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Unformatted text preview: Lecture 12 LU Decomposition In many applications where linear systems appear, one needs to solve A x = b for many different vectors b . For instance, a structure must be tested under several different loads, not just one. As in the example of a truss (9.2), the loading in such a problem is usually represented by the vector b . Gaussian elimination with pivoting is the most efficient and accurate way to solve a linear system. Most of the work in this method is spent on the matrix A itself. If we need to solve several different systems with the same A , and A is big, then we would like to avoid repeating the steps of Gaussian elimination on A for every different b . This can be accomplished by the LU decomposition , which in effect records the steps of Gaussian elimination. LU decomposition The main idea of the LU decomposition is to record the steps used in Gaussian elimination on A in the places where the zero is produced. Consider the matrix: A = 1- 2 3 2- 5 12 2- 10 . The first step of Gaussian elimination is to subtract 2 times the first row from the second row. In order to record what we have done, we will put the multiplier, 2, into the place it was used to make...
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