If you do not or are unable to dig these, you will need to either try smothering them, consistently cutting any new growth as it appears, or using an herbicide. If you don't want to use herbicide, you could put a plastic tarp over them and over it with mulch or arborist wood chips, but it may take a year or so for them to die, since they look fairly large and we would imagine that their root mass is rather large as well. Also, cutting new growth as it appears may take some time to deplete the energy stored in the roots. Plants with large root masses will have more energy storage and will take longer for that energy to be used up.

Alternatively, you can wait for growth to appear and apply a non-selective herbicide, which will help kill the roots. This may also take a few applications to have them die completely. 

Let us know if you have further questions,

Emily