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Subsection 6.6.1 Additional homework

Homework 6.6.1.1.

In Units 6.3.1-3 we analyzed how error accumulates when computing a dot product of \(x \) and \(y \) of size \(m \) in the order indicated by

\begin{equation*} \kappa = ( ( \cdots ( ( \chi_0 \psi_0 + \chi_1 \psi_1 ) + \chi_2 \psi_2 ) + \cdots ) + \chi_{m-1} \psi_{m-1} ). \end{equation*}

Let's illustrate an alternative way of computing the dot product:

  • For \(m = 2\text{:}\)

    \begin{equation*} \kappa = \chi_0 \psi_0 + \chi_1 \psi_1 \end{equation*}

  • For \(m = 4\text{:}\)

    \begin{equation*} \kappa = ( \chi_0 \psi_0 + \chi_1 \psi_1 ) + ( \chi_2 \psi_2 + \chi_3 \psi_3 ) \end{equation*}

  • For \(m = 8\text{:}\)

    \begin{equation*} \kappa = ( ( \chi_0 \psi_0 + \chi_1 \psi_1 ) + ( \chi_2 \psi_2 + \chi_3 \psi_3 ) ) + ( ( \chi_4 \psi_4 + \chi_5 \psi_5 ) + ( \chi_6 \psi_6 + \chi_7 \psi_7 ) ) \end{equation*}

and so forth. Analyze how under the SCM error accumulates and state backward stability results.  You may assume that \(m\) is a power of two.

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