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An Introduction to Numerical Analysis（数值分析引论，英文版）

样章
作者：
金小庆 刘伟辉 著
定价：
59.00 元
版面字数：
240千字
开本：
16开
装帧形式：
平装
版次：
1
最新版次
印刷时间：
2026年
ISBN：
978-7-04-066548-2
物料号：
66548-00
出版时间：
2026-01-19
读者对象：
学术著作
一级分类：
自然科学
二级分类：
数学与统计
三级分类：
计算数学
暂无
  • 目录
    • 前辅文
      • Preface
        • 1 Introduction
          • 1.1 Preliminary
            • 1.2 Errors
              • 1.2.1 Errors provided by computers
                • 1.2.2 Floating-point expression
                  • 1.2.3 Error bound and significant digit
                    • 1.2.4 Relative error and significant digit
                    • Exercises
                    • 2 Interpolation
                      • 2.1 Lagrange interpolation
                        • 2.1.1 Linear interpolation
                          • 2.1.2 Quadratic interpolation
                            • 2.1.3 General case
                              • 2.1.4 Lagrange remainder
                              • 2.2 Newton interpolation
                                • 2.2.1 Divided differences
                                  • 2.2.2 Properties of divided differences
                                    • 2.2.3 Newton formula
                                    • 2.3 Hermite interpolation
                                      • 2.4 Piecewise interpolation
                                        • 2.4.1 Piecewise linear interpolation
                                          • 2.4.2 Piecewise cubic Hermite interpolation
                                          • 2.5 Spline interpolation
                                            • Exercises
                                            • 3 Numerical Integration and Differentiation
                                              • 3.1 Numerical quadrature
                                                • 3.1.1 Mean value theorem for definite integrals
                                                  • 3.1.2 Basic forms
                                                    • 3.1.3 Algebraic accuracy
                                                      • 3.1.4 Interpolation-based quadrature
                                                      • 3.2 Newton-Cotes formulas
                                                        • 3.2.1 General formula
                                                          • 3.2.2 Algebraic accuracy of Newton-Cotes formula
                                                            • 3.2.3 Remainder of Newton-Cotes formula
                                                              • 3.2.4 Composite numerical integration
                                                              • 3.3 Gaussian quadrature rule
                                                                • 3.3.1 High algebraic accuracy quadrature
                                                                  • 3.3.2 Gaussian nodes
                                                                    • 3.3.3 Legendre polynomials
                                                                    • 3.4 Numerical differentiation
                                                                      • 3.4.1 Richardson extrapolation
                                                                        • 3.4.2 Interpolation-based differentiation
                                                                        • Exercises
                                                                        • 4 Finite Difference Methods for ODEs
                                                                          • 4.1 Euler method
                                                                            • 4.1.1 Deduction of Euler method
                                                                              • 4.1.2 Implicit Euler method
                                                                                • 4.1.3 Two-step Euler method
                                                                                • 4.2 Modified Euler method
                                                                                  • 4.2.1 The trapezoidal method
                                                                                    • 4.2.2 Modified Euler method
                                                                                    • 4.3 Runge-Kutta methods
                                                                                      • 4.3.1 Basic idea of Runge-Kutta method
                                                                                        • 4.3.2 Runge-Kutta method of order two
                                                                                          • 4.3.3 Runge-Kutta method of order three
                                                                                            • 4.3.4 Runge-Kutta method of order four
                                                                                              • 4.3.5 Step size of Runge-Kutta method
                                                                                              • 4.4 Adams methods
                                                                                                • 4.4.1 Adams-Bashforth methods
                                                                                                  • 4.4.2 Adams-Moulton methods
                                                                                                    • 4.4.3 Adams predictor-corrector method
                                                                                                    • 4.5 Convergence and stability
                                                                                                      • 4.5.1 Convergence
                                                                                                        • 4.5.2 Stability
                                                                                                        • 4.6 Boundary value problem
                                                                                                          • Exercises
                                                                                                          • 5 Iterative Root-finding Algorithms
                                                                                                            • 5.1 Fixed-point iteration
                                                                                                              • 5.1.1 Basic idea
                                                                                                                • 5.1.2 Convergence of an iterative method
                                                                                                                  • 5.1.3 Convergence rate of an iterative method
                                                                                                                  • 5.2 Acceleration of iterative methods
                                                                                                                    • 5.2.1 A modified iterative formula
                                                                                                                      • 5.2.2 Aitken’s acceleration method
                                                                                                                      • 5.3 Newton’s method
                                                                                                                        • 5.3.1 Deduction of the formula
                                                                                                                          • 5.3.2 An application
                                                                                                                          • 5.4 Secant method
                                                                                                                            • Exercises
                                                                                                                            • 6 Direct Methods for Linear Systems
                                                                                                                              • 6.1 Review
                                                                                                                                • 6.1.1 Basic knowledge
                                                                                                                                  • 6.1.2 Operation cost of Cramer’s rule and Gaussian elimination .
                                                                                                                                  • 6.2 LU factorization
                                                                                                                                    • 6.2.1 Triangular linear systems
                                                                                                                                      • 6.2.2 Gaussian transform matrix
                                                                                                                                        • 6.2.3 Computation of LU factorization
                                                                                                                                        • 6.3 LU factorization with pivoting
                                                                                                                                          • 6.3.1 LU factorization with permutations
                                                                                                                                            • 6.3.2 Pivoting technique
                                                                                                                                            • 6.4 Cholesky factorization
                                                                                                                                              • Exercises
                                                                                                                                              • 7 Norms and Perturbation Analysis
                                                                                                                                                • 7.1 Norms
                                                                                                                                                  • 7.2 Perturbation analysis of linear systems
                                                                                                                                                    • Exercises
                                                                                                                                                    • 8 Conjugate Gradient Method
                                                                                                                                                      • 8.1 Steepest descent method
                                                                                                                                                        • 8.2 Conjugate gradient method
                                                                                                                                                          • 8.2.1 Basic idea of CG method
                                                                                                                                                            • 8.2.2 Properties of CG method
                                                                                                                                                            • 8.3 Practical CG method and convergence analysis
                                                                                                                                                              • 8.3.1 Practical CG method
                                                                                                                                                                • 8.3.2 Convergence analysis of CG method
                                                                                                                                                                • 8.4 Preconditioning technique
                                                                                                                                                                  • 8.5 Three classes of preconditioners
                                                                                                                                                                    • 8.5.1 Diagonal preconditioner
                                                                                                                                                                      • 8.5.2 Incomplete Cholesky factorization preconditioner
                                                                                                                                                                        • 8.5.3 Optimal (circulant) preconditioner
                                                                                                                                                                        • Exercises
                                                                                                                                                                        • 9 Matrix Eigenvalue Computation
                                                                                                                                                                          • 9.1 Basic properties
                                                                                                                                                                            • 9.2 Power method and inverse power method
                                                                                                                                                                              • 9.3 QR method
                                                                                                                                                                                • Exercises
                                                                                                                                                                                • A Chebyshev Polynomial
                                                                                                                                                                                  • B Fast Fourier Transform
                                                                                                                                                                                    • Bibliography
                                                                                                                                                                                      • Index

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