Editing Toom–Cook multiplication (section)

=== Pointwise multiplication ===
Unlike multiplying the polynomials ''p''(·) and ''q''(·), multiplying the evaluated values ''p''(''a'') and ''q''(''a'') just involves multiplying integers &mdash; a smaller instance of the original problem. We recursively invoke our multiplication procedure to multiply each pair of evaluated points. In practical implementations, as the operands become smaller, the algorithm will switch to [[Multiplication algorithm#Long multiplication|schoolbook long multiplication]]. Letting ''r'' be the product polynomial, in our example we have:

:{|
|-
|''r''(0) || = || ''p''(0)''q''(0) || = || 56789012 × 54321098 || = ||align="right"| 3084841486175176
|-
|''r''(1) || = || ''p''(1)''q''(1) || = || 135813702 × 97639739 || = ||align="right"| 13260814415903778
|-
|''r''(−1) || = || ''p''(−1)''q''(−1) || = || −21988766 × 11199987 || = ||align="right"| −246273893346042
|-
|''r''(−2) || = || ''p''(−2)''q''(−2) || = || −100519632 × −31723594 || = ||align="right"| 3188843994597408
|-
|''r''(∞) || = || ''p''(∞)''q''(∞) || = || 123456 × 98765 || = ||align="right"| 12193131840.
|}

As shown, these can also be negative. For large enough numbers, this is the most expensive step, the only step that is not linear in the sizes of ''m'' and ''n''.