Scientific Computing

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CS205 - Class 3 1. When solving systems of equations, we want Ax=b . We define the residual as r=b-Ax , and note that the residual gives us some measure of the error. Of course the goal is to attain r= 0. 2. Both Gaussian Elimination and the y factorization method are examples of direct methods. The other kind of method is called an iterative method where one starts with an initial guess 1 x and iterates through a sequence 1 x , 2 x , 3 x … ending up with a final guess of m x . a. Issues here include both finding an initial guess 1 x and deciding on a stopping criterion that says that m x , for some m, is good enough. b. The best preconditioner, if we could guess it would be A -1 . Then we would get Ix=A -1 b . On a sparse matrix we might include incomplete factorizations such as an incomplete Cholesky preconditioner where we don’t allow fill-in of non-zero entries of the L matrix not in the orginal A matrix. 3. So far we have discussed solving Ax=b for square n n × matrices A . For more general n m × matrices, there are a variety of scenarios. a. When m < n , the problem is underdetermined since there is not enough information to determine a unique solution for all the variables. Usually m<n implies that there are infinite solutions . However, in some cases, there may be contradictory equations leading to the absence of any solutions. b. When m > n , the problem is overdetermined although this in itself doesn’t tell us everything about the nature of the solution. For example, if enough equations are linear combinations of each other, there can still be a unique solution or infinite solutions. i. We can use the rank of the matrix to enumerate the possibilities. Recall that the rank of a matrix is the number of linearly independent columns that it has. Thus a n m × matrix has at most a rank of n. ii. If the rank < n some columns are linear combinations of others and we say that the matrix is rank-deficient and there may be an infinite number of solutions.
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