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A) b) Show that in solving Tx = b, if b j ;c 0 then Xj ;c 0 for i :S;j :s; N. Show that T- 1 is a full lower triangular matrix. 1 depend upon the no-cancellation assumption? Explain. 4? 4 for upper triangular matrices. 6) Suppose you have numerous N by N lower triangular systems of the form Ly = b to solve, where Land b are both sparse. It is known that the solution y is also sparse for these problems. You have a choice of two storage schemes for L, as illustrated by the 5 by 5 example below; one is column oriented and one is row oriented.

F l. 2 8 2 2 2 4 This simple example illustrates that a judicious choice of P can result in dramatic reductions in fill-in and arithmetic requirements. Therefore, in solving a given linear-equation problem Ax -b. 24 Sec. 1: The Factorization Algorithm the general procedure involves first finding a permutation or ordering P of the given problem. Then the system is expressed as (PApT)(px) = Ph and Cholesky's method is applied to the symmetric positive definite matrix PAp T yielding the triangular factorization LL T.

0 As mentioned above, the attraction of this approach is its simplicity. However, it has some potentially serious weaknesses. 2 will be inefficient. 3). Thus, there are prol:ilems for which band methods are simply inappropriate. Perhaps the most persuasive reason for not being very enthusiastic about band schemes is that the envelope schemes discussed in the next section share all the advantages of simplicity enjoyed by band schemes, with very few of the disadvantages. 1) Suppose A is an N by N symmetric positive definite matrix with bandwidth~.

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