What's the difference between combinatorial and numerical problems

Question

Could you please give at least two examples of each. Thanks.

Answer 1

Numerical problems are those in which there is a calculation of some numerical quantity. The inputs, outputs, and states tend to range over the continuous sets, such as the real numbers. An examples would be: calculate how high this cannon ball will fly, given its angle and initial velocity. Numerical problems can often be solved by approximation. Because the variables are continuous, there is an assumption of "smoothness" in that if f(x-a) is too low, and f(x+a) is too high, then f(x) is likely to be closr to correct. (I may be missing the proper terminology here.)

Combinatorial problems are those in which the inputs, outputs and states tend to range over discrete sets. An example would be: calculate how many distinct paths from a to b there are in this graph.

Note that it's easy to combine aspects of each in a single problem. For example, what is the average length of the paths from a to b? Or how about: "The real part of any non-trivial zero of the Riemann zeta function is 0.5" http://en.wikipedia.org/wiki/Riemann%5Fhypothesis.

Answer 2

Combinatorial problems are effectively counting problems; these arise in the study of discrete mathematics. How many permutations are there of a finite set? Given n cereal boxes each containing one of k different prizes, how many ways are there to collect all k prizes?

Numerical problems are effectively calculation problems; these typically arise in engineering and the sciences in attempts to approximate solutions to equations (root finding or differential equations, for example) or in attempts to approximate numerical values (definite integrals or eigenvalues, for example).