# Numerical Approximation Task

A Numerical Approximation Task is an approximation task that requires the solution to a numerical function.

**AKA:**Approximate Mathematical Analysis.**Context:**- It can be solved by a Numerical Approximation System (that implements a numerical approximation algorithm).
- It can range from being a Stochastic Numerical Approximation Task to being a ...
- 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.

## References

### 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).One of the earliest mathematical writings is a Babylonian tablet from the Yale Babylonian Collection (YBC 7289), which gives a sexagesimal numerical approximation of [math]\sqrt{2}[/math], the length of the diagonal in a unit square. Being able to compute the sides of a triangle (and hence, being able to compute square roots) is extremely important, for instance, in astronomy, carpentry and construction.

^{[1]}Numerical analysis continues this long tradition of practical mathematical calculations. Much like the Babylonian approximation of [math]\sqrt{2}[/math], modern numerical analysis does not seek exact answers, because exact answers are often impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors.

Numerical analysis naturally finds applications in all fields of engineering and the physical sciences, but in the 21st century also the life sciences and even the arts have adopted elements of scientific computations. Ordinary differential equations appear in celestial mechanics (planets, stars and galaxies); numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.

Before the advent of modern computers numerical methods often depended on hand interpolation in large printed tables. Since the mid 20th century, computers calculate the required functions instead. These same interpolation formulas nevertheless continue to be used as part of the software algorithms for solving differential equations.

- ↑ The New Zealand Qualification authority specifically mentions this skill in document 13004 version 2, dated 17 October 2003 titled CARPENTRY THEORY: Demonstrate knowledge of setting out a building

### 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

- https://www.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.