Dealing with computations that are tedious and time consuming

Question

In attempting mathematics homework question one of the greatest redundancies I generally face is having to deal with long winded and tedious computations. For instance, in the field of Quantum mechanics there are often, simple integrals but which are extremely tedious and susceptible for anyone to fumble. In such similar scenarios, where computations can span more than a page, a single mistake can be costly. Assuming the use of Mathematica is prohibited how do you guy go about reducing such inefficiencies? I think the problem I generally face under such circumstance is a reluctance to 'let it go'. The steps are right but I feel very affected by not seeing the right solution. Has anyone face this? How do you deal with it?

Answer 1

My usual practice with things like that is to check my work as I go. The rule of thumb I tend to use is: After completing the next line, go back to the line 3rd from the bottom (which I just added) and assume that is true, and then check that the next two lines do indeed follow from there. In all, you will end up checking the connections between each line and the next three times (which makes it fairly unlikely that a simple mistake will go unnoticed).

Answer 2

There is a difference between a student homework assignment and a project for an employer or for publication.

In the homework assignment, you may be prohibited from using Mathematica or other software because one purpose of the problem is to give you practice with integration or another kind of computation. (Even then, I'd wonder why it would be prohibited to use software to check the accuracy of your work.)

There is always the caveat that answers from software are not always right--either because you made a mistake using the program or the programmers made a mistake writing it. Furthermore, digital computers do not deal with all real numbers, but with a very large and carefully chosen finite subset of the rational numbers. That distinction can lead to nasty surprises.

Once you get beyond drill problems, I think there is an obligation to verify results by using software whenever feasible. Sometimes other checks involving common sense are possible, for example making sure units make sense, dimensionalities match, and so on.