Rhodos are an extremely effort-intensive plant to eridicate. Best way is treat large cut stumps with strong glyphospate and spray any regrowth during the active period. The makers of Timbrel (triclopyr based- like a super concentrated SBK brushkiller) reckon its twice as effective as roundup/ glyphospate when sprayed on regrowth. An alternative to try, as Timbrel is very concentrated and expensive, you could use a high concentration of SBK, its the same stuff but can be had for about £17 for a litre of concentrate. I think its bad news for fish, so you wouldn't want to use it near ponds or running water, otherwise it breaks down harmlessly in a few weeks, supposedly.
Timbrel bumph...
http://msdssearch.dow.com/PublishedLiteratureDAS/dh_0061/0901b803800616d4.pdfThis FC experiment of different treatments also seems to sing the praises of Timprel vs glyphospate on Rhodo
http://www.forestry.gov.uk/fr/INFD-8F7C9YWhereas Timbrel is 44% Triclopyr, SBK is only 4% odd, so if substituting SBK for Timbrel you'd need ten times the quantity per treatment. Timbrel is recommended to be sprayed on Rhodo foliage at 130 ml (or cc) per 5 litres of water (or you can use paraffin/diesel), so SBK would need 1.3 litres per 5 litres, I think. (Check the maths and doses before trying yourself).
Alternatively you could ask the US Air force if they have some Napalm or Agent Orange going spare, they're quite effective at obliterating everything living in a given area.