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