《计算方法》教学大纲

课程代码

045101691

课程名称

计算方法

英文名称

Computation Methods

课程类别

专业基础课

选修课

课程性质

必修

选修

学时

总学时：48实验学时：8实习学时：0其他学时：0

学分

3.0

开课学期

6/4

开课单位

计算机科学与工程学院

适用专业

计算机科学与技术、网络工程、信息安全

授课语言

中文

先修课程

数学分析、线性代数与解析几何、高级语言程序设计

课程对毕业要求的支撑

本课程对学生达到如下毕业要求有如下贡献：

 1. 工程知识：能够将数学、自然科学、工程基础和专业知识用于解决计算机复杂工程问题。

 2. 问题分析：能够应用数学、自然科学和工程科学的基本原理，识别、表达、并通过文献研究分析计算机复杂工程问题，以获得有效结论。

 3. 研究：能够基于科学原理并采用科学方法对与计算机相关复杂工程问题进行研究，包括设计实验、分析与解释数据、并通过信息综合得到合理有效的结论。

4. 使用现代工具：能够针对与计算机相关复杂工程问题，开发、选择与使用恰当的技术、资源、现代工程工具和信息技术工具，包括对复杂工程问题的预测与模拟，并能够理解其局限性。

课程目标

完成课程后，学生将具备以下能力：

（1）掌握计算方法相关的基本概念[1/2/3]；

（2）掌握各种计算方法的基本思想、推导过程、计算步骤和编程实现[1/2/3/4]；

（3）掌握各种计算方法的误差估计和收敛性判断[1/2/3]。

课程思政

 1. 通过在教学中对学生的严谨性和逻辑性的严格要求，逐步培养学生坚持真理、一丝不苟、实事求是的科学态度和遵章守纪的诚信观念；

2. 通过数学的有序性、简明性、对称性和统一性，培养学生的审美意识和高尚情操.

课程简介

本课程主要介绍使用计算机解决某些数学问题的近似方法，课程实用性较强，在科学研究、科学实验和工程技术中都有很多的应用。通过本课程的学习，使学生不仅要掌握计算方法的基本概念、各种计算方法的基本思想、推导过程、计算过程和在计算机上如何实现，而且也要掌握某些计算方法的误差估计和收敛性判断，为今后使用计算机解决实际问题打下良好的基础。

教学内容与学时分配

（一）课程目的意义和误差（2学时）

主要内容：了解计算方法在科学研究、科学实验和工程技术中的重要性；介绍计算方法在我国很多工程实践中的应用；误差的来源；误差、误差限和有效数字；相对误差和相对误差限；数值计算中的误差估计；数值计算中应注意的一些问题。

（二）代数插值与数值微分（8学时）

主要内容：线性插值与二次插值；n次插值的Lagrange形式和Newton形式；分段线性插值；Hermite插值；分段三次Hermite插值；三次样条插值；数值微分。

（三）数据拟合（3学时）

主要内容：单变量数据拟合及最小二乘法；多变量数据拟合；非线性数据线性化。

（四）数值积分（6学时）

主要内容：梯形求积公式、Simpson求积公式和Newton-Cotes求积公式；求积公式的代数精确度；梯形求积公式和Simpson求积公式的误差估计；复化求积公式；自动选取积分步长梯形法；数值方法中的加速收敛技巧——Richardson外推算法；Romberg求积法。

（五）解线性代数方程组的直接法（6学时）

主要内容：高斯消去法；LU分解法；对称正定矩阵的平方根法和LDLT分解法；向量与矩阵范数。

（六）解线性代数方程组的迭代法（7学时）

主要内容：几种常用的迭代格式；迭代法收敛性理论。

（七）非线性方程和非线性方程组的解法（4学时）

主要内容：对分法；迭代法；Newton迭代法；割线法；解非线性方程组的迭代法和Newton法。

（八）常微分方程初值问题的数值解法（4学时）

主要内容：欧拉法；龙格—库塔法；线性多步法。

实验教学（包括上机学时、实验学时、实践学时）

有

教学方法

课程教学以课堂教学、课外作业、上机实验以及与授课教师的科研实践相结合等共同实施。

考核方式

本课程采用闭卷考试并结合实验、作业和考勤进行综合评估，其中闭卷考试占该课程总评成绩的70 %，实验、作业与考勤占该课程总评成绩的30 %。

教材及参考书

现用教材：

韩国强, 林伟健等编著. 数值分析, 华南理工大学出版社, 2005年

主要参考资料：

 [1].李庆扬, 王能超, 易大义. 数值分析（第5版）. 清华大学出版社, 2008

[2]. J.H. Mathews and K.D. Fink. Numerical Methods Using MATLAB (4th Edition). Photocopy, Beijing, China: Publishing House of Electronics Industry, 2005

