By Eugene Isaacson, Herbert Bishop Keller
First-class advanced-undergraduate and graduate textual content covers norms, numerical resolution of linear structures and matrix factoring, iterative strategies of nonlinear equations, eigenvalues and eigenvectors, polynomial approximation and extra. cautious research and rigidity on recommendations for constructing new equipment. Examples and difficulties. 1966 variation. Bibliography.
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Additional info for Analysis of Numerical Methods
We refer to this as the notion of a well-posed computing problem. First, we must clarify what is meant by a "computing problem" in general. , rounding rules) to be used on specified data. Such a computing problem may have as its object, for example, the determination of the roots of a quadratic equation or of an approximation to the solution of a nonlinear partial differential equation. How any such rules are determined for a particular purpose need not concern us at present (this is, in fact, what much of the rest of this book is about).
Rounding" in the statement. ] 2. (a) Find a representation for (b) If fl (~l C} Cl > C2 > ... > C> 0, ll in what order should fl(tl Ct) be cal- culated to minimize the effect of rounding? 3. 8 1 82 • . 8t ) Q 0 or 1? 3. WELL·POSED COMPUTATIONS Hadamard introduced the notion of well-posed or properly posed problems in the theory of partial differential equations (see Section 0 of Chapter 9). However, it seems that a related concept is quite useful in discussing computational problems of almost all kinds.
Then the error estimate (15) becomes simply Ilhll (17) liXf ~ fL118AII/IIAII 1 - fLI18AII/IIAII' and it is clear that only 118A II in this error bound depends upon the roundoff errors and method of computation. In the case of Gaussian elimination we have seen in Theorem 1, that exact calculations yield the factorization (5b), LV = A. Here Land U are, respectively, lower and upper triangular matrices determined by (5c) and (3a). £' and qt. £,(;7f == A + E. There are additional rounding errors committed in computing g defined by (3b) or (5d), and in the final back substitution (6b) in attempting to compute the solution x.
Analysis of Numerical Methods by Eugene Isaacson, Herbert Bishop Keller