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

Alfio Quarteroni, Riccardo Sacco, Fausto Saleri

Numerical Mathematics (Hardcover)

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Description 

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.

商品描述(中文翻譯)

**描述**

數值數學是數學的一個分支，提出、發展、分析並應用科學計算的方法於多個領域，包括分析、線性代數、幾何、近似理論、函數方程、優化及微分方程。其他學科，如物理學、自然科學和生物科學、工程學、經濟學及金融科學，經常會產生需要科學計算來解決的問題。

因此，數值數學是現代應用科學中幾個重要學科的交匯點，並且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，分析其基本理論特性（穩定性、準確性、計算複雜度），並通過示例和反例展示其性能，概述其優缺點。這是使用 MATLAB 軟體環境來完成的，該環境使用友好且廣泛採用。在任何特定類別的問題中，回顧最合適的科學計算算法，進行理論分析，並在 MATLAB 的計算實現上驗證預期結果。每一章都提供了示例、練習和將所討論理論應用於解決現實問題的應用。

本書針對高年級本科生和研究生，特別關注工程、數學、物理和計算機科學的學位課程。對於應用及相關軟體開發的重視，使其對於各種專業領域的科學計算研究者和使用者也具有價值。

在本第二版中，圖片、表格和程式標題的可讀性得到了改善。對於迭代方法和多項式近似的章節也進行了幾處修改。

**目錄**

系列前言.- 前言.- I 開始.- 1. 矩陣分析的基礎.- 2 數值數學的原則.- II 數值線性代數.- 3 線性系統的直接解法.- 4 解線性系統的迭代方法.- 5 特徵值和特徵向量的近似.- III 函數與泛函.- 6 非線性方程的根尋找.- 7 非線性系統與數值優化.- 8 多項式插值.- 9 數值積分.- IV 變換、微分與問題離散化.- 10 近似理論中的正交多項式.- 11 常微分方程的數值解.- 12 二點邊值問題.- 13 拋物線和雙曲線初邊值問題.- 參考文獻.- MATLAB 程式索引.- 索引.

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