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Stability in numerical analysis: Why is it crucial for mathematical algorithms?
In the field of numerical analysis, numerical stability is a highly desirable property of mathematical algorithms. The precise definition of stability depends on the context, especially in numerical linear algebra and algorithms for solving ordinary and partial differential equations via discrete approximations.
“Stability ensures that small changes to input data do not cause large fluctuations in the final result.”
In numerical linear algebra, one is primarily interested in instabilities that arise near nearly singular problems, such as very small or nearly coincident eigenvalues. In numerical algorithms for solving differential equations, the concern is that round-off errors or small fluctuations in the initial data can lead to large deviations between the final answer and the exact solution. Some numerical algorithms may smooth out small fluctuations and errors in the input data, while others may amplify these errors. Computations that can be proven not to amplify approximation errors are called numerically stable.
A common task in numerical analysis is to choose algorithms that are robust, meaning that the results do not vary drastically when the input data changes slightly. The opposite phenomenon is algorithmic instability. Often, an algorithm involves an approximation method, and in some cases it can be shown that the algorithm will approach the correct solution when using real numbers instead of floating point numbers. However, in this case, there is no guarantee that it will converge to the correct solution, since round-off or truncation errors in floating-point numbers may be magnified, causing the deviation from the correct solution to grow exponentially.
Stability in Numerical Linear Algebra
In numerical linear algebra, the concept of stability can be formalized in several different ways. The commonly used definitions of forward, backward, and mixed stability often appear in this field. Let us consider a problem solved by a numerical algorithm, where the function f maps data x to a solution y. The algorithm's result y* will usually deviate from the "true" solution y. The main reasons for the error are rounding error and truncation error.
“The forward error of an algorithm is the difference between its result and the solution; the backward error is the smallest Δx such that f(x + Δx) = y*.”
The forward error is the difference between y* and y; the backward error is the minimum Δx such that f(x + Δx) = y*. There is a condition number relationship between the forward error and the backward error: the size of the forward error is at most the product of the condition number and the backward error. In many cases, it is more natural to consider relative errors. When the backward error is small for all inputs x, we call the algorithm backward stable. Of course, "small" is a relative term and its definition will depend on the specific context.
Often a more general definition of numerical stability is given, called hybrid stability, which combines the forward error and the backward error. An algorithm is stable if it approximately solves a neighboring problem, that is, there exists a small Δx such that f(x + Δx) - y* is also small. Therefore, a backwards-stable algorithm is always stable. With respect to forward stability, an algorithm is forward stable if its forward error divided by the condition number of the problem is relatively small.
Stability in Numerical Differential Equations
In solving differential equations, stability is defined differently. In numerical ordinary differential equations, there are various notions of numerical stability, such as A-stability. These concepts are often related to certain notions of stability in dynamical systems, in particular Lyapunov stability. When solving stiff equations, it is very important to use a stable method.
"Stability is sometimes achieved by introducing numerical diffusion, which ensures that round-off errors in calculations do not accumulate to dangerous levels."
An algorithm for solving partial differential equations of the linear evolution type is considered stable if the total change in the numerical solution remains bounded as the step size approaches zero. The Lax equivalence theorem states that if an algorithm is consistent and stable, it will converge. However, for nonlinear partial differential equations, the definition of stability is much more complicated because many properties in nonlinear equations do not exist in their linear counterparts.
Example
Computing the square root of 2 (approximately 1.41421) is a well-defined problem. Many algorithms solve this problem by starting with an initial approximation x0, such as x0 = 1.4, and then continually computing improved guesses x1, x2, and so on. A typical method that might be used is the famous Babylonian method, which has the formula xk+1 = (xk + 2/xk) / 2.
The other method is called "Method X", and its formula is xk+1 = (xk^2 − 2)² + xk. A few iterations of each method are recorded below the table, and we see that the Babylonian method converges quickly regardless of the initial guess, while method X converges very slowly at x0 = 1.4 and diverges oddly at x0 = 1.42. Therefore, the Babylonian method is considered to be numerically stable, while method X is numerically unstable.
Numerical stability is also affected by the number of significant digits retained by the machine. A machine that retains only four significant digits would give a good example of the consequences that might result from a loss of significance. For example, consider the equivalent functions f(x) and g(x). When calculating f(500) and g(500), even though the two functions are equal, they produce completely different results, showing how small errors can lead to large variations.
In summary, numerical stability is crucial in numerical analysis, as it affects the accuracy and efficiency with which we solve problems. However, in your mind, what kind of algorithm or method can remain stable under unstable conditions?
