2000 Lexus ES 300 clicks but won't start

Last November, my car would not start so a mechanic installed another battery. It sometimes still would not start and would always make the clicking sound when I turned the ignition. So a mechanic installed an alternator, and, when the problem persisted, he thought that new alternator may be bad, so he installed another alternator. When I drove long distances over the holidays, the car was fine, but my relatives did all the local driving there. After the holidays the same problem recurred, not only the clicking when the ignition was turned but even stalling twice while idling. It could always be started with a jump. Oddly, after the car sat on the driveway for 2 days with frigid temperatures, the car actually started but stalled later after some local driving. Any suggestions for my mechanic? Also, is there any correlation between the car's computer and the alternator and battery?