I had my final exam today and the paper being easy,the exam was a breeze.So 3 weeks of rest follow for me after a busy month of classes.After my exam,I happened to read this:
http://21writersblock.blogspot.com/2007/08/youve-got-mail-again.html
Now the author happens to be someone I am very fond of,and her earnest appeal to people in general to find a way to explain the idea of arranged marriage didnt fall on deaf ears.I took upon my frail shoulders the onerous task of removing the stigma(if I may call it that) associated with an "arranged" marriage.I know that the tone of my writing puts the word "arranged" at about the same level as the word match "fixing" in cricket.A thousand pardons for that inadvertent error.At the outset, let me issue the usual disclaimers that am neither married nor in love,and any views that I may express may be taken with many grains of salt.
To any prospective girls who might have an outside chance of falling in love with yours truly--I agree that after reading this your feelings for me would probably be not far from that you share for a brick on the road,or the friendly neighbourhood bulldozer, for that matter.I hasten to add that I am a romantic at heart,and its just my other side taking over for a brief period.I may be guilty of bringing down something as intangible as love,to something as mundane as mathematics.However,there are similarities-both are abstract:)) to start with.But if you look beyond the obvious ,am sure the parable am going to narrate will prove most instructive.
I shall assume that most of my readers are familiar with some numerical techniques for finding roots of equations.But the person for whom the post is intended, though clever, almost certainly doesnt,so a quick introduction.
A function f(x) is said to have a root at c if f(c)=0.Now consider an equation like f(x,y). The point (i,j) is a root if f(i,j)=0.
Consider an equation like x-cos(x+4)=0 or better still sin(x)-log(y) +6=0
I ask you to find a root of the equation.
To come to my point,I need 2 names.Lets take,err...preetha and preethi,for want of any better names:)).I give this problem to both of them, and allow them use of an elementary calculator.
Preetha,being a highly whimsical girl and a good programmer,insists on the following idiosyncrasy.She wants to arrive at the solution by chance alone,so with the help of a random number generator,she tries out numbers one after the other and tries to find if any of them is a root.Sometimes its love,oops! sorry, root at first sight.But often the random numbers turn out too random,and try as she might,picking a root by trial seems too difficult for some equations.But she is delighted whenever she does stumble on the root by chance:)
Preethi on the other hand is a girl singularly devoid of such eccentricity.She consults a book of numerical recipes and hits upon tried and tested techniques like the Golden Search method,Newton Raphson method,bisection method or for more complicated functions,Steepest Descent method and so on.The way these methods work is as follows.
Instead of choosing at random,from the general behavior of the given function,a smaller region is identified in the search space,where a root is likely to occur.Once the region is identified,simply do what Preetha does at random-try out numbers in the region and see which one fits.Notice that in the end, you dont accept a number as a root without checking its functional value.So you are not compromising in any way:))
The general name for such techniques is quite appropriate-Directed Search methods:))
Some more comments on this technique are warranted.An equation can have more than one root.But they all might not lie in the same search space.Also let us recognise a tacit assumption we have made that the equation in question has a real root,which might even be false.(sorry!!)
Assuming that both Preetha and Preethi arrive at the same solution,lets now explore the matter further.For one thing,would you expect that Preetha is happier than Preethi just because she "found" her root by chance alone?Likely not.Now how viable is Preetha's method? If she has time till kingdom come,its a foolproof method.But if you are short of time, then trusting the God of Randomness is not highly advised.
Then there is the problem of "irrational" roots:) In such cases people stop their search once they find an integer value sufficiently close to the root.:)How close?That's decided by the Convergence Criterion -the mathematical term for this.
Also let us remind ourselves that the best method actually would be to graph the function and simply determine all the roots by inspection. But such a process is feasible if you are looking from above ,which we are not.
Having said all that,when I have an equation of my own to solve,I will give the random numbers a chance first:)To my dearest random root finder I have this to say-I do wish with all my heart that the correct number pops up soon.Randomly,or with an algorithm,does it really matter ?:)
I believe in the fact that if you wish something for others,it comes true.And I say this in all seriousness.
10 comments:
Wow, I am floored... brilliant analogy ... you quite hit the nail at a number of places :). Lets see if we can carry this offline. And btw and fyi I know how to find roots of an equation. :P
Of course you do!:)How dare I suggest otherwise:DMy point was about the numerical methods of finding roots,not the analytical methods:P
ss
Brilliant and Hilarious! The analogy to match "fixing" totally hit it!!!
I would probably never have recommended math as mode of explaining love. It tends to be a bit onerous on one's faculties. An extremely entertaining thought all the same! Iam going to guess that some of your future non-math readers are going to ask for an English version of it.
Excellent post buddy!
-A
Lets take up Preetha and Preethi's algorithm again.
First of all, saying that Preetha's solution algo finds the root by chance isnt really correct. In fact, Preetha's algo would be more like - "Thou shall keep iterating, till the root is found"
Now lets take up Preethi's case. She will do the following -
a smaller region is identified in the search space,where a root is likely to occur. Once the region is identified,simply do what Preetha does at random-try out numbers in the region and see which one fits.
So we could then conclude some of these -
1) As long as the search-space of Preetha and Preethi are the same, considering that the search algo is the same, the results should be the same
2) The 'chance' factor is, whether the root is found or not. Considering that the sample space is the same set, and the search algo is the same - the chance of finding a "good root" is hence the same
3) If we may consider the search-space to be different, I would say that Preethi is suffering from a disadvantage - of having to find a 'root' among a set, whose each and every element is 'trying to be at their best behaviour', cause they the know the algo being applied on them is for finding them as a root. On that count, Preetha stands a better chance, as she can silently apply her algo on the elements of her set, and thus getting a more natural and unpretended behavioural response pattern.
From the above points it seems as if Preethi is actually running a higher probability (risk) of picking an element of the set, based on qualities which were 'doctored', as the elements knew they were being searched for the sole purpose of being a root.
Further analysis most welcome :-D
LOL!!!! that was really entertaining! cud not help laughing all the way thru! the graph analogy "looking from above".. awesome man u've given this considerable thought!!!
just one thing.. u need to know the equation first right? figuring out what the equation is, is the tough part.. which u have assumed and bypassed most conveniently! :-P
and o yes! every number is a root of more than one equation.. is that ur explanation for love triangles?! lol!
It won't be long until yours truly attracts the whips.. interesting perspective, but I felt you overdid the math part.. esp after having sat thru with you in that lousy Opti course.. :D
@Anshuman-I must admit I didnt think of that problem-that the algo being applied would itself change the behaviour of the roots:D
It reminds me of the classical problem of measurement-which incidentally led to Heisenberg's uncertainty principle and all that.
For example-if you want to measure the force of attraction between 2 bodies,you MUST assume that the introduction of your instrument does not change the force between them-if it does then you are in a soup anyway and all point of measurement is lost:)At the atomic level,indeed this is true.At the macro level,this interaction may be neglected.
Now the question is which of these is true:)
I admit thats just a new way of saying "I dont know",but you must give me credit for deftly sidestepping your question:-)
And as for this-"On that count, Preetha stands a better chance, as she can silently apply her algo on the elements of her set"-Lol!!Perhaps once Preetha finishes with her iteration I can ask her about the behaviour of "her" "roots":)
ha ha.. this has become thoroughly entertaining :-D I guess the comments section in itself can become another post, for the greater good (entertainment. enlightenment ;-D) of the community ;-)
:) Interesting comment arguments, I just wish I wasn't the sacrificial bakra :)
man.. awesome post!! Great work!!
Post one off posts on my blogs!! :-)
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