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Advanced numerical methods are essential in making numerical weather prediction feasible.

The overall goal of the field of numerical analysis is the design and analysis of techniques to give approximate but accurate solutions to hard problems, the variety of which is suggested by the following:

3.4 Solving eigenvalue or singular value problems.

3.3 Solving equations and systems of equations.

3.2 Interpolation, extrapolation, and regression.

2.3 Numerical stability and well-posed problems.

2.2 Truncation and discretization error.

1.2.1 Discretization and numerical integration.

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. 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.

The numerical point of view goes back to the earliest mathematical writings.

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Since the mid 20th century, computers calculate the required functions instead, but many of the same formulas continue to be used in software algorithms. 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, and stochastic differential equations and Markov chains for simulating living cells in medicine and biology.Д«efore modern computers, numerical methods often relied on hand interpolation formulas, using data from large printed tables. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. 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. 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). The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures.

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