制定人及制定时间

何军辉，2019年4月10日

 “Computation Methods” Syllabus

Course Code

045101691

Course Title

Computation Methods

Course Category

 Specialty Basic Courses

Elective Courses

Course Nature

 Compulsory Course

Elective Course

Class Hours

Total hours: 48 Machine hours: 8Experimental hours: 0Practice hours: 0

Credits

3.0

Semester

6/4

Institute

School of Computer Science & Engineering

ProgramOriented

Computer science and technology, Network engineering, Information security

Teaching Language

Chinese

Prerequisites

Mathematics Analysis, Linear Algebra & Analytic Geometry, Advanced Language Programming

 Student Outcomes

 (Special Training Ability)

 This course contributes to the following graduation requirements for students:

 1. Engineering Knowledge: An ability to apply knowledge of mathematics, science, engineering fundamentals and engineering specialization to the solution of complex engineering problems.

 2. Problem Analysis: An ability to identify, formulate and analyze complex engineering problems, reaching to substantiated conclusions using basic principles of mathematics, science, and engineering.

 3. Research: An ability to conduct investigations of complex engineering problems based on scientific theories and adopting scientific methods including design of experiments, analysis and interpretation of data and synthesis of information to provide valid conclusions.

4. Applying Modern Tools: An ability to create, select and apply appropriate techniques, resources, and modern engineering and IT tools, including prediction and modelling, to complex engineering activities, with an understanding of the limitations.

Course Objectives

 Upon completion of the course, students will have the following abilities:

 (1) Master the basic concepts related to the computing method [1/2/3];

 (2) Master the basic ideas of various computing methods, derivation process, calculation steps and programming implementation [1/2/3/4];

(3) Master the various methods of error estimation and convergence judgment [1/2/3].

Course ideological

 1. Through the strict requirements of students' rigor and logic in teaching, we will gradually cultivate students' scientific attitudes of adhering to the truth, meticulousness, seeking truth from facts and obeying the rules of integrity;

 2. Through the order, conciseness, symmetry and unity of mathematics, cultivate students' aesthetic consciousness and noble sentiment.

Course Description

This course mainly introduces the approximate method of using computer to solve some mathematical problems. The course is practical and has many applications in scientific research, scientific experiment and engineering technology. Through the study of this course, the students will not only master the basic concepts of computing methods, the basic idea of various methods of calculation, derivation process, the calculation process and implementation on computer, but also master some of the calculation method of error estimation and convergence judgment.And it will lay a good foundationfor the future use of computers to solve practical problems.

Teaching Content and Class Hours Distribution

 (A) Course objective and significance, Error (2 hours)

 Main contents: understand the importance of computation methods in scientific research, scientific experiments and engineering techniques; introduce the application of computational methods in many engineering practices in China ; the source of the error; error, error limit and effective number; relative error and relative error limit; error estimatefor numerical calculation;some of the problems about numerical calculation should be paid attention to.

 (B) Algebraic interpolation and numerical differentiation (8 hours)

 Main contents: linear interpolation and quadratic interpolation, Lagrange form and Newton form of n-order interpolation, piecewise linear interpolation, Hermite interpolation, segmented cubic Hermite interpolation, cubic spline interpolation, numerical differentiation

 (C) Data fitting (3 hours)

 Main contents: Univariate data fitting and least squares method; multivariable data fitting; nonlinear data linearization.

 (D) Numerical integration (6 hours)

 Main contents: trapezoidal quadrature formula, Simpson quadrature formula and Newton-Cotes quadrature formula; algebraic accuracy of quadrature formula; trapezoidal quadrature formula and Simpson quadrature formula error estimation; complex quadrature formula; automatic selection of integral Step-by-step trapezoid method; accelerated convergence technique in numerical methods - Richardson extrapolation algorithm; Romberg quadrature method.

 (E) Direct method of solving linear algebraic equations (6 hours)

 Main contents: Gaussian elimination method; LU decomposition method; symmetric positive definite matrix square root method and LDLT decomposition method; vector and matrix norm.

 (F) Iterative method for solving linear algebraic equations (7 hours)

 Main contents: several commonly used iterative format; iterative method of convergence theory.

 (G) Solutions of nonlinear equations and nonlinear equations (4 hours)

 Main contents: Bisection; iterative method; Newton iterative method; secant method; solution of nonlinear equations of the iterative method and Newton method.

 (H) Numerical solution of initial value problem of ordinary differential equation (4 hours)

Main contents: Euler method; Longge-Kutta method; linear multi-step method.

