Analytical, Graphical, and Numerical


Quick
An analytical approach is used to find the exact solution to a problem but for complex systems the math can become too complicated.
A graphical approach is used to find an approximate solution to a problem by viewing an interpreting a graphical image accordingly.
A numerical approach is used to find an approximate solution to a problem but may be simpler than an analytical approach.


Details

To develop a solution graphically means to look at a graph and interpret it accordingly in order to determine a solution.
A graphical solution is a solution derived using graphical methods, or that was solved graphically.

To develop a solution analytically means to use calculus, trigonometry, and other math techniques to develop an exact solution.
An analytic or analytical solution is a solution derived using analytical methods, or that was solved analytically.

Sometimes the math involved with analytical analysis becomes too complicated. A numerical method can be a simpler approach than its equivalent analytical method. A numerical approach usually involves using equations to approximate between smaller and smaller intervals, in effort to develop a model that represents the exact solution.
A numerical solution is a solution derived using numerical methods, or that was solved numerically.

Some numerical methods are:
   ▪ accuracy
   ▪ bisection
   ▪ decimal answer
   ▪ error
       • left rule
       • midpoint rule
       • right rule
       • trapezoid rule
   ▪ Euler's method
   ▪ finding derivative
   ▪ integration
       • error
       • error and integrand
       • left rule
       • midpoint rule
       • right rule
       • Simpson's rule
       • trapezoid rule
   ▪ iterative
   ▪ Newton's method

The following is an example:

Imagine trying to solve an equation for an unknown variable, such as: y − 7 = 0. There is an analytic solution if the equation can be solved explicitly for the unknown variable. In this case, it is easy to see that the explicit analytic solution is y = 7, which is the exact solution. An alternative to this process is to develop an algorithm on a computer to solve the equation numerically.

The algorithm would test various values for y, and then stop with a solution when the equation was satisfied to some chosen tolerance. If it is designated for the computer to solve this equation to an accuracy of 0.5, then the computer would follow the algorithm until it found a solution to this degree of accuracy. Given an initial guess y = 1, depending on the algorithm, it might come up with the following guesses: y = 4.2 (no good), y = 5.3 (no good), y = 6.6 (good to the tolerance specified), and return the solution y = 6.6. Note that to be more accurate, a tolerance that is much smaller may be specified, such as 0.001. The computer might eventually get a result after the following sequence: 4.2, 5.3, 6.6, 7.2, 7.05, 6.98, 7.0005, and then return with the solution y = 7.0005.

For such a simple equation a computer wouldn't be necessary to arrive at a solution. But some equations contain such a large number of terms that an analytical process is very difficult to apply. Therefore, algorithms must be developed to solve these equations efficiently and accurately.