Stability Issues in Numerical Computation and Numerically Stable Algorithms

# 1. Introduction

## 1.1 Overview of Stability Issues

Stability is a crucial concept in numerical computation. When performing numerical calculations, the stability of algorithms is closely related to the accuracy and reliability of the results. The emergence of stability problems may lead to the continuous accumulation of computational errors, eventually resulting in serious deviations. Therefore, studying and applying stable algorithms is of great significance in numerical computation.

Stability issues mainly involve rounding errors and truncation errors. Rounding errors are caused by the limited precision of computer floating-point number representation, which leads to differences between the computed results and the true values. Truncation errors, on the other hand, result from truncating infinite sequences or functions, such as using Taylor expansions to compute functions.

## 1.2 Importance of Numerical Stable Algorithms

Numerical stable algorithms are effective solutions to stability issues. They can reduce the impact of rounding and truncation errors through a series of precise computational steps, thus ensuring the accuracy and reliability of the computed results. Numerical stable algorithms are not only applicable to solving equations, integration, and optimization problems but are also widely used in scientific computing, engineering simulation, and data processing.

In the following chapters, we will introduce stability issues in numerical computation, explore the fundamental principles of numerical stable algorithms, and present some common numerical stable algorithms and their evaluation methods. Finally, we will summarize numerical stable algorithms and look forward to their future development trends.

# 2. Stability Issues in Numerical Computation

Stability issues are a significant topic in numerical computation. They refer to the accumulation and magnification of errors during the computation process, leading to inaccurate or unreliable results. Stability issues frequently occur in scientific computing, engineering computation, financial computation, and many other fields.

### 2.1 Rounding and Truncation Errors

In numerical computation, rounding and truncation errors are two main factors causing stability issues. Rounding errors occur due to the approximation representation of floating-point numbers by computers, such as when converting an infinite decimal into a finite one. Truncation errors, on the other hand, arise from discarding parts of a mathematical problem during numerical approximation or numerical approximation, thereby introducing errors.

### 2.2 Ill-Posed Problems and Condition Number

Besides rounding and truncation errors, ill-posed problems and condition number are also significant factors contributing to stability issues. Ill-posed problems refer to those where small changes in input data lead to drastic changes in output results, meaning the solution is highly sensitive to changes in input data. The condition number measures the degree of ill-posedness, indicating how sensitive the input data of ill-posed problems is to the output results.

### 2.3 Impact of Stability Issues

The presence of stability issues can have many adverse effects on numerical computation. Firstly, stability issues can lead to increased computational errors, significantly reducing the accuracy of the results. Secondly, stability issues can make the computational process unreliable, as slight input errors may result in entirely different outcomes. Lastly, stability issues can also affect the efficiency of computation, as unstable computational processes usually require more operations and iterations to achieve precise results.

In the subsequent chapters, we will introduce some commonly used numerical stable algorithms to address stability issues in numerical computation.

# 3. Fundamental Principles of Numerical Stable Algorithms

Numerical stable algorithms refer to those that can maintain the accuracy and stability of results during numerical computation. They can effectively solve common stability issues such as rounding and truncation errors in numerical computation, ensuring the accuracy and reliability of the results. In this chapter, we will introduce the fundamental principles of numerical stable algorithms.

#### 3.1 Definition and Classification of Stability

In numerical computation, an algorithm is considered stable if and only if it has good tolerance for small disturbances in input data and can produce results close to the true solution. Stability is divided into two types: numerical stability and condition number stability.

- Numerical stability refers to the ability of an algorithm to minimize the cumulative effect of rounding and truncation errors during floating-point co