Numerical Approximation Theory and Its Applications in Practice

# 1. Overview of Numerical Approximation Theory

## 1.1 Basic Concepts and Principles of Numerical Approximation

Numerical approximation is a method that calculates mathematical problems using approximation techniques. It is based on numerical computing technology and aims to obtain sufficiently accurate solutions with finite computational resources. Numerical approximation methods require the use of numerical algorithms for approximate calculations instead of analytical solutions to obtain exact results.

The fundamental principle of numerical approximation is to transform a complex mathematical problem into an easily computable approximate problem. Numerical approximation methods can obtain solutions close to the true solution without considering infinite precision. Numerical approximation methods can be applied to various mathematical problems such as root finding, interpolation, curve fitting, etc.

## 1.2 The Importance of Numerical Approximation in Computer Science

Numerical approximation is of significant importance in computer science. In practical problems, it is often impossible to obtain accurate results through analytical solutions, hence numerical approximation methods are used for calculations. Numerical approximation methods can help us quickly solve various mathematical problems, including optimization problems, equation solving, and graphics processing.

In computer science, numerical approximation methods are widely applied in various fields. For example, in machine learning, we often need to optimize model parameters using numerical approximation methods; in image processing, numerical approximation methods can help us achieve smooth processing and contour extraction; in computer graphics, numerical approximation methods can be used for curve drawing and surface modeling.

## 1.3 Differences Between Numerical Approximation and Traditional Mathematical Methods

There are some differences between numerical approximation methods and traditional mathematical methods. Traditional mathematical methods are usually based on analytical derivation and mathematical formulas and can solve some mathematical problems precisely. However, for complex nonlinear problems or problems that cannot be solved by analytical methods, traditional mathematical methods often cannot provide accurate solutions.

In contrast, numerical approximation methods use approximate calculations and numerical algorithms to solve mathematical problems. Numerical approximation methods focus on obtaining sufficiently accurate solutions with finite computational resources instead of pursuing absolute accuracy. Numerical approximation methods can obtain solutions in a short time and can balance computational speed and result accuracy by adjusting the computational precision.

Summary:
This chapter introduces the basic concepts and principles of numerical approximation. Numerical approximation is a method that calculates mathematical problems using approximation techniques and has significant application value in computer science. Numerical approximation methods differ from traditional mathematical methods in solving complex problems and pursuing accuracy. In the next chapter, we will further introduce common numerical approximation methods and algorithms.

# 2. Numerical Approximation Methods and Algorithms

### 2.1 Common Numerical Approximation Methods

Numerical approximation methods are ways to construct numerical values using a finite number of approximate values. The following are common numerical approximation methods:

- **Interpolation Methods**: Construct a function through known data points so that the function passes through the given data points, ***mon interpolation methods include Lagrange interpolation, Newton interpolation, and Hermite interpolation.

- **Least Squares Method**: Approximate a model by fitting a dataset to minimize the error between the model and actual data. The least squares method is widely used in data fitting, regression analysis, and other fields, and can determine model parameters by minimizing the sum of squared residuals.

- **Fourier Approximation**: Fourier approximation utilizes Fourier series to approximate arbitrary functions, expressing any function as a sum of sine and cosine functions.

### 2.2 Newton's Iterative Method and Bisection Method

- **Newton's Iterative Method**: Newton's iterative method is an efficient numerical approximation algorithm. It uses the derivative of a function to approximate the real roots of the function. The basic principle of Newton's iterative method is to continuously improve the approximate solution through iterative approximation until the desired precision or convergence is achieved.

def newton_iteration(f, f_prime, x0, epsilon):
     x = x0
     while abs(f(x)) > epsilon:
         x = x - f(x) / f_prime(x)
     return x

Code Explanation:
- `f` and `f_prime` are the function f and its derivative
- `x0` is the initial approximate solution
- `epsilon` is the desired precision

- **Bisection Method**: The bisection method is a simple yet effective numerical approximation algorithm. It continually bisects an interval and then selects a new interval to approximate the root of a function. The prerequisite for the bisection method is that the function has monotonicity within the given interval.

public double binarySearch(double a, double b, double epsilon) {
     double left = a;
     double right = b;
     double mid = (left + right) / 2;
     while (Math.abs(f(mid)) > epsilon) {
         mid = (left + right) / 2;
         if (f(mid) * f(left) < 0) {
             right = mid;
         } else {
             left = mid;
         }
     }
     return mid;
}

Code Explanation:
- `a` and `b` are the left and right boundaries of the interval
- `epsilon` is the desired precision

### 2.3 Principles for Choosing Numerical Approximation Algorithms in Practical Problems

In practical problems, choosing the appropriate numerical approximation algorithm is crucial. Here are some principles for selection:

- Determine the required precision: According to the problem's requirements, determine the necessary precision to choose the appropriate algorithm.

- Consider the efficiency of the algorithm: Different algorithms have different efficiencies; some may be better suited for large problems, while others are more suitable for small problems.

- Consider the particularity of the problem: Some problems may have special structures that can be utilized to select a more efficient algorithm.

- Choose a reliable algorithm: Some algorithms may be unstable or non-convergent for specific types of problems, so choosing a stable and reliable algorithm is very important.

Thus, Chapter 2 introduces common numerical approximation methods, including interpolation methods, the least squares method, and Fourier approximation. It also introduces two common numerical approximation algo