This unique book provides a comprehensive introduction to computational mathematics, which forms an essential part of contemporary numerical algorithms, scientific computing and optimization. It uses a theorem-free approach with just the right balance between mathematics and numerical algorithms. This edition covers all major topics in computational mathematics with a wide range of carefully selected numerical algorithms, ranging from the root-finding algorithm, numerical integration, numerical methods of partial differential equations, finite element methods, optimization algorithms, stochastic models, nonlinear curve-fitting to data modelling, bio-inspired algorithms and swarm intelligence. This book is especially suitable for both undergraduates and graduates in computational mathematics, numerical algorithms, scientific computing, mathematical programming, artificial intelligence and engineering optimization. Thus, it can be used as a textbook and/or reference book.

From the winner of the Turing Award and the Abel Prize, an introduction to computational complexity theory, its connections and interactions with mathematics, and its central role in the natural and social sciences, technology, and philosophy Mathematics and Computation provides a broad, conceptual overview of computational complexity theory—the mathematical study of efficient computation. With important practical applications to computer science and industry, computational complexity theory has evolved into a highly interdisciplinary field, with strong links to most mathematical areas and to a growing number of scientific endeavors. Avi Wigderson takes a sweeping survey of complexity theory, emphasizing the field’s insights and challenges. He explains the ideas and motivations leading to key models, notions, and results. In particular, he looks at algorithms and complexity, computations and proofs, randomness and interaction, quantum and arithmetic computation, and cryptography and learning, all as parts of a cohesive whole with numerous cross-influences. Wigderson illustrates the immense breadth of the field, its beauty and richness, and its diverse and growing interactions with other areas of mathematics. He ends with a comprehensive look at the theory of computation, its methodology and aspirations, and the unique and fundamental ways in which it has shaped and will further shape science, technology, and society. For further reading, an extensive bibliography is provided for all topics covered. Mathematics and Computation is useful for undergraduate and graduate students in mathematics, computer science, and related fields, as well as researchers and teachers in these fields. Many parts require little background, and serve as an invitation to newcomers seeking an introduction to the theory of computation. Comprehensive coverage of computational complexity theory, and beyond High-level, intuitive exposition, which brings conceptual clarity to this central and dynamic scientific discipline Historical accounts of the evolution and motivations of central concepts and models A broad view of the theory of computation's influence on science, technology, and society Extensive bibliography

Computational Mathematics: Models, Methods, and Analysis with MATLAB and MPI explores and illustrates this process. Each section of the first six chapters is motivated by a specific application. The author applies a model, selects a numerical method, implements computer simulations, and assesses the ensuing results. These chapters include an abundance of MATLAB code. By studying the code instead of using it as a "black box, " you take the first step toward more sophisticated numerical modeling. The last four chapters focus on multiprocessing algorithms implemented using message passing interface (MPI). These chapters include Fortran 9x codes that illustrate the basic MPI subroutines and revisit the applications of the previous chapters from a parallel implementation perspective. All of the codes are available for download from www4.ncsu.edu./~white. This book is not just about math, not just about computing, and not just about applications, but about all three--in other words, computational science. Whether used as an undergraduate textbook, for self-study, or for reference, it builds the foundation you need to make numerical modeling and simulation integral parts of your investigational toolbox.

This fantastic and deep book about how to use Sage for learning and doing mathematics at all levels perfectly complements the existing Sage documentation. It is filled with many carefully thought through examples and exercises, and great care has been taken to put computational functionality into proper mathematical context. Flip to almost any random page in this amazing book, and you will learn how to play with and visualize some beautiful part of mathematics. --- William A. Stein, CEO, SageMath, and professor of mathematics, University of Washington SageMath, or Sage for short, is an open-source mathematical software system based on the Python language and developed by an international community comprising hundreds of teachers and researchers, whose aim is to provide an alternative to the commercial products Magma, Maple, Mathematica, and MATLAB. To achieve this, Sage relies on many open-source programs, including GAP, Maxima, PARI, and various scientific libraries for Python, to which thousands of new functions have been added. Sage is freely available and is supported by all modern operating systems. Sage provides a wonderful scientific and graphical calculator for high school students, and it efficiently supports undergraduates in their computations in analysis, linear algebra, calculus, etc. For graduate students, researchers, and engineers in various mathematical specialties, Sage provides the most recent algorithms and tools, which is why several universities around the world already use Sage at the undergraduate level.

EACM is a comprehensive reference work covering the vast field of applied and computational mathematics. Applied mathematics itself accounts for at least 60 per cent of mathematics, and the emphasis on computation reflects the current and constantly growing importance of computational methods in all areas of applications. EACM emphasizes the strong links of applied mathematics with major areas of science, such as physics, chemistry, biology, and computer science, as well as specific fields like atmospheric ocean science. In addition, the mathematical input to modern engineering and technology form another core component of EACM.

Computational Mathematics in Engineering and Applied Science provides numerical algorithms and associated software for solving a spectrum of problems in ordinary differential equations (ODEs), differential algebraic equations (DAEs), and partial differential equations (PDEs) that occur in science and engineering. It presents detailed examples, each

New mathematical insights and rigorous results are often gained through extensive experimentation using numerical examples or graphical images and analyzing them. Today computer experiments are an integral part of doing mathematics. This allows for a more systematic approach to conducting and replicating experiments. The authors address the role of