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1 Module 4 : Solving Linear Algebraic Equations Section 5 : Iterative Solution Techniques 5 Iterative Solution Techniques By this approach, we start with some initial guess solution, say for solution and generate an improved solution estimate from the previous approximation This method is a very effective for solving differential equations, integral equations and related problems Kelley. Let the residue vector be defined as (57) i.e. The iteration sequence is terminated when some norm of the residue becomes sufficiently small, i.e (58) is an arbitrarily small number (such as or ). Another possible termination criterion can be (59) It may be noted that the later condition is practically equivalent to the previous termination condition. A simple way to form an iterative scheme is Richardson iterations Kelley (60) or Richardson iterations preconditioned with approximate inversion (61) matrix is called approximate inverse of if A question that naturally arises is 'will the iterations converge to the solution of?'. In this section, to begin with, some well known iterative schemes are presented. Their convergence analysis is presented next. In the derivations that follow, it is implicitly assumed that the diagonal elements of matrix are nonzero, i.e. If this is not the case, simple row exchange is often sufficient to satisfy this condition. 5.1 Iterative Algorithms JacobiMethod Suppose we have a guess solution, say for To generate an improved estimate starting from consider the first equation in the set of equations, i.e., (62) Rearranging this equation, we can arrive at a iterative formula for computing, as (63)
2 Similarly, using second equation from, we can derive (64) Table 1: Algorithm for Jacobi Iterations INITIALIZE WHILE FOR END FOR END WHILE In general, using row of we can generate improved guess for the i'th element of as follows The above equation can also be rearranged as follows (65) is defined by equation (Ri). The algorithm for implementing the Jacobi iteration scheme is summarized in Table GaussSeidel Method When matrix is large, there is a practical difficulty with the Jacobi method. It is required to store all components of in the computer memory (as a separate variables) until calculations of is over. The GaussSeidel method overcomes this difficulty by using immediately in the next equation while computing This modification leads to the following set of equations (66) Table 2: Algorithm for GaussSeidel Iterations
3 INITIALIZE WHILE FOR END FOR END WHILE (67) (68) In general, for i'th element of, we have To simplify programming, the above equation can be rearranged as follows (69) The algorithm for implementing GaussSiedel iteration scheme is summarized in Table Relaxation Method Suppose we have a starting value say, of a quantity and we wish to approach a target value, say by some method. Let application of the method change the value from to. If is between and which is even closer to than, then we can approach faster by magnifying the change ( ) Strang. In order to achieve this, we need to apply a magnifying factor and get Table 3: Algorithms for OverRelaxation Iterations INITIALIZE WHILE FOR
4 END FOR END WHILE (70) This amplification process is an extrapolation and is an example of overrelaxaation. If the intermediate value tends to overshoot target, then we may have to use ; this is called underrelaxation. Application of overrelaxation to GaussSeidel method leads to the following set of equations (71) are generated using the GaussSeidel method, i.e., (72) The steps in the implementation of the overrelaxation iteration scheme are summarized in Table T O R. It may be noted that is a tuning parameter, which is chosen such that \caption{algorithms for OverRelaxation Iterations} 5.2 Convergence Analysis of Iterative Methods [3, 2] VectorMatrix Representati When is to be solved iteratively, a question that naturally arises is 'under what conditions the iterations converge?'. The convergence analysis can be carried out if the above set of iterative equations are expressed in the vectormatrix notation. For example, the iterative equations in the GaussSiedel method can be arranged as follows (73) Let and be diagonal, strictly lower triangular and strictly upper triangular parts of, i.e., (74) (The representation given by equation (74) should NOT be confused with matrix factorization
5 ). Using these matrices, the GaussSiedel iteration can be expressed as follows (75) or (76) Similarly, rearranging the iterative equations for Jacobi method, we arrive at and for the relaxation method we get (77) (78) Thus, in general, an iterative method can be developed by splitting matrix. If is expressed as (79) then, equation can be expressed as Starting from a guess solution (80) we generate a sequence of approximate solutions as follows (81) Requirements on and matrices are as follows [3] : matrix A should be decomposed into such that Matrix should be easily invertible Sequence should converge to is the solution of. The popular iterative formulations correspond to the following choices of matrices and [3,4] Jacobi Method: (82) Forward GaussSeidel Method (83) Relaxation Method: (84) Backward Gauss Seidel: In this case, iteration begins the update of rather than the first. This results in the following splitting of matrix [4] (85) with n'th coordinate In Symmetric Gauss Seidel approach, a forward GaussSeidel iteration is followed by a backward GaussSeidel iteration Iterative Scheme as a Linear Difference Equation In order to solve equation (LAE), we have formulated an iterative scheme
