Introduction to Numerical Analysis: Second Edition

This book by Hildebrand along with its companion Methods of Applied Mathematics are a perfect match for students of Science and Engineering. It is a must for this era where the blind use of mathematical software is prevalent masking the intricacies of the method. The details of the method employed in the software should be understood to have any faith in the software output.

This is an excellent numerical methods book ,covering many techniques especially the polynomials which are not used much by practising engineers today , but who knows,may find applications in odd places.Prof Hildebrand easily builds up so that one can understand the steps from a first course in Calculus...Glad that Dover has reprinted this classic book.

Although old, this is still an outstanding introduction to a wide range of topics in numerical analysis. I get impatient with the amount of detail Hildebrand devotes to some topics, but that's because those are topics where I already know the techniques and pitfalls. However, I have one serious criticism of this book. Hildebrand in very many places drags in the question of inherent errors in input data, but fails to distinguish the different views one must take depending on how one got involved with some topic. In 50+ years of doing numerical analysis and numerical software from time to time, I have come to realize that three quite different issues of inherent error occur. First, one may be working with scientists or engineers to derive results for a specific problem or set of problems. In this case, one must ask two pertinent questions, and keep asking until one gets clear answers: "How are you getting the input data?" and, "What are you going to do with the results?" Given answers to these two questions, one can do analysis and computation knowing from the start how accurate the input data is likely to be, and how much that matters to the results. Hildebrand pays little attention to the quite complicated problem of how one should do the analysis and programming in those situations. Second, it may happen that there is no input data from the real world, and hence no inherent error; the input data is conjured up out of whole cloth, as happens in many calculations in "computational physics". In those cases, one wants to produce results that accurately reflect the hypotheses provided by the people with the problem to be solved. Usually, one finds in such cases that the more accurately one can do the computation or analysis, the better one can serve one's users. Third, and most difficult, is the situation where one is writing a utility routine for use by large numbers of people, most of whom one will never encounter. Everyone who has done much numerical programming faces this issue from time to time. Here the problem is that the users are likely to place absolute faith in the results, even in cases where you, as the implementer of the software or originator of the analysis, may know all too well that the results are unstable with respect to very minor variations in input data. This occurs with monotonous regularity, for example, in routines that manipulate matrices to derive such quantities as eigenvalues and eigenvectors. In my own experience, a high proportion of the actual matrices that users present to "utility packages" are ill-conditioned, and there's a reason for this. If the problem were well-conditioned, it wouldn't be a problem for the scientists or engineers or financial types who need a solution; they would know a priori from experience what the answers are. I have no good answer for how one should think about such "utility software" and neither does Hildebrand. The way I deal with it myself is to ensure that mathematically accurate

This is a reprint of the 1974 2nd edition. So what is Numerical Analysis? It's the down-and-dirty methods of approximation and interpolation of equations that don't have closed-formed mathematical solutions. Just about every real-world problem in material engineering from pipe flow, wing design, convection currents to multivariate econometrics have to resort to numerical approximations. You'll find all the familiar names from your undergrad Math and Physics courses here (Newton, Gauss, Largrange, etc.); however, advanced methods in Numerical Analysis has changed tremendously since this book was published. Since NA is dependent on present computing power, what was once too expensive or unthinkable in the 70s can be done today. However, it's still a great introduction and a great bargain from Dover. The writing style is informal and conversationally peer-to-peer, rather than teacher-to-student. There is no historic consciousness placing methods and men in context. You won't find programming algorithms here (not even Fortran or Pascal). There are probably better books out there for what ever your specific speciality is, but at five times the price of this Dover reprint. You'll will find the old favorites here. The book covers the various finite difference approximations (forward, backward and central differences). It uses the operational approach for these. The later chapters cover splines, continued fraction and iterative methods. More importantly it covers the difference between round-off error v. truncation of divergent series in approximations -- something that still confuses practicing professionals. Be warned there have been many improvements in theory and methods in finite element methods of Fluid Dynamics and other 3D modeling (bezier and NURBS); And, the whole world of Complexity and Chaos theory happened well AFTER this book was published. Calculus and Differential EQs are prerequisite, there's no attempt at introduction in the text.