Experimental Teaching

Yes

Teaching Method

Teaching methods include classroom teaching, extracurricular homework, experiments and the combination of teaching and lecturer’s research practice.

Examination Method

This course will be evaluated with comprehensive assessment, including closed book examinations, experiments, assignments and attendance. The closed book exams account for 70% of the total score of the course. Experiments, homework and attendance account for 30% of the total score of the course.

Teaching Materials and Reference Books

 Present textbook:

韩国强, 林伟健等编著. 数值分析, 华南理工大学出版社, 2005年

 The main references:

 [1].李庆扬, 王能超, 易大义. 数值分析（第5版）. 清华大学出版社, 2008

[2]. J.H. Mathews and K.D. Fink. Numerical Methods Using MATLAB (4th Edition). Photocopy, Beijing, China: Publishing House of Electronics Industry, 2005

Prepared by Whom and When

Junhui He, April 10, 2019


《计算方法》实验教学大纲

课程代码

045101691

课程名称

计算方法

英文名称

Computation Methods

课程类别

专业基础课

选修课

课程性质

必修

选修

学时

总学时：48实验：8实习：0其他：0

学分

3.0

开课学期

6/4

开课单位

计算机科学与工程学院

适用专业

计算机科学与技术、网络工程、信息安全

授课语言

中文

先修课程

数学分析、线性代数与解析几何、高级语言程序设计

毕业要求（专业培养能力）

本课程对学生达到如下毕业要求的贡献：

  1. 工程知识：能够将数学、自然科学、工程基础和专业知识用于解决计算机复杂工程问题。

  2. 问题分析：能够应用数学、自然科学和工程科学的基本原理，识别、表达、并通过文献研究分析计算机复杂工程问题，以获得有效结论。

  3. 研究：能够基于科学原理并采用科学方法对与计算机相关复杂工程问题进行研究，包括设计实验、分析与解释数据、并通过信息综合得到合理有效的结论。

  4. 使用现代工具：能够针对与计算机相关复杂工程问题，开发、选择与使用恰当的技术、资源、现代工程工具和信息技术工具，包括对复杂工程问题的预测与模拟，并能够理解其局限性。

课程培养学生的能力（教学目标）

完成课程后，学生将具备以下能力：

  1. 掌握计算方法相关的基本概念[1/2/3]；

  2. 掌握各种计算方法的基本思想、推导过程、计算步骤和编程实现[1/2/3/4]；

  3. 掌握各种计算方法的误差估计和收敛性判断[1/2/3]。

课程简介

本课程主要介绍使用计算机解决某些数学问题的近似方法，课程实用性较强，在科学研究、科学实验和工程技术中都有很多的应用。通过本课程的学习，使学生不仅要掌握计算方法的基本概念、各种计算方法的基本思想、推导过程、计算过程和在计算机上如何实现，而且也要掌握某些计算方法的误差估计和收敛性判断，为今后使用计算机解决实际问题打下良好的基础。

主要仪器设备与软件

PC机、编程环境（C++、Java、Python等）

实验报告

每次实验需提交实验报告，实验报告的内容应包括实验目的及要求、实验环境、实验过程和实验小结等。

考核方式

本实验课程成绩将结合出勤、实验操作以及实验报告等进行综合评估，其中出勤占实验课程总评成绩的10%，实验操作占实验课程总评成绩的60%，实验报告占实验课程总评成绩的30%。

教材、实验指导书及教学参考书目

实验指导书与参考书：

  1. 韩国强, 林伟健等编著，数值分析, 华南理工大学出版社, 2005年

  2. 李庆扬, 王能超, 易大义. 数值分析（第5版）. 清华大学出版社, 2008

  3. J.H. Mathews and K.D. Fink. Numerical Methods Using MATLAB (4th Edition). Photocopy, Beijing, China: Publishing House of Electronics Industry, 2005