6 Let the true solution equation (LAE) be (86) Defining error vector (87) (88) and subtracting equation (true) from equation (Itr), we get (89) Thus, if we start with some then after iterations we have (90) (91) (92) (93) The convergence of the iterative scheme is assured if (94) (95) for any initial guess vector Alternatively, consider application of the general iteration equation (86) k times starting from initial guess At the k'th iteration step, we have
7 (96) If we select such that (97) represents the null matrix, then, using identity we can write for large The above expression clearly explains how the iteration sequence generates a numerical approximation to provided condition (97) is satisfied Convergence Criteria for Iteration Schemes It may be noted that equation (89) is a linear difference equation of form (98) with a specified initial condition. Here, and is a matrix. In Appendix A, we analyzed behavior of the solutions of linear difference equations of type (98). The criterion for convergence of iteration equation (89) can be derived using results derived in Appendix A. The necessary and sufficient condition for convergence of (89) can be stated as i.e. the spectral radius of matrix should be less than one. The necessary and sufficient condition for convergence stated above requires computation of eigenvalues of which is a computationally demanding task when the matrix dimension is large. For a large dimensional matrix, if we could check this condition before starting the iterations, then we might as well solve the problem by a direct method rather than using iterative approach to save computations. Thus, there is a need to derive some alternate criteria for convergence, which can be checked easily before starting iterations. Theorem 14 in Appendix A states that spectral radium of a matrix is smaller than any induced norm of the matrix. Thus, for matrix we have is any induced matrix norm. Using this result, we can arrive at the following sufficient conditions for the convergence of iterations Evaluating 1 or norms of is significantly easier than evaluating the spectral radius of. Satisfaction of any of the above conditions implies However, it may be noted that these are only sufficient conditions. Thus, if or we cannot conclude anything about the convergence of iterations. If the matrix has some special properties, such as diagonal dominance or symmetry and positive definiteness, then the convergence is ensured for some iterative techniques. Some of the important convergence results available in the literature are summarized here. Definition 1 :A matrix is called strictly diagonally dominant if
8 (99) Theorem 2 [2] A sufficient condition for the convergence of Jacobi and GaussSeidel methods is that the matrix of linear system is strictly diagonally dominant. Proof: Refer to Appendix B. Theorem 3 [5] The GaussSeidel iterations converge if matrix is symmetric and positive definite. Proof: Refer to Appendix B. Theorem 4 [3] For an arbitrary matrix, the necessary condition for the convergence of relaxation method is Proof: Refer to appendix B. Theorem 5 [2] When matrix is strictly diagonally dominant, a sufficient condition for the convergence of relaxation methods is that Proof: Left to reader as an exercise. Theorem 6 [2] For a symmetric and positive definite matrix, the relaxation method converges if and only if Proof: Left to reader as an exercise. The Theorems 3 and 6 guarantees convergence of GaussSeidel method or relaxation method when matrix is symmetric and positive definite. Now, what do we do if matrix in is not symmetric and positive definite? We can multiply both the sides of the equation by the original problem as follows and transform (100) If matrix is nonsingular, then matrix is always symmetric and positive definite as (101) Now, for the transformed problem, we are guaranteed convergence if we use the GaussSeidel method or relaxation method such that. Example 7 [3] Consider system (102) For Jacobi method (103) (104) Thus, the error norm at each iteration is reduced by factor of 0.5For GaussSeidel method
9 (105) (106) Thus, the error norm at each iteration is reduced by factor of 1/4. This implies that, for the example under consideration For relaxation method, (107) (108) (109) (110) (111) (112) Now, if we plot v/s, then it is observed that at From equation (110), it follows that (113) at optimum Now, (114) (115) (116) (117) This is a major reduction in spectral radius when compared to GaussSeidel method. Thus, the error
10 norm at each iteration is reduced by factor of 1/16 ( ) if we choose Example 8 Consider system (118) If we use GaussSeidel method to solve for the iterations do not converge as (119) (120) Now, let us modify the problem by premultiplying by on both the sides, i.e. the modified problem is The modified problem becomes (121) The matrix is symmetric and positive definite and, according to Theorem 3, the iterations should converge if GaussSeidel method is used. For the transformed problem, we have (122) (123) and within 220 iterations (termination criterion ), we get following solution (124) which is close to the solution (125) computed as
11 Table 4: Rate of Convergence of Iterative Methods Example 9 Consider system (126) If it is desired to solve the resulting problem using Jacobi method / GaussSeidel method, will the iterations converge? To establish convergence of Jacobi / GaussSeidel method, we can check whether A is strictly diagonally dominant. Since the following inequalities hold matrix A is strictly diagonally dominant, which is a sufficient condition for convergence of Jacobi / GaussSeidel iterations Theorem 2. Thus, Jacobi / GaussSeidel iterations will converge to the solution starting from any initial guess. From these examples, we can clearly see that the rate of convergence depends on A comparison of rates of convergence obtained from analysis of some simple problems is presented in Table 4[2]
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