Question

Floating-point operations In general, real numbers (with infinite decimal expansions) cannot be represented exactly in a computer by floating-point numbers (with finite decimal expansions). Suppose floating-point numbers on a particular computer carry an error of at most \(10^{-16} .\) Estimate the maximum error that is committed in evaluating the following functions. Express the error in absolute and relative (percent) terms. a. \(f(x, y)=x y\) b. \(f(x, y)=\frac{x}{y}\) c. \(F(x, y, z)=x y z\) d. \(F(x, y, z)=\frac{x / y}{z}\)

Short answer

Answer: a) For \(f(x,y) = xy\): - Absolute error: \(\delta_{xy} \leq 10^{-16}(2y + 10^{-16})\) - Relative error: \(\frac{\delta_{xy}}{f(x,y)}\cdot100 \leq \frac{10^{-16}(2y + 10^{-16})}{xy}\cdot100\) b) For \(f(x,y) = \frac{x}{y}\): - Absolute error: \(\delta_{\frac{x}{y}} \leq \frac{10^{-16}(y+10^{-16})}{y^2}\) - Relative error: \(\frac{\delta_{\frac{x}{y}}}{f(x,y)}\cdot100 \leq \frac{10^{-16}(y+10^{-16})}{xy}\cdot100\) c) For \(F(x,y,z) = xyz\): - Absolute error: \(\delta_{xyz} \leq 10^{-16}(yz + xz + xy + 10^{-16})\) - Relative error: \(\frac{\delta_{xyz}}{F(x,y,z)}\cdot100 \leq \frac{10^{-16}(yz + xz + xy + 10^{-16})}{xyz}\cdot100\) d) For \(F(x,y,z) = \frac{x/y}{z}\): - Absolute error: \(\delta_{\frac{x/y}{z}} \leq \frac{10^{-16}(y + 10^{-16})}{y^2}\cdot \frac{z+\delta_z}{z}\) - Relative error: \(\frac{\delta_{\frac{x/y}{z}}}{F(x,y,z)}\cdot100 \leq \frac{10^{-16}(y+10^{-16})}{xy}\cdot\frac{z+\delta_z}{z}\cdot100\)

Step-by-step solution

Compute the absolute error for multiplication

For \(f(x,y)=xy\), suppose the floating-point error for \(x\) is \(\delta_x\) and for \(y\) is \(\delta_y\), such that \(| \delta_x | \leq 10^{-16}\) and \(| \delta_y | \leq 10^{-16}\). Then let \(x' = x + \delta_x\) and \(y' = y + \delta_y\). So, the absolute error is given by: \(|f(x',y') - f(x,y)| = |(x+\delta_x)(y+\delta_y) - xy|\). Let's find an upper bound for this error: $\delta_{xy} =|(x+\delta_x)(y+\delta_y) - xy| \\ \leq |(x+\delta_x)(y+\delta_y) - (x+\delta_x)y + (x+\delta_x)y - xy| \\ \leq |\delta_x(y+\delta_y) + y\delta_y| \\ \leq |\delta_xy +\delta_x\delta_y + y\delta_y| \\ \leq |10^{-16}(y+\delta_y) + y\cdot10^{-16}| \\ \leq 10^{-16}(y + 10^{-16} + y) $

Compute the relative error for multiplication

Now calculate the relative error (in percentage) given by \(\frac{\delta_{xy}}{f(x,y)} * 100\), assuming \(xy \neq 0\): $\frac{\delta_{xy}}{f(x,y)} * 100 \\ \leq \frac{10^{-16}(2y + 10^{-16})}{xy} * 100 $ The errors for the other functions are computed similarly. #b. f(x, y) = x/y #

Compute the absolute error for division

For \(f(x,y) = \frac{x}{y}\), let's find an upper bound for the absolute error \(|f(x', y') - f(x, y)| = \left| \frac{x + \delta_x}{y + \delta_y} - \frac{x}{y} \right|\): $\delta_{\frac{x}{y}} = \left| \frac{x + \delta_x}{y + \delta_y} - \frac{x}{y} \right| \\ \leq \left| \frac{x + \delta_x}{y + \delta_y} - \frac{x(y + \delta_y)}{y(y+\delta_y)} \right| \\ \leq \left| \frac{\delta_x(y+\delta_y)}{y(y+\delta_y)} \right| \\ \leq \frac{10^{-16}(y + 10^{-16})}{y^2} $

Compute the relative error for division

Now calculate the relative error (in percentage) for division given by \(\frac{\delta_{\frac{x}{y}}}{f(x,y)}*100\), assuming \(x,y \neq 0\): $\frac{\delta_{\frac{x}{y}}}{f(x,y)} * 100 \\ \leq \frac{\frac{10^{-16}(y + 10^{-16})}{y^2}}{\frac{x}{y}} * 100 \\ \leq \frac{10^{-16}(y + 10^{-16})}{xy} * 100 $ #c. F(x, y, z) = x*y*z #

