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**Numerical analysis** is making use of equations to calculate
quantities, not by solving for those values so as to carry out the
equation's specified arithmetic, but by devising a way to get closer
and closer to the answer by repeating some arithmetic. Essentially,
you devise a different set of equations that are solvable, and which
through repeated use, bring you closer to the original equation's
answer. This can be useful or vital if algebraically solving the
original equation is difficult or impossible.

Calculating a square root offers a straight-forward example: guess an answer, square it, and adjust your guess up or down as the resulting square indicates. Systematic methods of adjustment can be devised, e.g., if one trial was too big and another too small, use their mean as your next guess. Hitting an exact answer may require luck, but mere persistence gets you as close as you need.

Another numerical analysis technique, termed **numerical integration**,
is of use if you have no straight-forward formula for some function
but do have such a formula for the function's slope, and have at
least one point, i.e., a number and the functional value associated
with that number. Given that value and the calculable slope, the
strategy consists of using them to estimate the value of a nearby
point (the functional value of a number near the original one),
then repeating the process to map out values of the function over
an interval of interest. The slope is calculated repeatedly
throughout this interval, each time a small step further, and the
smaller the steps (i.e., the more calculation), the more accurate
the estimates (or rather, the procedure is only practical if the
function has this property).

Numerical analysis can require orders-of-magnitude more
arithmetic than more straight-forward methods: a tiny change in a
solvable equation that makes it not so, might easily make a calculation
require a million times more arithmetic. Currently it is natural
and common to use computers to do this, with programs that devise
appropriate new guesses. Often the term **computation** is used to
mean doing numerical analysis with a computer (because it requires
so much computation) and in fact, numerical analysis was a key
motivator in the development of the computer as we know it. The
methods of numerical analysis are referred to as **numerical methods**,
and their development is an entire science of much interest, because
no matter how much computing capacity is available, more efficient
and well-suited methods can solve more problems with that capacity.
Models using these methods are termed **numerical models**.

In contrast to numerical analysis, using equations by manipulating
them to solve them algebraically for the quantities you need
is called solving **analytically**, or using **analytical methods**.
With the growth of computer capacity, numerical methods are often
used even if a problem could be solved analytically.
A third method of tackling math problems, which numerical analysis
has displaced to a degree, is **analog computers**, such as **slide rules**.

The use of numerical analysis predates computers, but a much more
limited set of problems could be tackled in such a manner, even
using weeks or years of people doing the arithmetic. Numerical
analysis was often used for deriving generally useful answers, such
as digits of irrational numbers like π and *e*, values of
trigonometric functions for many possible inputs, and of roots, the
answers commonly published as tables.

Some types of methods that have been developed:

- adaptive mesh refinement.
- discontinuous Galerkin method.
- finite difference method.
- finite element method.
- fast Fourier transform.
- finite volume method.
- Krylov subspace method.
- Markov chain Monte Carlo.
- spectral method.
- smoothed-particle hydrodynamics.

adaptive mesh refinement (AMR)

astrometry

Bayesian statistics

Bernstein polynomial

finite difference method (FDM)

Fourier transform

flux reconstruction (FR)

general circulation model (GCM)

GW detection (GW)

high resolution shock capture (HRSC)

Lane-Emden equation

mass shell

Markov chain Monte Carlo (MCMC)

Mie scattering

Navier-Stokes equations (NS equations)

numerical relativity (NR)

numerical weather prediction (NWP)

PSF fitting

Riemann problem

RODEO

radiative transfer model (RTM)

stellar structure

stencil

task-based parallelism (TBP)

three dimensional model

transit timing variations (TTV)