数值分析复习提纲

数值分析（英）复习提纲

考试以基本概念为主，书上以前布置的计算机题目都不作要求。

第一章Solving equations

1.1 THE BISECTION METHOD

(a) 熟练掌握二分法;

(b) 对于给定解的误差精度要求能够熟练计算所需二分法步数，参考书上28页内容。

习题5,6

1.2 FIXED-POINT ITERATION

(a) 熟练掌握不动点迭代方法求方程的根；掌握不动点迭代方法的线性收敛性与收敛率; 此节书后习题不作要求。

1.4 NEWTON’S METHOD

（a）熟练掌握方程求根的NEWTON’S METHOD：Example 1.11, 1.12, 1.13

（b）对于重根熟练掌握Theorem 1.12, Theorem 1.13

习题2,5,7

第二章Systems of Equations

2.2 THE LU FACTORIZATION

（a）掌握矩阵LU分解方法；

（b）会使用LU分解方法求线性方程组的解:Example 2.5, 2.6, 2.7

2.3 SOURCES OF ERROR

本节只要掌握矩阵范数的定义，参阅90页

2.4 THE PA = LU FACTORIZATION

熟练掌握2.4.2 Permutation matrices, 2.4.3 PA = LU factorization: Example 2.16, 2.17, 2.18

习题4

2.5 ITERATIVE METHODS

熟练掌握Jacobi Method，Gauss–Seidel Method. 习题2

第三章Interpolation

3.1 DATA AND INTERPOLATING FUNCTIONS：

（a）熟练掌握Lagrange interpolation

（b）熟练掌握Newton’s divided differences

习题1,2,5

3.2 INTERPOLATION ERROR

熟练掌握定理3.4, Example 3.8, 习题1,2,4

第四章Least Squares

4.1.1 Inconsistent systems of equations

熟练掌握Normal equations for least squares：Example 4.1, Example 4.2

习题1,2

第五章Numerical Differentiation and Integration

5.1 NUMERICAL DIFFERENTIATION

熟练掌握一阶导数的Two-point forward-difference formula，Three-point centered-difference formula

熟练掌握二阶导数的Three-point centered-difference formula for second derivative

习题1,2,5,8,9

5.2 NEWTON–COTES FORMULAS FOR NUMERICAL INTEGRATION

熟练掌握Composite Trapezoid Rule，Example 5.8，习题1

第六章Ordinary Differential Equations

6.1.1 Euler’s Method

(a) 熟练掌握Euler方法（6.7）: Example6.2 习题5

6.2.2 The explicit Trapezoid Method

熟练掌握The explicit Trapezoid Method（6.29）：Example6.10 习题1