Numerical Mathematics

Numerical mathematics is the branch of mathematics that proposes, develops, analyzes and applies methods from scientific computing to several fields including analysis, linear algebra, geometry, approximation theory, functional equations, optimization and differential equations. Other disciplines, such as physics, the natural and biological sciences, engineering, and economics and the financial sciences frequently give rise to problems that need scientific computing for their solutions.
As such, numerical mathematics is the crossroad of several disciplines of great relevance in modern applied sciences, and can become a crucial tool for their qualitative and quantitative analysis.
One of the purposes of this book is to provide the mathematical foundations of numerical methods, to analyze their basic theoretical properties (stability, accuracy, computational complexity) and demonstrate their performances on examples and counterexamples which outline their pros and cons. This is done using the MATLAB software environment which is user-friendly and widely adopted. Within any specific class of problems, the most appropriate scientific computing algorithms are reviewed, their theoretical analyses are carried out and the expected results are verified on a MATLAB computer implementation. Every chapter is supplied with examples, exercises and applications of the discussed theory to the solution of real-life problems.
This book is addressed to senior undergraduate and graduate students with particular focus on degree courses in Engineering, Mathematics, Physics and Computer Sciences. The attention which is paid to the applications and the related development of software makes it valuable also for researchers and users of scientific computing in a large variety of professional fields.

Inhalt
Preface Part I: Getting Started 1. Foundations of Matrix Analysis 2. Principles of Numerical Mathematics Part II. Numerical Linear Algebra 3. Direct Methods for the Solution of Linear Systems 4. Iterative Methods for Solving Linear Systems 5. Approximation of Eigenvalues and Eigenvectors Part III: Around Functions and Functionals 6. Rootfinding for Nonlinear Equations 7. Nonlinear Systems and Numerical Optimization 8. Polynomial Interpolation 9. Numerical Integration Part IV: Transforms, Differentiation and Problem Discretization 10. Orthogonal Polynomials in Approximation Theory 11. Numerical Solution of Ordinary Differential Equations 12. Two-Point Boundary Value Problems 13. Parabolic and Hyperbolic Initial Boundary Value Problems References Index of MATLAB Programs Index