computational mathematics
Computational mathematics is the art of getting a computer to solve mathematical problems that have no tidy formula for the answer — or whose formula is too slow or too delicate to use directly. Most real problems are like this: you cannot write down the integral of e^(-x^2) in elementary functions, you cannot solve a general polynomial of degree five by a formula, and you cannot store the exact value of pi. So instead of an exact answer you compute an approximate one, on purpose, and you keep the error small and understood.
The core idea is to replace an infinite or continuous operation by a finite, mechanical one a machine can actually carry out. A derivative (a limit) becomes a difference quotient over a small step h; an integral (an area) becomes a sum of slices; the solution of a nonlinear equation becomes a sequence of guesses that march toward the root. Each replacement introduces error, and the whole subject is organized around three eternal questions: how accurate is the answer, how much work and memory does it cost, and is the procedure stable (do small disturbances in the input or the rounding stay small in the output).
This differs from pure and symbolic mathematics, which seek exact closed forms and proofs. Symbolic computation gives you sqrt(2) as the literal symbol sqrt(2); numerical computation gives you 1.41421356 and a bound on how wrong that is. The two are complementary — symbolic is exact but can explode in size and does not scale, numerical is approximate but fast and scales to millions of unknowns. Computational mathematics is the engine room behind weather forecasts, aircraft design, drug discovery, image processing, and machine learning.
To find sqrt(2), you do not look up a formula — you iterate x_{n+1} = (x_n + 2/x_n)/2 starting from x_0 = 1, giving 1.5, 1.41666..., 1.414215..., which doubles its correct digits each step until it matches the machine's precision.
An exact, unanswerable-in-closed-form question becomes a short, convergent, mechanical recipe — the heart of the subject.
A common misconception is that the computer gives the true answer. It almost never does: every numerical result is approximate, and the goal is a controlled, understood error — not exactness.