Numerical analysis

Numerical analysis is the study of <a href="/facts/Algorithm/fnl5NmRt">algorithms</a> that use numerical <a href="/facts/Approximation/LvYHsXzq">approximation</a> (as opposed to <a href="/facts/Symbolic_computation/hjun1KAX">symbolic manipulations</a>) for the problems of <a href="/facts/Mathematical_analysis/XXeR26Ih">mathematical analysis</a> (as distinguished from <a href="/facts/Discrete_mathematics/XdRUnzUL">discrete mathematics</a>). It is the study of numerical methods that attempt to find approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences like economics, 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: <a href="/facts/Ordinary_differential_equation/ygKGQ8kD">ordinary differential equations</a> as found in <a href="/facts/Celestial_mechanics/FA3uR78J">celestial mechanics</a> (predicting the motions of planets, stars and galaxies), <a href="/facts/Numerical_linear_algebra/jnS6Tteb">numerical linear algebra</a> in data analysis, and <a href="/facts/Stochastic_differential_equation/ol9a9WNq">stochastic differential equations</a> and <a href="/facts/Markov_chain/5oP5hntk">Markov chains</a> for simulating living cells in medicine and biology.
Before modern computers, <a href="/facts/Numerical_method/MrQIMSIY">numerical methods</a> often relied on hand <a href="/facts/Interpolation/8PyetX4f">interpolation</a> 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.
The numerical point of view goes back to the earliest mathematical writings. A tablet from the <a href="/facts/Yale_Babylonian_Collection/Exc4a3Ck">Yale Babylonian Collection</a> (<a href="/facts/YBC_7289/wA4I1Ke3">YBC 7289</a>), gives a <a href="/facts/Sexagesimal/bspNRWdR">sexagesimal</a> numerical approximation of the <a href="/facts/Square_root_of_2/SD0sag94">square root of 2</a>, the length of the <a href="/facts/Diagonal/39z9sIRb">diagonal</a> in a <a href="/facts/Unit_square/6aJUASbv">unit square</a>.
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.

Numerical analysis open-in-new

Numerical analysis