I experienced the same thing on my HP Dv6000 series notebook. Your compaq has the same mobo as mine. HP has a recall on certain items and your may be eligable. The extension is only for ONE year after the original ONE year warranty, so you have to take care of this in a timely manner. The recall is for the very issue you are having.
MY WORD OF ADVICE: I will be blunt, If you want this fixed, stop looking at the internet for answers, and call HP!!!!!! If you are too late, then you will have to replace the motherboard yourself!
I have seen this issue so many times and people do not listen! Take 10 mins to call HP and see if you are still eligable for the free fix.
If you are not eligable, they will tell you!
What has happened is the graphics card connector inside the laptop become hot and un-solders itself. The PC turns on but does not detect a graphics card so the display does nothing. Mine did the same thing until HP fixed it for FREE!
Check the link below and see if your model is listed.
Mine is a HP DV6000 (dv6045nr) and I luckily called with 3 days left on the extended warranty/recall HP put out.
http://h10025.www1.hp.com/ewfrf/wc/document?docname=c01087277&lc=en&dlc=en&cc=us&lang=en&os=228&product=3203586
Hope this helps and good luck! I hope your not too late! and please let me know what they say!