Computing for Numerical Methods Using Visual C++

Chapter 1. Modeling and Simulation

• 1.1 Numerical Approximations

• 1.2 C++ for Numerical Modeling

• 1.3 Mathematical Modeling

• 1.4 Simulation and Its Visualization

• 1.5 Numerical Methods

• 1.6 Numerical Applications

  • References

NUMERICAL APPROXIMATIONS

Numerical methods is an area of study in mathematics that discusses the solutions to various mathematical problems involving differential equations, curve fittings, integrals, eigenvalues, and root findings through approximations rather than exact solutions. This discussion is necessary because the exact solutions to these problems are difficult to obtain through the analytical approach. For example, it may be wise to evaluate

Equation 1.1. 

as the exact solution can be obtained through a well-known technique in calculus called integration by parts. However, it is not possible to apply the same method or any other analytical method to solve

Equation 1.2. 

The given equation in the above integral is difficult to solve as it is not subject to the exact methods discussed in ordinary calculus. Therefore, a numerical method is needed to produce a reasonably good approximated solution. A good approximated solution, whose value may differ from the exact solution by some fractions, is definitely better than nothing.

Numerical methods are also needed in cases where ...