Numerical analysis makes use of the Lagrange polynomial, which shows that in four different data points there is only one polynomial that interpolates all of the points. Since it is not possible to solve the equation using a linear system, the Lagrange polynomial makes use of the identity matrix, which is a fundamental concept in numerical analysis. Exercises using the Lagrange polynomial often ask students to determine which approximations of the polynomial are best (the most precise) given the exact values calculated.
Another type of numerical analysis exercise involves Euler's Method, which is a tool to help students solve first order differential equations. For differential equations where the students are asked to compute more than the directional field of the equation, Euler's Method allows students to determine some of the values of the solution. Even in systems where there is still ambiguity, knowing some of the values assigned to the equations' solution will allow students to infer more about the behavior of the equation when subjected to new restrictions or parameters.
There is an entire family of numerical analysis exercises that make use of the Runge-Kutta Methods. Runge-Kutta help students make approximations regarding ordinary differential equations. The Method is also a tool to help mathematicians correct the next predicted solution point of a numerical system. By examining the slope of the function curve, the Runge-Katta Method allows students to observe the effects of incremental changes along the differential equation curve. Solutions are not restricted to predetermined intervals, hence the predictive value of the Method.