B737-700 FLT CREW HAS TRACK HDG DEV DURING TRALR2 DEP LAS DUE TO FMC MALFUNCTION.

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TKOF LAS; TRALR2; NAV ENGAGED 400 FT AND AUTOPLT AT 1000 FT. DURING INITIAL CLB; I WAS CLRING OUTSIDE THE ACFT. UPON COMING INSIDE THE COCKPIT; NOTICED THE ACFT DID NOT TURN AT RBELL AND INSTEAD WENT STRAIGHT. SOMEHOW DURING THE CLB; SEVERAL WAYPOINTS DROPPED OUT OF THE FMC. THE ACFT OVERFLEW RBELL AND APCH ASKED US TO TURN IMMEDIATELY TO ROPPR AND CLB TO 8000 FT. DURING THIS TIME; THE NEW FO WAS TASK SATURATED WITH TRYING TO FIGURE OUT WHY NO CLB N1'S WERE DISPLAYED ON THE CLB PAGE. I WAS TRYING TO FIGURE OUT WHY WE HAD NO WAYPOINTS ON THE CDU. AFTER HEARING AND SENSING THE URGENCY AND TONE OF THE CTLR'S VOICE; I TOLD THE FO TO TELL APCH 'UNABLE RNAV.' APCH STARTED GIVING US VECTORS AND HDGS. DURING THIS TIME; THE CDU STARTED ACTING VERY WEIRD. SEVERAL BOGUS MESSAGES CAME UP; SUCH AS 'USING RESEARCH FUEL;' 'INSUFFICIENT FUEL.' ALSO THERE WAS NO CLB INFO ON THE CLB PAGE AND NO CRUISE INFO ON THE CRUISE PAGE. APCH PASSED ON OUR REQUEST FOR VECTORS TO THE NEXT CTLR. EVENTUALLY; WE GOT DIRECT TO A NAVAID DOWN THE ROAD AND RE-ENGAGED THE NAV BUTTON. IT APPEARED TO WORK FINE AFTER THAT. WE WROTE UP THE FMC AFTER THE FLT. SEVERAL THINGS CONTRIBUTED TO THIS DEV: 1) THE FMC COMPUTER WAS GENERATING BOGUS NAV INFO ON THE CDU. BECAUSE THERE IS NO RAW NAVAID BACKUP AND NOWHERE TO FLY; THERE WAS A PERIOD OF TIME WHERE WE WERE TRYING TO FIGURE OUT WHAT HAPPENED. 2) THE FO; BEING ON HIS SECOND TRIP EVER AT COMPANY; HAD CHANNELIZED ATTN ON THE CLB PAGE TRYING TO FIGURE OUT WHY NO CLB N1'S WERE PRESENT. THIS PREVENTED HIM FROM HELPING ME FIX THE PROB. 3) THE LNAV DEPS HAVE AN EXTREMELY TIGHT TOLERANCE AND THERE IS NO ROOM FOR DEV OR ERROR. 4) BOTH THE FO AND MYSELF HAD A 9 HR 45 MIN OVERNIGHT THE PREVIOUS NIGHT. DUE TO THE EXTREME SHIFT IN OUR BODY CLOCKS AND THE LACK OF SLEEP; OUR REACTIONS AND CRITICAL THINKING WERE SLOWED DOWN. IN RETROSPECT; I ONLY WOULD HAVE DONE ONE THING DIFFERENTLY. I SHOULD HAVE TOLD THE FO TO IMMEDIATELY TELL THE APCH CTL; 'UNABLE RNAV.' BEING A PLT; MY FIRST REACTION WAS TO TRY TO SOLVE THE PROB. THIS TOOK UP VALUABLE TIME DURING WHICH OUR DEV FROM COURSE FURTHER INCREASED. THIS CAUSED US TO HAVE A POTENTIAL CONFLICT WITH ANOTHER ACFT. I BELIEVE THE LNAV DEPS IN VEGAS HAVE TOO TIGHT A TOLERANCE FOR ANY ERRORS. IN THIS CASE; THE FMC COMPUTER MALFUNCTION WAS SOMETHING TOTALLY OUT OF OUR CTL. THE DISTR OF BEING HEADS DOWN; TRYING TO SOLVE THE PROB AND THE HIGH DENSITY TFC INTO LAS; CREATES A POTENTIALLY HAZARDOUS SITUATION. TO PREVENT THIS FROM HAPPENING AGAIN; I WOULD RECOMMEND A CHANGE TO THE FOM BE CREATED. I WOULD RECOMMEND THAT ANY TIME A PLT IS EITHER UNSURE OF THEIR POS ON AN RNAV DEP OR MISSES ANY TURN WHATSOEVER; THAT APCH IS TOLD; 'UNABLE RNAV;' PRIOR TO ANY PROB SOLVING. HAD WE DONE THIS; INSTEAD OF TRYING TO TROUBLESHOOT IT; WE WOULD HAVE AVOIDED THE TFC CONFLICT THAT WE HAD INADVERTENTLY CREATED.