic Computing II Relaxation MethodsMichael Bader
Summer term 2012

26. Working scenario

27. Demonstration