By Eugene Isaacson, Herbert Bishop Keller
Very good advanced-undergraduate and graduate textual content covers norms, numerical answer of linear structures and matrix factoring, iterative ideas of nonlinear equations, eigenvalues and eigenvectors, polynomial approximation and extra. cautious research and rigidity on suggestions for constructing new equipment. Examples and difficulties. 1966 version. Bibliography.
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Extra info for Analysis of Numerical Methods
Since A -1 and f(j) need not have any zero or unit elements, it requires in general n2 ops. to compute A -If(j). [Sec. 2] A PRIORI ERROR ESTIMATES; CONDITION NUMBER 37 Thus, to solve m systems requires mn 2 ops. and if we include the n 3 operations to compute A-I, we get the result: (II) ops. are required to solve the m systems (7), when using the inverse matrix. Upon comparing (9) and (II) it follows that for any value ofm the use of the inverse matrix is less efficient than using direct elimination.
But if (b) holds for some norm, then Theorem 2' implies there exists an M such that Hence, (a) holds. Next we show that (b) and (c) are equivalent. Note that by Theorem 2' there is no loss in generality if we assume the norm to be a natural norm. But then, by Lemma 2 and the fact that '\(Am) = ,\m(A), we have IIAml1 ~ p(Am) = pm(A), [Sec. 1] CONVERGENT MATRICES 15 so that (b) implies (c). On the other hand, if (c) holds, then by Theorem 3 we can find an f' > 0 and a natural norm, say N(·), such that N(A) ::; peA) + f' == 8 < I.
I, in the lower triangular part of the original matrix, A. (That is, mi. D. Thus, no operations in (3a) ever need to be repeated. From (3) and (4) we see that in eliminating Xk-b a square submatrix of order n - k + I is determined and the last n - k + 1 components of each right-hand side are modified. Each element of the new submatrix and subvectors is obtained by performing a multiplication (and an addition which we ignore), but the quotients which appear as factors in (3) are computed only once.
Analysis of Numerical Methods by Eugene Isaacson, Herbert Bishop Keller