In an earlier discussion, we assumed

• error originates on the operands (i.e., numbers).
• arithmetic operations introduce no additional error.

How would things change if arithmetic operations introduced additional error?

1 Multiplication

Suppose we were multiplying two numbers:

$$ x * y = z $$

To denote finite precision, we need to use the star/chop notation:

$$ z^{*} = (x^{*} * y^{*})^* $$

where

• $x^{*} = x(1 + \epsilon_{x})$
• $y^{*} = y(1 + \epsilon_{y})$
• $z^{*} = z(1 + \epsilon_{z})$

Let us start by expanding all the terms (and cleaning everything up).

\begin{array}{rl} z^{*} &=& (x^{*} * y^{*})^{*} \\ &=& (x^{*} y^{*})^{*} \\ &=& (x (1 + \epsilon_{x}) y (1 + \epsilon_{y}))^{*} \\ &=& (xy (1 + \epsilon_{x})(1 + \epsilon_{y}))^{*} \\ &\approx& (xy (1 + \epsilon_{x} + \epsilon_{y}))^{*} \\ &\approx& (xy (1 + \epsilon_{x} + \epsilon_{y}))(1 + \epsilon_{mult}) \\ &\approx& xy (1 + \epsilon_{x} + \epsilon_{y})(1 + \epsilon_{mult}) \\ &\approx& xy (1 + \epsilon_{x} + \epsilon_{y} + \epsilon_{mult}) \\ \end{array}

We know how to compute the relative error.

\begin{array}{rl} z^{*} - z &=& \frac{| xy (1 + \epsilon_{x} + \epsilon_{y} + \epsilon_{mult}) - xy |}{xy} \\ &=& \frac{| xy (\epsilon_{x} + \epsilon_{y} + \epsilon_{mult})|}{xy} \\ &=& |\epsilon_{x} + \epsilon_{y} + \epsilon_{mult}| \\ \end{array}

Interesting… the only difference is an additional epsilon term (i.e., $\epsilon_{mult}$)

$$ \begin{array}{rl} |\epsilon_{x} + \epsilon_{y}| &vs& |\epsilon_{x} + \epsilon_{y} + \epsilon_{mult}| \\ \end{array} $$

2 Other Operations

How do division, addition, and subtraction change? Those can be practice problems!