Exploiting Errors for Efficiency: A Survey from Circuits to Algorithms <s> Scientific computing <s> The World of Scientific Computing. Letting It Fly. Initial Value Problems. Pinning It Down: Boundary Value Problems. More on Linear Systems of Equations. Life Is Really Nonlinear. Is There More Than Finite Differences? N Important Numbers. Space and Time. The Curse of Dimensionality. Appendixes: Analysis of Differential Equations. Linear Algebra. Bibliography. Index. <s> BIB001 </s> Exploiting Errors for Efficiency: A Survey from Circuits to Algorithms <s> Scientific computing <s> A standard for the notation of the most used quantities and operators in interval analysis is proposed. <s> BIB002 </s> Exploiting Errors for Efficiency: A Survey from Circuits to Algorithms <s> Scientific computing <s> Preface Part I. Pitfalls in Numerical Computation: 1. What can go wrong in scientific computing? Bo Einarsson 2. Assessment of accuracy and reliability Ronald F. Boisvert, Ronald Cools and Bo Einarsson 3. Approximating integrals, estimating errors and giving the wrong solution for a deceptively easy problem Ronald Cools 4. An introduction to the quality of computed solutions Sven Hammarling 5. Qualitative computing Francoise Chaitin-Chatelin and Elisabeth Traviesas-Cassan Part II. Diagnostic Tools: 6. PRECISE and the quality of reliable numerical software Francoise Chaitin-Chatelin and Elisabeth Traviesas-Cassan 7. Tools for the verification of approximate solutions to differential equations Wayne H. Enright Part III. Technology for Improving Accuracy and Reliability: 8. General methods for implementing reliable and correct software Bo Einarsson 9. The use and implementation of interval data types G. William Walster 10. Computer-assisted proofs and self-validating methods Siegfried M. Rump 11. Hardware assisted algorithms Craig C. Douglas and Hans Petter Langtangen 12. Issues in accurate and reliable use of parallel computing in numerical programs William D. Gropp 13. Software reliability engineering of numerical systems Mladen A. Vouk Bibliography Index. <s> BIB003
Scientific computing can be defined as "the collection of tools, techniques, and theories required to solve on a computer mathematical models of problems in science and engineering" BIB001 . Most of these models are real-valued, and exact analytical solutions rarely exist or are costly to compute   . As a result, numerical approximations and their associated quality-efficiency tradeoffs have always been important in scientific computing BIB003 . These numerical approximations are introduced at different levels of abstraction. Because the real-world is too complex to be represented exactly, practical considerations require resorting to models, incurring modeling errors  . Even with a model in hand, analytical solutions may not exist and numerical solutions are needed to approximate the exact answers [26, 46] , introducing further deviations from the expected result. And finally, most models are real-valued and thus have to be approximated by finite-precision arithmetic, adding roundoff errors  . Roundoff errors can be bounded to some extent automatically using techniques such as interval arithmetic BIB002 . Dealing with most of the errors introduced by modeling, numerical approximation, and finite-precision arithmetic, is rarely automated by software tools. The state of the art in dealing with modeling and numerical errors often requires manual intervention of the programmer or domain expert and is typically on a per-application basis. Because of the resulting complexity of the error analysis, the resulting error bounds are often only asymptotic.