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1 Mathematical Preliminaries
2 Solutions of Equations in One Variable
2.1 Newton’s Method
2.2 Modified Newton’s method
3 Solving Linear Systems
3.1 Direct Methods
3.1.1 Gaussian Elimination
the same as Linear Algebra
3.1.2 Elimination with Maximal Column Pivoting(additional comparisons)
- Determine the smallest such that
就是说找到列最大的换行到还没消元的第一行
- Then Gaussian Elimination
3.1.3 Elimination with Scaled-Column Pivoting(additional comparisons and divisions)
- Define a scale factor for each row as
- Determine the smallest such that
the scaled factors must be computed only once
3.1.4 Elimination with Complete Pivoting(maximal pivoting)( additional comparisons)
- Search all the entries for to find the entry with the largest magnitude.
- let it turn to
3.2 Decomposition
Method of Decomposition
For Tridiagonal Matrix
3.3 Iterative Techniques
3.3.1 Jacobi Methods
3.3.2 Gauss-Seidel Methods
3.3.3 Error
3.3.4 Relaxation Methods（SOR）
4 Approximating Eigenvalues
4.1 Power Method
when .then.
However, maybe when . So it needs to be normalized.
Normalization
Let . Then and .
.
4.2 Inverse Power Method
to get with
5 Interpolation and Polynomial Approximation
5.1 Lagrange Polynomial
error:
5.2 Divided Differences
5.3 Newton’s Interpolation
the last item is error
5.4 Hermite Interpolation
6 Approximation Theory
6.1 Least Squares Approximation
let ,then
6.2 Chebyshev Polynomials and Economization of Power Series
7 Numerical Differentiation and Integration
7.1 Numerical Differentiation
7.2 Elements of Numerical Integration
7.3 Trapezoidal Rule
7.4 Simpson’s Rule
7.5 Composite Numerical Integration
7.6 Gaussian Quadrature
8 Initial-Value Problems for Ordinary Differential Equations
to solve the equations
8.1 Runge-Kutta Methods
8.2 Multistep Methods
8.3 Higher-Order Equations and Systems of Differential Equations
- 作者:Tannin Rachel
- 链接:https://www.tanninrachel.ren/article/na-1
- 声明:本文采用 CC BY-NC-SA 4.0 许可协议，转载请注明出处。
