數值計算詳細筆記（一）：基礎數學知識回顧

文章目錄

• 1. Mathematical Preliminaries
• 1.1 Concept introduction
• 1.1.1 Equivalent description
• 1.1.2 Main Contents
• 1.1.3 Reason for learning
• 1.1.4 Book content
• 1.1.5 Steps for solving
• 1.2 Calculus Review
• 1.2.1 Limit of a Function
• 1.2.2 Continuous of a Function
• 1.2.3 Limit of a Sequence
• 1.2.4 Derivative of a Function
• 1.2.5 Mean Value Theorem
• 1.2.6 Extreme Value Theorem
• 1.2.7 Riemann Integral
• 1.2.8 Weighted Mean Value Theorem for the Integral（Importent）
• 1.2.9 Generalized Rolle's Theorem
• 1.2.10 Intermediate Value Theorem
• 1.2.11 Taylor's Theorem
• 1.3 Roundoff Errors in Computer
• 1.3.1 Roundoff Error
• 1.3.2 Measurement of Error
• 1.3.3 Growth of Error
• 1.3.4 Propagation of Error
• 1.4 Algorithms and Convergence
• 1.4.1 Basic Definitions
• 1.4.2 Some Important Concepts
• 1.4.3 Rate of Convergence
• 1.4.4 Well-Posed or Ill-posed Problem
• 1.4.5 Approximation of Solution

1. Mathematical Preliminaries

1.1 Concept introduction

1.1.1 Equivalent description

• Numerical analysis
• Numerical Methods or Computational Methods
• Numerical Mathematics
• Numerical Computing
• Scientific Computing

1.1.2 Main Contents

• Methods using computer to solve mathematical problems in science and engineering, which mostly are continuous.
• design and analysis of algorithms for different mathematical problems.
• Theory and Application of Numerical Approximation Techniques.

1.1.3 Reason for learning

With the fast development of PC techniques, many mathematical problems arising in the science and engineering, such as derivatives (導數)、integrals (積分)、nonlinearities (非線性)、Linear Algebra problems (線性代數問題)、differential equations (微分方程) is possible to be solved.

Deals continuous quantities with numerical approximate techniques and considers effects of approximations, such as error (誤差), convergence (收斂), uniqueness (唯一性), existence (存在性).

Assessment on algorithm: efficiency, reliability, accuracy, etc.

1.1.4 Book content

1. Approximation Methods for solving equation(s)
2. Polynomial (多項式) and interpolation approximation (插值逼近)
3. Numerical Differentiation and Integration
4. Numerical methods for ODE and PDE
5. Eigenvalues and Eigenvectors

1.1.5 Steps for solving

1. Mathematical modeling, usually equations (方程).
2. Design algorithms to solve these equations.
3. Implement algorithms in computer software and run.
4. Represent the computed results in forms or graphical visualization.
5. Interpret and validate (解釋和驗證) the computed results.

summary: MAIRI

1.2 Calculus Review

1.2.1 Limit of a Function

Definition

Let $f$f be a function defined on a set X of real numbers. Then $f$f has the limit L at $x_0$x0​, written as

$\lim_{x \rightarrow x_0} f(x)=L,$x→x0​lim​f(x)=L,

if, given any real number $\varepsilon>0$ε>0, there exists a real number $\xi>0$ξ>0 such that

$|f(x)-L|<\varepsilon$∣f(x)−L∣<ε

whenever $x\in X$x∈X and $0<|x-x_0|<\xi$0<∣x−x0​∣<ξ.

1.2.2 Continuous of a Function

Condition

Let $f$f be a function defined on a set X of real numbers and $x_0 \in X$x0​∈X.

Theorem

Then $f$f is continuous at $x_0$x0​ if

$\lim_{x\rightarrow x_0} f(x)=f(x_0 ).$x→x0​lim​f(x)=f(x0​).

Extended Theorem

The function $f$f is continuous on the set X if it is continuous at each number in X.

Definition of $C(X)$C(X)

Let $C(X)$C(X) denote the set of all functions that are continuous on the set X.

1.2.3 Limit of a Sequence

Difinition

Let $\{x_n\}_{n=1}^{\infty}${xn​}n=1∞​ be an infinite sequence of real or complex number. The sequence converges to a number $x$x (Limit) if, for any $\varepsilon >0 $, there exists a positive integer $N(\varepsilon)$N(ε), such that implies

$|x_n-x|<\varepsilon,$∣xn​−x∣<ε,

whenever $n>N(\epsilon)$n>N(ϵ).

Extended Theorem（Obviously）

If $f$f is a function defined on a set of real numbers and $x_0\in X$x0​∈X, then the following statements are equivalent:

1. f is continuous at $x_0$x0​
2. if $\{x_n\}_{n=1}^\infty${xn​}n=1∞​ is any sequence in X converging to $x_0$x0​, then $\lim_{n\rightarrow\infty}f(x_n)=f(x_0)$limn→∞​f(xn​)=f(x0​).

1.2.4 Derivative of a Function

Definition

If $f$f is a function defined in an open interval (開區間) containing $x_0$x0​, the $f$f is differentiable at $x_0$x0​, if
$f'(x_0) =\lim_{x\rightarrow x_0 }\displaystyle\frac{f(x)-f(x_0) }{x-x_0 }$f′(x0​)=x→x0​lim​x−x0​f(x)−f(x0​)​
exists.

Remark

The number $f’(x_0) $ is called the derivative of $f(x)$f(x) at $x_0$x0​.

$C^n(X)$Cn(X) denote the set of all functions that have $n$n continuous derivatives on $X$X.

Especially $C^\infty(X) $ denote the set of all functions that have derivatives of all orders on X.

Extended Theorem

If the function $f$f is differentiable at $x_0$x0​, the $f$f is continuous at $x_0$x0​.

Rolle’s Theorem（羅爾定理）

Suppose $f\in C[a,b]$f∈C[a,b] and $f$f is differentiable on $(a,b)$(a,b).

If $f(a)=f(b)=0$f(a)=f(b)=0, then a number $$ $f(a)=f(b)=0$ f(a)=f(b)=0 , then a number $c$c in $(a,b)$(a,b) exists with $f'(c)=0$f′(c)=0.

1.2.5 Mean Value Theorem

Suppose $f\in C[a,b]$f∈C[a,b] and $f$f is differentiable on $(a,b)$(a,b), then a number $c$c in $(a,b)$(a,b) exists with

$f'(c)=\displaystyle\frac{f(b)-f(a)}{b-a}$f′(c)=b−af(b)−f(a)​

1.2.6 Extreme Value Theorem

If $f\in C[a,b]$f∈C[a,b], then $c_1,c_2\in[a,b]$c1​<