numerical computation and symbolic computation

[PG1995]

Hi

In simple words, in numerical computation you get a numeric value, i.e. a number, as a solution while in symbolic computation you can get a general solution/formula for the problem at hand. For example, TI-84 can only do numerical computation and TI-89 is capable of symbolic computation. On the other hand, software packages such as Matlab also perform numerical computation but have very rich library of built-in algorithms/methods to handle a variety of different problems.

It looks like numerical computation is more preferred way of doing real life problems or problems in general as compared to using symbolic computation. Why is so? Why do we need numerical computation or analysis? Why don't we use symbolic computation method (computer algebra system) to get a general formula? I understand doing a complex problem by hand can become unwieldy and complex and we would become lost in symbols. But computers don't have such limitations and I think they can get a general formula even for some equation with degree of 100 within a matter of few blinks whereas for human it becomes cumbersome even to solve an equation with degree of '4'. I hope you get what I'm trying to ask. Please help me with it. Thank you.

Regards
PG

 

[panic mode]

solving something using symbolic form can be very difficult. on the other hand cpu time is cheap so it is not big delay to run several iterations to get satisfactory result.

 

[MrAl]

Hi PG,

You just can not do everything with symbolic calculations. You'll find this out when you read your other thread with the integration of e^(x^2)

Also, you can not get a computer to do an equation of degree above 4 or 5, and you're lucky if you get it to do a degree of 4. That's because to factor some equations you have to be able to find the roots, and it's not possible to find the roots symbolically if the degree is over 5, and probably too much for most software packages to do a degree of 5 too. You may not even get one to do degree of 3. Most can do up to a degree of 2 though.
Numerically however, the roots can be found within a second of even higher orders like 7 or 8, but there is even a limit to this as the numerical roundoff and truncation takes a toll on the accuracy so going higher means specialized software that can handle big floats.

Computers can do a lot symbolically, but since there is a limit to what can be done by ANY means we cant do everything like that.