制定人及发布时间

何军辉，2019年4月30日


《计算方法》实验教学内容与学时分配

实验项目编号

实验项目名称

实验学时

实验内容提要

实验类型

实验要求

每组人数

主要仪器设备与软件


多项式插值与曲线拟合

2

  1. 拉格朗日插值和牛顿插值

  2. 最小二乘法曲线拟合

设计性

必做

1

PC机、编程环境


数值积分与线性方程组求解

2

  1. 自动选取步长梯形积分、龙贝格积分

  2. 高斯消去法、LU直接分解法、对称正定矩阵的平方根法

设计性

必做

1

PC机、编程环境


线性方程组和非线性方程迭代求解

2

  1. Jacobi迭代法和Seidel迭代求解线性方程组

  2. 对分法、松弛法和牛顿法求非线性方程根

设计性

必做

1

PC机、编程环境


特征值和特征向量/微分方程

2

  1. 幂法和雅可比方法求特征值和特征向量

  2. 龙格-库塔法求解微分方程

设计性

必做

1

PC机、编程环境


 “Computation Methods” Syllabus

Course Code

045101691

Course Title

Computation Methods

Course Category

 Specialty Basic Courses

Elective Courses

Course Nature

 Compulsory Course

Elective Course

Class Hours

Total hours: 48 Experimental hours: 8 Practice hours: 0 Other hours:0

Credits

3.0

Semester

6/4

Institute

School of Computer Science & Engineering

Program Oriented

Computer science and technology, Network engineering, Information security

Teaching Language

Chinese

Prerequisites

Mathematics Analysis, Linear Algebra & Analytic Geometry, Advanced Language Programming

Student Outcomes (Special Training Ability)

This course contributes to the following graduation requirements for students:

  1. Engineering Knowledge: An ability to apply knowledge of mathematics, science, engineering fundamentals and engineering specialization to the solution of complex engineering problems.

  2. Problem Analysis: An ability to identify, formulate and analyze complex engineering problems, reaching to substantiated conclusions using basic principles of mathematics, science, and engineering.

  3. Research: An ability to conduct investigations of complex engineering problems based on scientific theories and adopting scientific methods including design of experiments, analysis and interpretation of data and synthesis of information to provide valid conclusions.

  4. Applying Modern Tools: An ability to create, select and apply appropriate techniques, resources, and modern engineering and IT tools, including prediction and modelling, to complex engineering activities, with an understanding of the limitations.

Teaching Objectives

Upon completion of the course, students will have the following abilities:

  1. Master the basic concepts related to the computing method [1/2/3];

  2. Master the basic ideas of various computing methods, derivation process, calculation steps and programming implementation [1/2/3/4];

  3. Master the various methods of error estimation and convergence judgment [1/2/3].

Course Description

This course mainly introduces the approximate method of using computer to solve some mathematical problems. The course is practical and has many applications in scientific research, scientific experiment and engineering technology. Through the study of this course, the students will not only master the basic concepts of computing methods, the basic idea of various methods of calculation, derivation process, the calculation process and implementation on computer, but also master some of the calculation method of error estimation and convergence judgment. And it will lay a good foundation for the future use of computers to solve practical problems.

Instruments and Equipments

PC, programming environment (C ++, Java, Python, etc.)

Experiment Report

Each experiment must submit an experimental report, the experimental report should include the contents of the experimental requirements, experimental environment, experimental process and experimental summary.

Assessment

The experimental result will be evaluated with a comprehensive assessment, including attendance, experimental operation and experimental reports. Attendance accounts for 10% of the total score of the experimental course, experimental operations account for 60% of the total score of the experimental course, the experimental reports account for 30% of the total score.

Teaching Materials and Reference Books

Experimental Guidance and Reference:

  1. 韩国强, 林伟健等编著，数值分析, 华南理工大学出版社, 2005年

  2. 李庆扬, 王能超, 易大义. 数值分析（第5版）. 清华大学出版社, 2008

  3. J.H. Mathews and K.D. Fink. Numerical Methods Using MATLAB (4th Edition). Photocopy, Beijing, China: Publishing House of Electronics Industry, 2005

Prepared by Whom and When

Junhui He, April 30, 2019

 “Computation Methods” Experimental Teaching Arrangements

No.

Experiment Item

Class Hours

Content Summary

Category

Requirements

Number of StudentsEach Group

Instruments, Equipments and Software

 1

Polynomial Interpolation and Curve Fitting

2

  1. Lagrangian interpolation and Newton interpolation

  2. Least square curve fitting

Design

Compulsory

1

PC, programming environment (C ++, Java, Python, etc.)

2

Numerical Integral and Solution of Linear Equations

2

  1. Automatically select the step length trapezoidal integral, Longberg integral

  2. Gaussian elimination method, LU direct decomposition method, symmetric positive definite matrix square root method

Design

Compulsory

1

PC, programming environment (C ++, Java, Python, etc.)

3

Solving Linear Equations and Nonlinear Equations

2

  1. Jacobi iterative method and Seidel iteration to solve linear equations

  2. Bisection, relaxation method and Newton method to find the nonlinear equation root

Design

Compulsory

1

PC, programming environment (C ++, Java, Python, etc.)

4

Eigenvalue and eigenvector / differential equation

2

  1. Power law and Jacobian method for eigenvalues and eigenvectors

  2. Runge-Kutta method for solving differential equations

Design

Compulsory

1

PC, programming environment (C ++, Java, Python, etc.)