DCMS say in a statement:

It is with great regret that we learn of the death of the King Abdullah bin Abdulaziz, King of Saudi Arabia.

It is requested that all flags be half-masted from 8am today until 8pm this evening.

Any other UK national flags flown alongside the Union Flag when it is at half-mast should also be at half-mast. If a flag of a foreign nation is normally flown on the same stand as the Union Flag, it should be removed.

For shame.

Pic via @suttonnick

UPDATE:

Flying flags at half mast on gov buildings for the death of Saudi king is a steaming pile of nonsense. That is all.

— Ruth Davidson (@RuthDavidsonMSP) January 23, 2015