Root finding
Many problems come down to solving f(x) = 0 for x. Root-finding methods produce successively better approximations of a root. The bisection method brackets a root between two points where the function changes sign and halves the interval each step — slow but dependable. The Newton-Raphson method uses the derivative to move quickly toward a root when the starting guess is good, but it can fail when the guess is poor or the derivative is awkward to evaluate. A fuller treatment is in Root Finding Methods.
Linear systems
Linear systems of the form Ax = b appear everywhere in science and engineering, and a large share of scientific computing time is spent solving them. Direct methods such as Gaussian elimination transform the system into a triangular form and then solve it by substitution; iterative methods repeatedly improve a starting guess. The choice depends on the size and structure of the matrix. See Solving Linear Systems.
Interpolation and approximation
When data is known only at points, interpolation fits a function through those points so values can be estimated in between. Polynomial interpolation in its Lagrange and Newton divided-difference forms is the classical approach; piecewise methods such as splines fit separate low-degree polynomials on each interval and are often more stable for many data points. Approximation problems — finding a simple function close to a complicated one — are handled by related least-squares techniques.
Differential equations
Differential equations describe how systems change over time. Methods for initial value problems step forward from a starting condition: the Euler method uses the slope at the current point, while Runge-Kutta methods evaluate the slope at several points within each step and combine them for higher accuracy. Stiff problems need specially designed methods. See Ordinary Differential Equations.
Every method family has strengths and weaknesses; the right choice depends on the problem, the accuracy required, and the available computing resources.