Lecture 12 handout.pdf - Math 472 Lecture 12 Chapter 2...

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Math 472 Lecture 12 Chapter 2 Systems of Equations Ruoyu Wu U Michigan 1 / 11
2.5 Iterative methods Gaussian elimination is called a direct method for solving systems of linear equation. In theory, it gives the exact solution within a finite number of steps. Indirect methods are almost always iterative in nature: a simple process is applied repeatedly to generate a sequence of approximated solution vectors, which ideally converges to the actual solution. The iteration is terminated when an approximate solution is obtained within a specific accuracy or after a certain number of iterations. But why do we care about iterative methods? Polishing: when we have obtained the solution x to Ax = b but later on A or b changes a little. Sparse matrix: when most entries of the matrix A are 0 . If interested in details, see Section 2.5.4 in the textbook. 2 / 11
2.5.1 Jacobi method The Jacobi method is a form of FPI for Ax = b . To find the iteration, we simply solve the i -th equation for the i -th unknown , then iterate as in the FPI. Example 2.19 Apply the Jacobi method to solve the system 3 u + v = 5 , u + 2 v = 5 . Solving the first equation for u and the second one for v yields u = 5 - v 3 and v = 5 - u 2 .

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