Numerical Mathematics (Hardcover) (數值數學 (精裝版))

Alfio Quarteroni, Riccardo Sacco, Fausto Saleri

Numerical Mathematics (Hardcover)

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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.

In this second edition, the readability of pictures, tables and program headings have been improved. Several changes in the chapters on iterative methods and on polynomial approximation have also been added.

 

 
Table of Contents
Series Preface.- Preface.- I Getting Started.- 1. Foundations of Matrix Analysis.- 2 Principles of Numerical Mathematics.- 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.- III Around Functions and Functionals.- 6 Rootfinding for Nonlinear Equations.- 7 Nonlinear Systems and Numerical Optimization.- 8 Polynomial Interpolation.- 9 Numerical Integration.- 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.

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數值數學是數學的一個分支，提出、發展、分析和應用科學計算方法於多個領域，包括分析、線性代數、幾何、逼近理論、函數方程、最佳化和微分方程。其他學科，如物理學、自然和生物科學、工程學、經濟學和金融科學，經常出現需要科學計算解決的問題。

因此，數值數學是現代應用科學中幾個相關學科的交叉點，可以成為其定性和定量分析的關鍵工具。

本書的目的之一是提供數值方法的數學基礎，分析它們的基本理論性質（穩定性、準確性、計算複雜性）並在例子和反例中展示它們的優點和缺點。這是通過使用廣泛使用且用戶友好的MATLAB軟件環境來完成的。在任何特定類別的問題中，將回顧最適合的科學計算算法，進行理論分析並在MATLAB計算機實現中驗證預期結果。每個章節都附有例子、練習和將討論的理論應用於解決實際問題。

本書針對工程、數學、物理和計算機科學等學位課程的高年級本科生和研究生。對於各種專業領域的科學計算研究人員和用戶，本書所關注的應用和相關軟件開發也具有價值。

在第二版中，改善了圖片、表格和程序標題的可讀性。還增加了對迭代方法和多項式逼近章節的一些變更。

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