  We describe an algorithm to find ribbon disks for alternating knots, and the
results of a computer implementation of this algorithm. The algorithm is
underlain by a slice link obstruction coming from Donaldson's diagonalisation
theorem. It successfully finds ribbon disks for slice two-bridge knots and for
the connected sum of any alternating knot with its reverse mirror, as well as
for 662,903 prime alternating knots of 21 or fewer crossings. We also identify
some examples of ribbon alternating knots for which the algorithm fails to find
ribbon disks, though a related search identifies all such examples known.
Combining these searches with known obstructions, we resolve the sliceness of
all but 3,276 of the over 1.2 billion prime alternating knots with 21 or fewer
crossings.
