Numerical Analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis,^[2][3][4] and stochastic differential equations and Markov chains for simulating living cells in medicine and biology.

Before modern computers, numerical methods often relied on hand interpolation formulas, using data from large printed tables. Since the mid 20th century, computers calculate the required functions instead, but many of the same formulas continue to be used in software algorithms.^[5]

The numerical point of view goes back to the earliest mathematical writings. A tablet from the Yale Babylonian Collection (YBC 7289), gives a sexagesimal numerical approximation of the square root of 2, the length of the diagonal in a unit square.

Numerical analysis continues this long tradition: rather than giving exact symbolic answers translated into digits and applicable only to real-world measurements, approximate solutions within specified error bounds are used.

数值分析是对使用数值近似（与符号操作相反）来解决数学分析问题（与离散数学不同）的算法的研究。数值分析在工程和物理科学的所有领域都有应用，在 21 世纪也适用于生命和社会科学、医学、商业甚至艺术。当前计算能力的增长使得能够使用更复杂的数值分析，在科学和工程中提供详细和现实的数学模型。数值分析的例子包括：天体力学中的常微分方程（预测行星、恒星和星系的运动）、数据分析中的数值线性代数、[2][3][4] 以及随机微分方程和马尔可夫链在医学和生物学中模拟活细胞。

在现代计算机之前，数值方法通常依赖于手动插值公式，使用来自大型印刷表格的数据。自 20 世纪中叶以来，计算机改为计算所需的函数，但许多相同的公式继续用于软件算法。 [5]

数值的观点可以追溯到最早的数学著作。 Yale Babylonian Collection (YBC 7289) 中的一块平板电脑给出了 2 的平方根的六十进制数值近似值，即单位正方形中对角线的长度。

数值分析延续了这一悠久的传统：不是将精确的符号答案转换为数字并仅适用于实际测量，而是使用指定误差范围内的近似解。