Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 10.2

Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 10.2

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10.2 ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS As a numerical technique, Gaussian elimination is rather unusual because it is direct . That is, a solution is obtained after a single application of Gaussian elimination. Once a “solu- tion” has been obtained, Gaussian elimination offers no method of refinement. The lack of refinements can be a problem because, as the previous section shows, Gaussian elimination is sensitive to rounding error. Numerical techniques more commonly involve an iterative method. For example, in calculus you probably studied Newton’s iterative method for approximating the zeros of a differentiable function. In this section you will look at two iterative methods for approxi- mating the solution of a system of n linear equations in n variables. The Jacobi Method The first iterative technique is called the Jacobi method, after Carl Gustav Jacob Jacobi (1804–1851). This method makes two assumptions: (1) that the system given by has a unique solution and (2) that the coefficient matrix A has no zeros on its main diago- nal. 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. 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, Initial approximation and substitute 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 x i ( x 1 , x 2 , x 3 , . . . , x n ), x n 5 1 a nn ( b n 2 a n 1 x 1 2 a n 2 x 2 2 . . . 2 a n , n 2 1 x n 2 1 d . . . x 2 5 1 a 22 ( b 2 2 a 21 x 1 2 a 23 x 3 2 . . . 2 a 2 n x n ) x 1 5 1 a 11 ( b 1 2 a 12 x 2 2 a 13 x 3 2 . . . 2 a 1 n x n ) x 2 , x 1 , a 11 , a 22 , . . . , a nn a n 1 x 1 1 a n 2 x 2 1 . . . 1 a nn x n 5 b n . . . . . . . . . . . . a 21 x 1 1 a 22 x 2 1 . . . 1 a 2 n x n 5 b 2 a 11 x 1 1 a 12 x 2 1 . . . 1 a 1 n x n 5 b 1 578 CHAPTER 10 NUMERICAL METHODS
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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. This procedure is illustrated in Example 1. EXAMPLE 1 Applying the Jacobi Method 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 as a convenient initial approximation. So, the first approximation is Continuing this procedure, you obtain the sequence of approximations shown in Table 10.1.
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This note was uploaded on 02/24/2012 for the course MATH 310 taught by Professor Staff during the Spring '08 term at VCU.

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Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 10.2

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