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#1 2012-11-22 15:51:30

Raabi
Member
Registered: 2012-11-22
Posts: 18

How the Numerical methods help with higher Accuracy & Precision

Hi

As an absolute novice in Numeric Analysis, I have another query - How the Numerical methods (Newton, Euler, Taylor, False Positioning etc.) help with higher Accuracy & Precision.
I mean, if I calculate an equation


with a quality scientific calculator, and then with Newton's method; wouldn't I get the same accuracy to 15 places of decimal, for example? Or, there are some other reasons for using these methods instead of using MS Excel, Scientific calculator, or just Microsoft Math?

Please pardon me for asking too basic questions. They may help me grasp the true essence of the subject.

Thanks in anticipation.

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#2 2012-11-22 20:57:07

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 82,913

Re: How the Numerical methods help with higher Accuracy & Precision

Hi Raabi;

Generally to get higher precision you will need to work with higher precision.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#3 2012-11-23 01:41:29

Raabi
Member
Registered: 2012-11-22
Posts: 18

Re: How the Numerical methods help with higher Accuracy & Precision

Sorry bobbym, I could not get your point. If it is not inconvenient, please elaborate it to an absolute novice. I want to understand, why and when we take refuge with Numerical methods. The example, I had quoted was for a comparison to understand well.

Raabi

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#4 2012-11-23 05:21:56

debjit625
Member
Registered: 2012-07-23
Posts: 89

Re: How the Numerical methods help with higher Accuracy & Precision

Well a program should behave what its programmed for. If it's programmed to use  Newton's method, it will  and its the same what you do on paper. But at the same time computers have limitations. In this case the limitation is on decimal points ,computers work only with binary numbers they don't understand decimal points they don’t even understand negative numbers
in binary we can only have positive whole numbers {0,1,2,...} and even whole numbers are limited ,the range will depend on the bits you are using to describe the number.

So how come we can calculate decimal and negative stuff on our computers?
Computers use a special format to represent decimal points, normally in the world of programming we say floating point numbers for decimal point numbers ex .. {0.25,0.55,...}.It is important to know that binary floating point system can only represent a finite number of floating point numbers in exact form, its because the number of bits used to represent the  floating point system, if you exceed that limit then you will loose the precision.

http://en.wikipedia.org/wiki/Floating_point

You have to need programming knowledge to understand these...

If you want to know more about this format search "Floating-Point " IEEE is the most commonly used format.

Here is an example floating point numbers and their precision on my 32bits computer programmed in a C++ compiler with ANSI/ISO C++ standards
float type data gives 6 digit precision.
double type data gives 10 digit precision.

Good Luck

Last edited by debjit625 (2012-11-23 05:24:20)


Debjit Roy
___________________________________________________
The essence of mathematics lies in its freedom - Georg Cantor

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#5 2012-11-23 05:23:42

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 82,913

Re: How the Numerical methods help with higher Accuracy & Precision

Hi;

MS Excel, Scientific calculator, or just Microsoft Math?

Those three tools as well as all the rest use numerical algorithms like Newton's, secant, interval bisection to get their answers. They are all well written and will generally get you 15 or 16 digits of precision.

When you are studying numerical analysis you are studying the algorithms like Newton's that helps those programs get their answers.

If you only work with say 16 digits of precision in those algorithms then that is what you hope you will get your answer in.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#6 2012-11-23 05:55:59

Raabi
Member
Registered: 2012-11-22
Posts: 18

Re: How the Numerical methods help with higher Accuracy & Precision

Thanks Bobbym. I got some hint. Some people, like myself, are not good at asking their questions well - I'll try to improve.

Is it one of the main reasons for using Numerical methods, called improved Approximation?

I have some idea on Single Precision & Double Precision Floating Point numbers; but still could not understand; why should we use Numerical methods for the problems; which can be solved with plain algebra (probably called Analytical methods?). A real life example; comparing the two methods may resolve my problem for ever.

I am afraid, you people may feel off for my too basic questions. I can request to bear with me just for a little while; until I start walking, from crawling, on the path of you geeks. Thanks.

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#7 2012-11-23 05:58:48

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 82,913

Re: How the Numerical methods help with higher Accuracy & Precision

Hi;

We use numerical methods for problems that there are no analytical methods for. The problems you are being given in textbooks are only toy type problems. Yes, they have other ways to get the answer. Most problems require numerical techniques.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#8 2012-11-23 06:09:59

Raabi
Member
Registered: 2012-11-22
Posts: 18

Re: How the Numerical methods help with higher Accuracy & Precision

I see! We are actually getting redied for a bigger altar.  It is time to run away :-) Thank you Bobbym and have a good time.

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#9 2012-11-23 06:14:11

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 82,913

Re: How the Numerical methods help with higher Accuracy & Precision

Hi;

Hope I have not scared you away. I am working on your other question in the other thread.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

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