Approximate Mathematical Analysis Task
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A Approximate Mathematical Analysis Task is a mathematical analysis task that is an approximation task (which requires the solution to a mathematical statement).
- AKA: Numerical Analysis.
- Context:
- It can be solved by a Numerical Approximation System (that implements a numerical approximation algorithm).
- It can be the subject of a Numerical Analysis Domain.
- …
- Example(s):
- Interpolation Task: We have observed the temperature to vary from 20 degrees Celsius at 13:00 to 14 degrees at 15:00. What was the temperature at 13:30?
- Extrapolation Task: We have observed that a country's gross domestic product has been growing an average of 5% per year and was 100 billion dollars last year, what will the GDP be this year?
- Regression Task: given n points, we compute a line that passes as close as possible to those n points.
- Optimization Task: Say you sell lemonade at a lemonade stand, and notice that at $1, you can sell 197 glasses of lemonade per day, and that for each increase of $0.01, you will sell one glass of lemonade less per day. If you could charge $1.485, you would maximize your profit, but due to the constraint of having to charge a whole cent amount, charging $1.49 per glass will yield the maximum income of $220.52 per day.
- Integral Approximation Task.
- Differential Equation Approximation: If you set up 100 fans to blow air from one end of the room to the other and then you drop a feather into the wind, what happens? The feather will follow the air currents, which may be very complex. One approximation is to measure the speed at which the air is blowing near the feather every second, and advance the simulated feather as if it were moving in a straight line at that same speed for one second, before measuring the wind speed again. This is called the Euler method for solving an ordinary differential equation.
- …
- Counter-Example(s):
- See: Function Approximation, Numerical Optimization, Mathematical Analysis, Ordinary Differential Equation, Numerical Approximation, Symbolic Computation, Mathematical Analysis.
References
2023
- (Wikipedia, 2023) ⇒ https://en.wikipedia.org/wiki/numerical_analysis Retrieved:2023-4-26.
- 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). It is the study of numerical methods that attempt at finding 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, 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, 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.[1]
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.
- 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). It is the study of numerical methods that attempt at finding 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, 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, 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.[1]
- ↑ Brezinski, C.; Wuytack, L. (2012). Numerical analysis: Historical developments in the 20th century. Elsevier. ISBN 978-0-444-59858-5.
2015
- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/numerical_analysis Retrieved:2015-2-1.
- Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).
2013
- (Wkipedia, 2013) ⇒ http://en.wikipedia.org/wiki/Numerical_analysis#Areas_of_study
- The field of numerical analysis is divided into different disciplines according to the problem that is to be solved.
- 3.1 Computing values of functions.
- 3.2 Interpolation, extrapolation, and regression.
- 3.3 Solving equations and systems of equations.
- 3.4 Solving eigenvalue or singular value problems.
- 3.5 Optimization.
- 3.6 Evaluating integrals.
- 3.7 Differential equations
- The field of numerical analysis is divided into different disciplines according to the problem that is to be solved.
2011
- (SIAM, 2011) ⇒ https://siam.org/journals/sinum.php
- QUOTE: The SIAM Journal on Numerical Analysis contains research articles on the development and analysis of numerical methods. Topics include the rigorous study of convergence of algorithms, their accuracy, their stability, and their computational complexity. Also included are results in mathematical analysis that contribute to algorithm analysis, and computational results that demonstrate algorithm behavior and applicability.