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 finding approximate solutions to mathematical problems).
- AKA: Numerical Analysis, Computational Mathematics.
- Context:
- It can (typically) require Approximation Elements, such as:
- It can need error bound calculation through numerical methods.
- It can involve iterative processes for solution refinement.
- It can demand convergence analysis of numerical sequences.
- It can (typically) utilize Numerical Tools, such as:
- It can employ discretization methods for continuous problems.
- It can use optimization algorithms for solution finding.
- It can leverage computational techniques for complex calculations.
- It can (often) face Approximation Challenges, such as:
- It can encounter numerical stability issues requiring stabilization methods.
- It can confront computational complexity needing efficient algorithms.
- It can address accuracy requirements through error analysis.
- It can range from being a Simple Approximation Task to being a Complex Approximation Task, depending on its problem complexity.
- It can range from being a Direct Approximation to being an Iterative Approximation, depending on its solution method.
- It can range from being a Single Variable Approximation to being a Multiple Variable Approximation, depending on its problem dimension.
- ...
- It can (typically) require Approximation Elements, such as:
- Examples:
- Function Approximation Tasks, such as:
- Interpolation Tasks, such as:
- Estimating intermediate values from known data points.
- Computing temperature values between measurement times.
- Extrapolation Tasks, such as:
- Predicting future values from historical trends.
- Forecasting economic indicators beyond observed periods.
- Interpolation Tasks, such as:
- Equation Solving Tasks, such as:
- Differential Equation Solutions, such as:
- Using Euler method for ordinary differential equations.
- Applying numerical integration for partial differential equations.
- System of Equations Solutions, such as:
- Solving linear systems through iterative methods.
- Finding roots using Newton method.
- Differential Equation Solutions, such as:
- Optimization Tasks, such as:
- Constrained Optimizations, such as:
- Maximizing profit functions under price constraints.
- Finding optimal solutions with boundary conditions.
- Unconstrained Optimizations, such as:
- Minimizing error functions in data fitting.
- Finding local minimums of complex functions.
- Constrained Optimizations, such as:
- Numerical Integration Tasks, such as:
- Definite Integral Approximations, such as:
- Computing areas using trapezoidal rule.
- Evaluating volumes through numerical quadrature.
- Multiple Integral Approximations, such as:
- Calculating surface integrals using mesh methods.
- Approximating volume integrals through discretization.
- Definite Integral Approximations, such as:
- ...
- Function Approximation Tasks, such as:
- Counter-Examples:
- Symbolic Computation Tasks, which find exact solutions through algebraic manipulation.
- Exact Mathematical Analysis Tasks, which require precise answers rather than approximations.
- Discrete Mathematics Tasks, which deal with countable sets rather than continuous functions.
- Formal Logic Tasks, which use logical operations rather than numerical methods.
- See: Numerical Method, Approximation Theory, Computational Mathematics, Error Analysis, Algorithm Complexity.
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.