DEV FROM CLRED RNAV (DP) RTE DUE TO MALFUNCTION OF SINGLE FMC SYS.

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COURSE DEV ON LNAV DEP. FO WAS THE PF ON THE FLT TO LAX. HE WAS ON DAY 2 OF HIS UOE; AND I WAS THE CHK AIRMAN FLYING IN THE R SEAT. THE FLT WAS BEING OBSERVED BY FAA INSPECTOR. WE WERE CLRED VIA THE IDALE 2 LNAV DEP; WHICH FO THOROUGHLY BRIEFED AND RTE WAS CLOSED AND VERIFIED CORRECT IN THE MCDU. THE FLT PUSHED AND WAS OFF. COMPANY LNAV DEP PROCS WERE FOLLOWED TO THE LETTER; AND WE LEVELED AT 7000 FT MSL. BTWN LODZY AND IDALE; DEP CTL CLRED US TO BOACH. FO ASKED ME TO ENTER BOACH IN THE MCDU; AND I DID. I ASKED HIM IF HE WANTED ME TO EXECUTE BOACH. HE VERIFIED BOACH WAS IN THE PROPER POS AND SAID TO EXECUTE. I EXECUTED BOACH AND IT WAS HIGHLIGHTED IN THE MCDU; AND A MAGENTA LINE FORMED ON THE CFIS DISPLAY FROM PRESENT POS TO BOACH; AND THE ACFT BEGAN A L TURN TO BOACH. SHORTLY AFTER THAT; BOTH MCDU'S BLANKED AND FLASHED BACK ON. BOACH WAS NO LONGER HIGHLIGHTED. SEVERAL RTE DISCONTINUITY BLOCKS APPEARED ON THE LEGS PAGE; AND BOACH WAS NO LONGER DISPLAYED ON THE EFIS DISPLAY. THE ACFT SEQUENCED ITS BANK AND CONTINUED TURNING L TOWARD AN UNKNOWN FIX. I TOLD FO THAT 'THE BOX HAD DROPPED BOACH; AND WE NEEDED TO TURN R.' FO SELECTED HDG MODE ON THE MCP AND BEGAN AN IMMEDIATE R TURN. AT THIS TIME; DEP CTL ASKED 'COMPANY NUMBER WHERE ARE YOU GOING?' I ANSWERED; WE WERE IN A R TURN TOWARDS HECTOR. WE COULD NO LONGER IDENT BOACH; AND HECTOR WAS THE NEXT FIX. DEP CTL INSTRUCTED US TO GO DIRECT BOACH. BY THIS TIME; FO HAD RE-ESTABLISHED BOACH IN THE MCDU AND EXECUTED IT AND ENGAGED LNAV. WE PROCEEDED TOWARDS BOACH; BUT THE EFIS DISPLAY NOW SHOWED A MAGENTA LINE TO BOACH; HECTOR AND BACK BOACH. FO CORRECTED THE PROB; AND NO FURTHER PROBS OCCURRED. INSPECTOR INDICATED THAT IT APPEARED THAT WE HAD A SOFTWARE PROB. WE PROCEEDED TO LAX AND LANDED. DURING THE DEBRIEF OF THE FLT; INSPECTOR; AGAIN; STATED THAT HE WAS OF THE OPINION THAT THE SOFTWARE HAD FAILED; AND IT APPEARED TO HIM THAT WE WERE DOING EVERYTHING POSSIBLE TO CORRECT THE PROB. HE STATED THAT HE THOUGHT FO'S SITUATIONAL AWARENESS COULD HAVE BEEN BETTER; BUT HE REACTED AS RAPIDLY AS POSSIBLE; AND HE WOULD BE AVAILABLE TO VERIFY THE FACTS OF OUR PROB. INSPECTOR SIGNED FO'S UOE FORMS; AND TOLD US TO CONTINUE OUR UOE. I CALLED THE NUMBER REQUESTED OF US TO CALL; AND WAS ANSWERED WITH A RECORDED MESSAGE FROM 'NAME.' I LEFT A BRIEF MESSAGE OF THE SIT. I THEN CALLED CHIEF PLT AND RELAYED THE EVENTS. I MADE A WRITE-UP IN THE ACFT LOGBOOK; AND LAX MAINT PERFORMED SEVERAL TESTS AND SIGNED OFF THE ACFT FOR SVC. I CONTACTED COMPANY ATC SPECIALIST AND RELAYED THE EVENTS TO HIM. FO AND I BOTH MADE PHONE CALLS TO COMPANY SAFETY COMMITTEE AND ANSWERED THE QUESTIONS. IN RETROSPECT; I FEEL WE WERE REACTING TO IDENTING A PROB AND CORRECTING THE PROB AS FAST AS HUMANLY POSSIBLE. I FEEL I SHOULD HAVE IMMEDIATELY ASKED FOR A VECTOR DUE TO AN LNAV FAILURE. I AM CONFIDENT THE PROB WAS NOT INDUCED BY PLT ACTION. I HAVE NEVER SEEN THIS KIND OF PROB BEFORE; AND HAVE NOT HAD ANY PROBS WITH ANY LNAV DEPS. I THINK FO DID A VERY NICE JOB IN THE BRIEFING AND EXECUTION OF OUR LNAV DEP PROCS. THIS WAS THE SECOND DAY OF HIS UOE AND UNDER THE CIRCUMSTANCES; I THINK HE DID AN ADMIRABLE JOB. RESULT: A GROWING DISTRUST OF LNAV DEPS. RECOMMENDATION: EXTREME VIGILANCE OF ALL LNAV PROCS AND IMMEDIATE DECLARATIONS OF LNAV FAILURE TO ATC AND REQUEST VECTORS.