Relaxation method
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The relaxation method is an iterative technique used to solve systems of linear equations by successively improving the solution vector to minimize the residual error. It involves calculating the residual vector and adjusting the approximation to reduce this residual to zero through a series of iterations. Although it is a fundamental component of linear algebra with applications in various fields, the relaxation method is typically slower than other solution methods and serves more as a preconditioning technique.
![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](https://image.slidesharecdn.com/relaxationmethod-130624231244-phpapp01/85/Relaxation-method-25-320.jpg)
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
- 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.
- 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