» Why are we taught solving Numerical methods manually!
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bobbym
2012-11-24 05:48:04
Ask all the questions now and you will not have to ask them later. There really are no better questions.
Raabi
2012-11-24 05:44:53
Got it. Thank you Bobbym for your patience and getting me some confidence. Next time, I may bother you with a better question. Keep smiling.
bobbym
2012-11-24 05:36:45
Hi;
(a)- Couldn't we find the root with classic algebra (Analytically)?
Yes, but remember this is only an example book problem. It is there to teach the new technique. In the real world you rarely get such an easy problem.
How to decide which method (Newton's, Bisection, Secant etc) should we use for a particular problem?
That is something even the best books do not adequately cover. Finding the best one is an art form and is what separates one person from another.
Raabi
2012-11-24 05:29:14
Yes Bobbym. Actually, I have shot both questions in dark - looking at some example problems. I calculated the Root geometrically, by drawing the curve and then tangents at f(xn) repeatedly (4 iterations only). I still wonder:
(a)- Couldn't we find the root with classic algebra (Analytically)? (b)- How to decide which method (Newton's, Bisection, Secant etc) should we use for a particular problem?
Your encouragement really helped me moving ahead. Thank you so much.
Raabi
bobbym
2012-11-24 04:35:32
Hi;
Okay Question 1:
The above is the formula for Newton's iteration or Newton Raphson method.
In many numerical methods we start with a guess! This will be very disturbing to many math types who will actually begin to whine and moan about that. Do you understand so far?
Raabi
2012-11-24 01:02:29
This is very encouraging response, Bobbym. Here I have just copied the original question for solution:
Question 1: Use Newton's Method to find the only real root of the equation
correct to 9 decimal places.
Question 2: Use the Newton-Raphson method, with 3 as starting point, to find a fraction that is within 10^−8 of . Show that your answer is within 10^−8 of the truth.
I have copied the above questions from the question paper, given in an exam. It will be so nice to get my confusion cleared.
Raabi
bobbym
2012-11-23 19:58:32
Hi;
Post problems and we will do them together. Many others will join in. They will be seduced by the numerical side of math.
Raabi
2012-11-23 14:30:46
I feel so grateful to Bob Bundy and Bobbym for their help and references. I will try to benefit from these ideas, and wish the best for those; who try to help others like me.
bobbym
2012-11-23 08:03:44
Hi Raabi;
There is no mention of using any of these methods using computers.
The sad truth of the matter is most mathematicians hate computers and won't use them. Many textbook writers are only imitating their heroes.
If you find this hard to believe then you only have to listen to Sir Michael Atiyah, Andy Wiles and many more. Further proof is provided by the homepage Of Doron Zeilberger.
My own personal experience has been to watch them debate over whether they should use a calculator or not, computer or not for my entire life!
Well industry and labs have already solved that problem. Students coming out of MIT are unable to function in a lab or in industry because of their lack of computer skills. I am not talking about how to use Word, I am talking about dealing with subtractive cancellation, smearing, iteration, the proper use of Taylor series, stiff DE's, the proper way to use the quadratic formula, over dependence on Newton's iteration, why there is no such thing as the number line, why algebra lies...
I know I worked with them but if you need more authoritative proof Read "Numerical Methods that Usually Work," by the great Forman S. Acton. The best practical numerical mathematician of them all and he is chemist!
I once watched a mathematician computing Moebius numbers by hand, by hand! Because he hates a calculator or a computer! Of course he made a mistake, many of them.
Saw the best teacher over here trying to solve a 4 x 4 simultaneous set using Cramers rule! All those determinants filled with negative numbers. She insisted on doing it by hand! She never did get the answer.
Punchline:
Whenever I learn a new numerical technique, I do it by hand! So you see they are right for the wrong reason. We say,"even a blind squirrel finds an acorn now and then!"
bob bundy
2012-11-23 05:35:32
I tried to get a better answer by googling. Don't think I really got anywhere but I did find some articles/books/courses on the subject that may interest you.
Thought I might as well give you the links ... you can decide if they useful to you.
Well the only other thought I have on this is that a computer method for solving a problem has to depend on some technique or other, often from an area of maths. So you have to study the maths in order to get anywhere with the computerised solution.
Does it say if you will have an exam that expects you to use a computer as part of the assessment?
Bob
Raabi
2012-11-23 04:18:28
Sorry Bob, I did not see your reply before. Our syllabus, for this semester, is divided into three parts:
Part 1: General Concepts on Computer Science; which include Computer Architecture, Operating Systems and Data Structures etc. Part 2: System Analysis & Design Part 3: Numerical Methods. Complete solution of each problem is solved on papers, of course calculator is used just for basic operations.
Each part has a separate paper in the exam. We have already studied Programming in Pascal and C, in our previous semester. So Numerical Methods is an exclusive topic; not just a casual discussion.
Please help.
Raabi
2012-11-23 04:10:51
Thanks for your attention, Agnishom.
I have searched the topic on the Net and found that these methods are taught to be solved manually in US and Europe, as well. On the Web, I found so many questions/answers (solutions) in PDF, for these methods, to help prepare for the college exams, in US and Europe. There is no mention of applying these methods on computers. Then what do they have to do with computer science?
I just want to straighten my perception on the subject; otherwise I will not be able to study it with confused mind. Further help will be highly appreciated.
bob bundy
2012-11-23 03:58:33
hi Raabi
When you do a problem using, say, Newton's method, I'm guessing that you use a calculator for the iterations rather than working the sums on paper! I agree with Agnishom that this will help you understand the process first after which you can start to code it. Maybe you haven't got to that chapter yet?
Also, is the book general in the sense that it doesn't promote a particular computing language. If so, then you've got to expect that it won't go into the code in detail. Perhaps the book is intended for students who haven't even met the techniques before.
Bob
Agnishom
2012-11-23 03:36:15
I think these methods are taught to you people so that you have a better view of programming.
For example, solving the Towers of Hanoi Puzzle gives you an idea of recursive relations
Perhaps your teachers believe that solving manually helps to capture the concept clearly. I personally regret doing some problems without understanding them at all