Hi gilby, It sounds like its a Pampas grass you have.
I do know they are a devil to dig out and will take a month of sundays to dig it all out!
Luckely there are a couple of weedkillers that are available from garden centres that will do the hard wok for you, use either a product named- 'Root Out' or 'SBK Brushwood Killer and abide by instructions. You can apply it now,seeing as yuo've cut it down but it usually more effective in the spring. Maybe treat it now, then see if a further application is required in the spring. Good Luck Tbird+