This tag is for various question on numerical calculus / numerical analysis which concerned with all aspects of the numerical solution of a problem, from the theoretical development and understanding of numerical methods to their practical implementation as reliable and efficient computer programs.
Numerical calculus / Numerical analysis provides the foundations for a major paradigm shift in what we understand as an acceptable “answer” to a scientific or technical question. In classical calculus we look for answers like $\sqrt{\sin x}$, that is,answers composed of combinations of names of functions that are familiar. This presumes we can evaluate such an expression as needed, and indeed numerical analysis has enabled the development of pocket calculators and computer software to make this routine. But numerical analysis has done much more than this. Most numerical analysts specialize in small subfields, but they share some common concerns, perspectives, and mathematical methods of analysis.
Here is some issues that numerical analysis is used in:
$1.\quad$ Solving linear/non-linear equations and finding the real roots, many methods exist like: Bisection, Newton-Raphson ... etc.
$2.\quad$Fit some points to curve, good approximation and simple solution.
$3.\quad$Interpolation, great to get any value in between a table of values. It can solve the equally spaced readings for unequally spaced methods, Newton general method is implied.
$4.\quad$Solve definite integration, simple methods is used to compute an integration based on idea that the definite integration is the bounded area by the given curve, these methods approximate the area with great approximation. Many methods there, like Simpson’s rule.
$5.\quad$Solving initial value 1st and 2nd order differential equations, good approximation and simpler than normal analysis.
$6.\quad$Solving partial differential equations like Laplace equation for wave equation, very fast solution.
Applications:
Numerical analysis / Numerical calculus is the area of mathematics and computer science that creates, analyzes, and implements algorithms for solving numerically the problems of continuous mathematics. Such problems originate generally from real-world applications of algebra, geometry, and calculus, and they involve variables which vary continuously. These problems occur throughout the natural sciences, social sciences, medicine, engineering, and business. Beginning in the 1940's, the growth in power and availability of digital computers has led to an increasing use of realistic mathematical models in science, medicine, engineering, and business; and numerical analysis of increasing sophistication has been needed to solve these more accurate and complex mathematical models of the world. The formal academic area of numerical analysis varies from highly theoretical mathematical studies to computer science issues involving the effects of computer hardware and software on the implementation of specific algorithms.
References:
https://en.wikipedia.org/wiki/Numerical_analysis
"Numerical Methods for Scientific and Engineering Computation" by M. K. Jain, S.R.K. Iyengar, R. K. Jain.
" Introduction to Numerical Analysis" by F. B. Hildebrand
"Numerical Mathematical Analysis" by James B. Scarborough
