Symbolic .vs Numerical

Symbolic and numerical computation represent two fundamentally different ways of working with mathematical problems. Let's discuss each of these methods and their differences:

1. Symbolic Computation:

   • In symbolic computation, mathematical expressions are manipulated in their symbolic form. That is, they aren't immediately evaluated or approximated to a particular number.
   • Instead, the variables remain as symbols and the computations are carried out in terms of these symbols.
   • Symbolic computing is typically used for deriving formulas, simplifying expressions, solving equations exactly, and other tasks where the exact form of an equation or expression is required.
   • Example: When you ask a symbolic computation system like Wolfram Mathematica to compute the integral of ( \sin(x) ) with respect to ( x ), it will return the exact result, ( -\cos(x) + C ), where ( C ) is an integration constant.

2. Numerical Computation:

   • In numerical computation, mathematical problems are solved using numerical methods and approximations, typically yielding a specific number or a set of numbers as the result.
   • This approach is especially useful for problems that are difficult or impossible to solve symbolically.
   • Numerical methods are used to obtain approximate solutions to complex problems using iterative processes, approximations, and other techniques.
   • The precision of numerical solutions can often be controlled by setting the desired level of accuracy or the number of significant digits.
   • Example: If you want to solve a complex differential equation that doesn't have a known symbolic solution, you might use a numerical method to get an approximation of the solution for specific conditions or over a specific interval.

Differences between Symbolic and Numerical Computation:

• Exactness: Symbolic methods give exact answers (like formulas), whereas numerical methods provide approximations.
• Scope: Some problems can only be solved numerically because they don't have an exact symbolic solution.
• Speed: Numerical methods, especially for certain problems, can be faster than trying to solve something symbolically. This is especially true for very complex problems or those with no known closed-form solutions.
• Utility: Symbolic solutions are especially useful in theoretical work, where the form of the answer can give insights into the problem. Numerical solutions are practical for real-world problems where an exact symbolic answer either doesn't exist or isn't necessary.

In practice, many advanced software packages, including Wolfram Mathematica, offer tools for both symbolic and numerical computation, allowing users to choose the best method for a given problem.