Very likely GPU solder failure which is very common with iMac G5's with ATI Radeon graphic card, such as yours if it is 2005 model. In this state, this basically the case, there are of course more than one connections under the graphic chip: http://pcdestiny.webs.com/bga_sc2.jpg
If it is solder failure, there is couple options:
1) Get it reflown, in reflow process, the chip is heated to a point where the solder melts and reconnects with the logic board. This method is not the most reliable.
2) Get it reballed, in reballing the chip is removed and old solder balls which used to connect the chip are removed and new ones are placed. There is very much variety in solder material quality but usually and ideally the new material contains lead. Again the chip is then heated to form connection. This method is more reliable than reflow, of course only if the person who performs it knows what he's doing.
3) Get "new" logic board. Prices of these boards when sold separatedly is usually ridicously high.
2004 model usually suffered death by capacitors going bad:
Could be the case with your machine also, but usually it is the solder cracks which kill ATI iMac G5 models. Bad capacitors can be on the motherboard or in the power supply.
Have you tried resetting the power management controls and so on? First, disconnect all cables and add-ons to the machine, then disconnect the power cable, and press the power on button for about 5 seconds, wait another 5 to 10 seconds, then reconnect the power cable while keeping the power button pressed (the computer should not power on till you release the power button and press it again.) Next, when you power on the machine, keep the Command-Option/Alt-P-R keys pressed together till you hear the start-up chime at least twice more, then shift to pressing the Command-Option/Alt-O-F keys together to get into open firmware control. You should get a greyish-white screen with black text telling you to release the four keys. Then, enter "reset-nvram", "set-defaults" and "reset-all" (without the quotation marks) in sequence, when the computer should restart with basic parameters reset. Hope this works out for you. This has worked for me whenever I have otherwise inexplicable problems in running my own iMac 20" PPC G5 bought in 2005, now running Leopard.
I have an iMac G5 that I purchased in 2005.