Question

Why do the results obtained using a numerical method differ from the exact results obtained analytically? What are the causes of this difference?

Short answer

Answer: The main causes of differences between results obtained using numerical methods and the exact results obtained analytically are approximations, discretization, rounding errors, truncation errors, and sensitivity to initial conditions.

Step-by-step solution

Understanding Numerical Methods and Analytical Methods

Numerical methods are approximation techniques used to find the solutions of complex mathematical problems, generally because an analytical solution is too difficult, if not impossible, to obtain. Analytical methods, on the other hand, involve algebraic manipulations that lead to exact solutions or general formulas.

Main Causes of Difference between Numerical and Analytical Results

There are several reasons why results obtained using numerical methods might differ from the exact results obtained analytically. Some of the primary causes are: 1. **Approximations:** Numerical methods rely on approximations, which may result in an inaccurate solution. Analytical methods, however, involve solving problems exactly using mathematical operations. 2. **Discretization:** Numerical methods often involve converting continuous problems into discrete ones, which may not precisely capture the behavior of the underlying function. For example, when performing numerical integration or differentiation, we often divide the continuous function into discrete intervals and approximate the result. These approximations may lead to errors. 3. **Rounding Errors:** Computers have finite precision, which means they cannot represent all real numbers exactly. As a result, when using a numerical method, the computer must round off some decimal points, which can introduce small errors. Moreover, the accumulation of these rounding errors through repetitive calculations can yield unreliable results. 4. **Truncation Errors:** Numerical methods usually involve iterations until a stop condition is met, often based on a predetermined, acceptable error level. When reaching the maximum number of iterations or meeting the stop condition, the obtained result is only an approximation of the exact solution. Therefore, there might be a truncation error when stopping the process. 5. **Sensitivity to Initial Conditions:** Some numerical methods, such as those used to solve ordinary differential equations or iterative methods, are sensitive to the initial conditions and starting points chosen. The accuracy of the results, therefore, might be affected by the choice of these conditions. In conclusion, the main causes of the difference between results obtained using numerical methods and the exact results obtained analytically are approximations, discretization, rounding errors, truncation errors, and sensitivity to initial conditions.

Key concepts

Approximations in Numerical Methods

When diving into the realm of mathematics, specifically when solving equations, we encounter numerical methods. These are strategies we use to approach nearly insolvable problems analytically; they are our mathematical approximations. In essence, approximations are educated guesses but grounded in mathematical procedures—like trying to sketch a curve when your line can only be straight.

For instance, calculating the square root of 2 cannot be perfectly pinned down—analytically, it's an endless decimal. Numerical methods help by giving a value that's close enough for practical use. However, the beauty of their practicality is shadowed by their inherent inaccuracy: they are approximations, not exact values. The difference between the approximation and the true value is an error we need to reckon with in numerical computations.

Discretization Error

Envision you're trying to measure the smooth curve of a hill using only a staircase. This is the core idea behind discretization error in numerical methods. We convert continuous functions into a series of steps or discrete points. A classic case is numerical integration—think of it as summing up the areas of numerous tiny rectangles under a curve to estimate its integral. The problem here is that rectangles are not curves; they can't capture every subtle turn and twist of the hill.

The gap between the actual curve and our 'staircase' approximation is known as discretization error. The finer our 'steps' or intervals, the closer we get to the true value, but some degree of error is almost always present. This is a fundamental limitation we face when transitioning from the smooth world of calculus to the pixelated one of computational mathematics.

Rounding Errors in Computation

While working with computers, we step into a world where numbers are often rounded because of limited precision. Think of a computer like a person who, when asked to carry a huge pile of coins, can only hold a set number. They'll inevitably drop some, those lost coins representing the rounding errors in computation.

These errors stem from a computer's inability to precisely convey every real number, leading to unavoidable approximations. For example, the computer cannot endlessly replicate the value of π; it rounds it. Over time, these rounding 'dropped coins' can accumulate and distort the final result—especially in iterative processes where calculations stack upon each other, magnifying the impact of these seemingly insignificant rounding errors.

Truncation Errors

Imagine you're telling a story but get cut off before you can finish. The part you didn't tell? That's akin to truncation errors in numerical methods. Specifically, these occur when we halt an infinite process—like an infinite series or iterative method—after a finite number of steps.

For our calculations, we often set a stopping point, a moment when we decide the result is 'good enough', even though the process isn't truly complete. This introduces a truncation error because we truncate, or chop off, the remaining part of the process. How significant this error is depends on where we stop the story—do we stop before the climax, or just leave out a few final details?

Sensitivity to Initial Conditions

Some numerical methods are particularly delicate, sensitive to the starting position in a way similar to how the starting block position of a runner can influence the entire race. This sensitivity to initial conditions means that small changes at the start can lead to significantly different outcomes—even divergent paths—from what we'd expect if we began somewhere else.

This is especially true for iterative methods and those solving differential equations where choosing the starting value resembles planting a seed—the results grow from it, and different seeds can lead to varying mathematical 'plants', this variation is part of the challenges when applying numerical methods to real-world problems.