Solve linear algebra problems with direct methods. Working with linear systems in the form Ax=b, where A is the matrix of coefficients for the system, x is the column vector for the unknown variables x1 through xn, and b is a given column vector, solve for an exact solution x in a finite number of steps. One direct method is Gaussian elimination, which is a precise elimination algorithm similar to that found in elementary algebra. The problem with finding an exact solution is that rounding errors will produce imperfect results.
Solve linear algebra problems with iterative methods. An alternative to direct methods, iterative methods create a sequence of increasingly accurate approximating solutions.
Reduce non-linear problems to a sequence of linear problems. An example of this often occurs in business applications dealing with optimization, where f(x) is a function with x being a vector of unknowns. The problem may call for finding values of x that minimize f(x), where x can vary freely or be constrained. Optimization problems seek to find the most efficient way of allocating resources. This could involve determining the best investment strategies, inventory control, scheduling procedures and locating manufacturing facilities.
Approximate by interpolating. Interpolation is a way of extending a given definition of a known function that has discrete data points and applying it to nearby points and thus approximating them. This can be done with polynomials, rational functions, trigonometric polynomials and spline functions, which are smooth piecewise polynomial functions with low oscillation, used in computer graphics and statistics. You can also approximate integrals and derivatives numerically through interpolation by constructing an interpolating function p(x) that approximates function f(x), then integrating or differentiating p(x) to approximate f(x)'s integrals and derivatives.
Integrate or differentiate numerically. When dealing with ordinary and partial differential equations and integral equations from a numerical analytic standpoint, you have two methods at your disposal: finite element and finite difference methods. In the first method, you are approximating unknown functions using simpler functions to arrive at ballpark estimates of, for example, partial differential equations. In finite difference methods, used often in initial value problems in ordinary and partial differential equations, you approximate derivatives or integrals of equations by working with a discrete sets of points.