Multiplication & Convolution on a Linear Array

Tom Kelliher, CS 315

Apr. 14, 1999

Administrivia

Announcements

Assignment

Read 3.3.

From Last Time

Linear array multiplication example, run time calculation.

Outline

1. Returning to the run time.

2. Three optimizations.

3. Convolution, connections.

Coming Up

Shuffle-exchange networks.

Multiplication on a Linear Array

1. Run time, efficiency, work.

   What did we forget about in the run time calculation?

Improvement 1

1. Merging adjacent cells.

2. Reworked example.

3. Changes to run time, efficiency, work?

Improvement 2

1. All input from one side of array.

2. Reworked example.

3. Changes to run time, efficiency, work?

Improvement 3

Using the original array to perform two multiplications in parallel.

Optimality

Show that this algorithm is optimal for a linear array.

Connecting to Convolution

The convolution of the N-vectors and is the 2N - 1 vector where

where .

• How does this relate to integer multiplication?

• How does this relate to polynomial multiplication?

• Use in signal processing.