Numerical analysis builds algorithms when exact answers won’t come
Numerical analysis studies algorithms for continuous-mathematics problems—real or complex variables—using approximation alongside symbolic manipulation. When closed forms fail, stable discrete procedures still deliver usable digits. Its uses range from weather forecasting and spacecraft trajectories to car-crash simulation and airline scheduling.
Numerical analysis designs and studies methods that give approximate but accurate answers to hard problems, many impossible to solve symbolically, with solutions kept inside stated error bounds. The approach is ancient: the Babylonian tablet YBC 7289 gives a sexagesimal approximation of the square root of 2, the diagonal of a unit square, and linear interpolation was in use more than 2,000 years ago. Algorithm names such as Newton's method, the Lagrange interpolation polynomial and Gaussian elimination show how many great mathematicians worked on it, though some date the modern field to E. T. Whittaker in 1912.
Before computers, people looked up values in large printed tables, some calculated to 16 or more decimal places, and plugged them into interpolation formulas; mechanical calculators evolved into electronic computers in the 1940s, and many of the old formulas survive inside software. Applications now include weather prediction, spacecraft trajectories computed from systems of ordinary differential equations, car-crash simulations that solve partial differential equations, pricing of stocks and derivatives, and airline decisions on fares, crews and fuel. The Institute of Mathematics and its Applications launched the Leslie Fox Prize for the field in 1985.
Direct methods such as Gaussian elimination, QR factorisation and the simplex method would finish exactly in a finite number of steps given infinite precision, whereas iterative methods start from a guess and converge only in the limit, stopping when a test such as a small residual is met. Iterative methods are more common, and even the conjugate gradient method and GMRES, direct in principle, are used iteratively. Bisection applied to 3x³ − 24 starting between 0 and 3 narrows the root to between 1.875 and 2.0625.
Errors come from rounding, since finite machines cannot store every real number, and from cutting an iteration short or replacing a continuous problem with a discrete one, and they propagate through later steps. Conditioning matters too: evaluating 1/(x − 1) near 1 is ill-conditioned, as moving x from 1.1 to 1.001 swings the output from 10 to 1,000, while near 10 the same function is well-conditioned. A stable algorithm keeps errors from growing, and finding one for a well-posed problem is a central art of the field.
Source: Numerical analysis
