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Unformatted text preview: CS205 Class 2 Linear Systems Continued Covered in class: all sections 1. When constructing k M we needed to divide by k a which is the element on the diagonal. This could pose difficulties if the diagonal element was zero. For example, consider the matrix equation 4 2 2 4 9 3 8 2 3 7 10 x y z- - = -- where forming 1 M would lead to a division by 11 a = . In general, LU factorization fails if a small number shows up on the diagonal at any stage. a. When small numbers occur on the diagonal, one can change rows. In general, one can switch the current row k with a row j below it with j > k in order to get a larger diagonal element. This is called pivoting. Partial pivoting is the process of switching rows to always get the largest diagonal element, and full pivoting consists of switching both rows and columns to always obtain the largest possible diagonal element. Note that when switching columns, the order of the elements in the unknown vector needs to be changed in the obvious way, i.e. there is a corresponding row switching in the vector of unknowns. b. A permutation matrix can be used to switch rows in a matrix. Permutation matrices can be constructed by performing the desired row switch on the identity matrix. For example, a permutation matrix that switches the first and third rows of a matrix is obtained by switching the first and third rows of the identity matrix. For a 3x3 matrix, 1 1 1 P = ....
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- Spring '07