Question

For the following expressions, state the numerical difficulties that may occur, and rewrite the formulas in a way that is more suitable for numerical computation: (a) \(\sqrt{x+\frac{1}{x}}-\sqrt{x-\frac{1}{x}}\), where \(x \gg 1\). (b) \(\sqrt{\frac{1}{a^{2}}+\frac{1}{b^{2}}}\), where \(a \approx 0\) and \(b \approx 1\).

Short answer

Question: Rewrite the given expressions to avoid numerical difficulties. (a) \(\sqrt{x+\frac{1}{x}}-\sqrt{x-\frac{1}{x}}\), where \(x \gg 1\). (b) \(\sqrt{\frac{1}{a^{2}}+\frac{1}{b^{2}}}\), where \(a \approx 0\) and \(b \approx 1\). Answer: (a) Rewrite the expression as \(\frac{2\frac{1}{x}}{2\sqrt{1-\frac{1}{x^{2}} }}\). (b) Rewrite the expression as \(\frac{1}{|a|} \sqrt{1 + \frac{a^2}{b^2}}\).

Step-by-step solution

(a) Identify the issue

For the expression \(\sqrt{x+\frac{1}{x}}-\sqrt{x-\frac{1}{x}}\), the difference of two very close square roots may lead to cancellation errors.

(a) Simplify expression

To eliminate the potential cancellation error from the difference, we can multiply and divide the expression by the conjugate, as follows: $$ \frac{\sqrt{x+\frac{1}{x}}-\sqrt{x-\frac{1}{x}}}{\sqrt{x+\frac{1}{x}}+\sqrt{x-\frac{1}{x}}} $$

(a) Rewrite formula

By multiplying and dividing by the conjugate, we can rewrite the expression as: $$ \frac{2\frac{1}{x}}{2\sqrt{1-\frac{1}{x^{2}} }} $$ This expression is more suitable for numerical computation, since it eliminates the cancellation error due to the difference of near-equal square roots.

(b) Identify the issue

For the expression \(\sqrt{\frac{1}{a^{2}}+\frac{1}{b^{2}}}\), the term \(\frac{1}{a^{2}}\) may lead to numerical difficulties when \(a \approx 0\).

(b) Simplify expression

Instead of computing each term separately and then adding them, we can factor out the term \(\frac{1}{a^{2}}\) from both terms as follows: $$ \sqrt{\frac{1}{a^{2}} (1 + \frac{a^2}{b^2})} $$

(b) Rewrite formula

By factoring out \(\frac{1}{a^{2}}\), we avoid the numerical issue associated with dividing by a very small value. Thus, the expression can be rewritten as: $$ \frac{1}{|a|} \sqrt{1 + \frac{a^2}{b^2}} $$ This expression is more suitable for numerical computation since it does not involve direct calculation of \(\frac{1}{a^{2}}\) when \(a \approx 0\).

Key concepts

Cancellation Error

When performing calculations, a cancellation error occurs when subtracting nearly equal numbers, which can lead to a significant loss of precision in the result. This is due to the fact that during subtraction, the significant digits the numbers share are canceled out, leaving behind the less significant, and often less accurate, digits.

As we've seen in the exercise with the expression \( \sqrt{x+\frac{1}{x}}-\sqrt{x-\frac{1}{x}} \), when \(x \gg 1\), the two square roots become very similar, and subtracting them directly causes cancellation error. To mitigate this, the solution leveraged the mathematical technique of multiplying and dividing by the conjugate, which cleverly rearranges the terms to prevent this problematic subtraction.

By understanding and preventing cancellation errors, we can ensure more accurate numerical computations, particularly when dealing with floating-point arithmetic where precision is often limited.

Numerical Stability

The concept of numerical stability refers to the sensitivity of an algorithm's output to small changes or errors in its input. Numerically stable algorithms produce consistent and reliable results, even in the presence of rounding errors or when computing with numbers of vastly different magnitudes.

The exercise shows the importance of numerical stability when computing the square root of the sum of two fractions where \(a \approx 0\) and \(b \approx 1\). Directly evaluating such expressions can result in unstable computations due to the small value of \(a\). To overcome this, factoring out the term \(\frac{1}{a^{2}}\) from both parts of the expression sidesteps potential issues. Rather than computing values that could lead to overflow or underflow, we rearrange the expression to maintain stability.

Whether you're programming algorithms or solving mathematical problems, maintaining numerical stability is key to obtaining accurate, real-world applicable solutions.

Square Root Computations

Computing square roots is a common operation in many fields of science and engineering, but it can also present numerical challenges. When we compute square roots, we must be wary of how the operation is applied, especially when the input numbers might cause numerical instability or cancellation errors.

In both parts of the exercise, we saw that computing square roots directly from the given expressions could lead to inaccuracies. The solution to improve the computation involves algebraic manipulation to reframe the expressions before taking the square root. By using simplification techniques like factoring out common terms or multiplying by a conjugate, the stability of the square root computation is enhanced, and the possibility of errors is reduced.

As with any computation, understanding the underlying numeric behavior when taking square roots is crucial to prevent errors. This can involve checking the size of the input values, rearranging terms to avoid subtraction of similar numbers, and simplifying expressions to a form that's more numerically forgiving before performing the square root operation.