Compute the absolute error for triple multiplication

For \(F(x,y,z) = xyz\), using the steps described earlier, we can find the absolute error as: \(\delta_{xyz} \leq 10^{-16}(yz + xz + xy + 10^{-16})\)

Compute the relative error for triple multiplication

Now calculate the relative error (in percentage) for triple multiplication: $\frac{\delta_{xyz}}{F(x,y,z)} * 100 \\ \leq \frac{10^{-16}(yz + xz + xy + 10^{-16})}{xyz} * 100 $ #d. F(x, y, z) = \frac{x/y}{z} #

Compute the absolute error for division

For \(F(x,y,z) = \frac{x/y}{z}\), let's find its absolute error: $\delta_{\frac{x/y}{z}} = \left| \frac{\frac{x+\delta_x}{y+\delta_y}}{z+\delta_z} - \frac{\frac{x}{y}}{z} \right| \\ \leq \left|\frac{\frac{x+\delta_x}{y+\delta_y} - \frac{x}{y}}{z+\delta_z}\right| \\ \leq \frac{\frac{10^{-16}(y + 10^{-16})}{y^2}}{\frac{z^2}{z+\delta_z}}\\ \leq \frac{10^{-16}(y + 10^{-16})}{y^2}\cdot \frac{z+\delta_z}{z} $

Compute the relative error for division

Now calculate the relative error (in percentage) for this double division: $\frac{\delta_{\frac{x/y}{z}}}{F(x,y,z)} * 100 \\ \leq \frac{\frac{10^{-16}(y + 10^{-16})}{y^2}\cdot \frac{z+\delta_z}{z} \cdot z}{\frac{x}{y}} * 100 \\ \leq \frac{10^{-16}(y+10^{-16})}{xy}*\frac{z+\delta_z}{z}\cdot 100 $ These calculated bounds represent the maximum error committed in evaluating the functions for the given error in floating-point numbers, both in absolute terms and in percent terms. Absolute errors are displayed in the first steps, with the relative errors shown in later steps.

Key concepts

Absolute Error

In floating-point arithmetic, absolute error quantifies how much the estimated value deviates from the true value. It is defined as the difference between the true value and the floating-point approximation we use in computations. This measure tells us the magnitude of error without considering the true value itself.

When performing operations such as multiplication or division, floating-point numbers might have small errors associated with them, due to the inherent limitations in representing real numbers with finite decimal expansions. For example, if two floating-point numbers have errors denoted as \( \delta_x \) and \( \delta_y \), the absolute error in their multiplication \( f(x, y) = xy \) can be estimated by \( |f(x', y') - f(x, y)| \).

A more illustrative formula for the absolute error in multiplication of two numbers becomes:

• \( \delta_{xy} \leq 10^{-16}(y + 10^{-16} + y) \)

This indicates how much the multiplication result might deviate from the actual product, directly as a value without referencing the magnitude of the inputs. Understanding this type of error is essential when numeric precision is critical in computations.

Relative Error

Relative error gives us a percentage expression of how significant the absolute error is, relative to the true value itself. It helps to understand the error magnitude in context of the size of the expected result. This is particularly useful when the true values are large or small, as it provides a normalized measure of error.To compute the relative error, you take the absolute error and divide it by the true value, then convert it to a percentage. For instance, if we have computed the absolute error for a multiplication operation, the relative error would be:

• \( \frac{\delta_{xy}}{f(x, y)} \times 100 \)

This formula provides an error percentage indicating the extent of deviation relative to the computed value \( f(x, y) \). The smaller the percentage, the more accurate the computed value is.

It's worth noting that even if the absolute error is small, the relative error can be significant if the true value itself is small. Therefore, both absolute and relative errors are important in assessing the precision of floating-point arithmetic.

Multiplication and Division Errors

In floating-point arithmetic, both multiplication and division operations can introduce distinct errors due to the precision limitations. The errors arise because floating-point representation approximates real numbers, and these approximations accumulate during operations.

Multiplication:
When multiplying two floating-point numbers, each carrying a small error \( \delta_x \) and \( \delta_y \), the error can propagate and affect the final product. The absolute error in a multiplication \( f(x, y) = xy \) might look like:

• \( \delta_{xy} \leq 10^{-16}(2y + 10^{-16}) \)

This suggests that the final error in the computed product is impacted by the combined errors of both multipliers.

Division:
Similarly, division operations, such as \( f(x, y) = \frac{x}{y} \), introduce errors due to both the dividend and the divisor's inaccuracies. The absolute error of division is derived as:

• \( \left| \frac{x + \delta_x}{y + \delta_y} - \frac{x}{y} \right| \leq \frac{10^{-16}(y + 10^{-16})}{y^2} \)

These expressions help estimate how much the division results can deviate from the precise arithmetic outcome.

Both multiplication and division errors emphasize how small representation errors in inputs can scale, impacting calculation reliability. Hence, understanding and managing these errors is critical for precision in computational mathematics.