Re: Applications of num-analysis
From: Jon Harrop (usenet_at_jdh30.plus.com)
Date: 02/23/05
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Date: Wed, 23 Feb 2005 19:56:06 +0000
neptunesan@gmail.com wrote:
> Hi, I am very new to this field of mathematics, and am interested in
> the applications of numerical analysis. From speaking to some collegues
> i understand that it can be used in analysing a truss, buy looking at
> the eigenvalues, and also in weather forecasting. Could anyone here
> please explain any other uses of it to me?
Numerical analysis has a huge number of practical applications, far too many
to list here.
Many practically important problems (or at least, simplified models) can be
solved or approximated analytically.
Firstly, numerical analysis can be used to test such analytic results to
ensure that a mistake has not been made in the derivation. If the result of
a numerical computation differs significantly from the theoretical
expectation then one or both of the methods was wrong.
For example, the following Mathematica code computes the analytic result of
an indefinite integral before computing the numeric value of the result:
In[1]:= FullSimplify[Integrate[Sqrt[Tan[x]], {x, 0, Pi/4}]]
Pi + Log[3 - 2 Sqrt[2]]
Out[1]= -----------------------
2 Sqrt[2]
In[2]:= N[%, 50]
Out[2]= 0.48749549439936104835995016726460542200296010999736
The analytic derivation can be tested numerically, to evaluate the integral
without taking it analytically. This gives a similar result because the
analytic and numerical results are both accurate:
In[3]:= NIntegrate[Sqrt[Tan[x]], {x, 0, Pi/4}]
Out[3]= 0.487495
Often, numerical analysis is then applied to similar problems in order to
assess the applicability of the theoretical derivation, e.g. to see if the
theoretical derivation is actually more widely applicable than the rigorous
assumptions made in the derivation would suggest.
For example, it can be shown that the distribution of the eigenvalues of
Gaussian random matrices is a semi-circle. Furthermore, numerical
computations indicate that this law is a good approximation for many other
forms of random matrix. Such results are very important to a great many
disciplines because the eigenvalues of many practical matrices determine
physical properties.
My book gives several examples of numerical computations as well as
efficient implementations in OCaml:
http://www.ffconsultancy.com/products/ocaml_for_scientists/complete
-- Dr Jon D Harrop, Flying Frog Consultancy http://ffconsultancy.com
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