Why do numerical analysts replace a difficult problem with a nearby easier problem?

Numerical analysts replace difficult problems with "nearby" easier problems primarily because many mathematical models in science, engineering, and business are too complex to be solved explicitly. By substituting a challenging problem with a more tractable approximation, analysts can obtain useful numerical solutions using algorithms that computers can execute.123

This process often involves:

• Approximation Theory: Replacing complex functions with simpler ones, such as polynomials or rational functions, which are easier for computers to evaluate using basic arithmetic operations.4
• Discretization: Converting continuous problems—such as those involving differential equations or integral equations—into systems that can be solved at a discrete set of points.5
• Practical Implementation: Utilizing methods like the finite element method to approximate complex surfaces or processes with a finite number of simple elements.1

Ultimately, this approach allows for the practical application of sophisticated mathematical models, provided that the analyst carefully considers the error introduced by the approximation and ensures the numerical method remains stable and efficient.3