A B737-700 FLT CREW OVERSHOT THE FINAL APCH COURSE INBOUND WHEN THEIR FMC WOULD NOT DISPLAY THE ASSIGNED RWY IN THE DEP-ARR PAGE.

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BEFORE TOP OF DSCNT INTO LAX; PROGRAMMED THE FMC FOR SEAVU ONE ARR FOR RWY 24R. UPON HDOF TO SOCAL APCH; WE WERE TOLD TO 'DSND VIA SEAVU ONE ARR FOR RWY 25L.' I CHANGED THE RWY IN THE FMC TO RWY 25L. SEVERAL MOMENTS LATER WE WERE GIVEN A FREQ CHANGE TO THE FINAL CTLR AND ALSO TOLD TO CHANGE RWY BACK TO RWY 24R. I ATTEMPTED TO CHANGE THE RWY IN THE FMC TO RWY 24R; BUT ON THE DEP/ARR PAGE ALL THAT WAS LISTED WAS RWY 25L. THE FMC WOULD PROVIDE NO OPTION TO SELECT RWY 24R. AFTER SEVERAL ATTEMPTS TO CORRECT THE FMC; THE ACFT FLEW BEYOND LUVYN AND WAS OFF OF THE STAR. THE DISPLAY STILL HAD ALL THE FIXES FOR THE ARR TO RWY 25L; AND I WAS UNABLE TO ENTER THE FIXES FOR THE ARR TO RWY 24R. ATTEMPTED SEVERAL CALLS TO SOCAL APCH BUT HE WAS BUSY WITH OTHER ACFT AND DID NOT RESPOND. HIS FIRST CALL TO US WAS; 'COMPANY NUMBER; WHERE ARE YOU GOING?' WE IMMEDIATELY SAID; 'WE HAVE LOST OUR FMC AND REQUIRE VECTORS TO INTERCEPT THE LOC FOR RWY 24R; WE ARE LEVELING AT 11000 FT.' THE CTLR RESPONDED BY GIVING US A VECTOR TO JOIN THE LOC. MINS LATER WE RPTED RWY 24R IN SIGHT AND WERE CLRED FOR A VISUAL APCH. WE LANDED UNEVENTFULLY. I HAVE NO IDEA WHY I WAS ABLE TO SET UP FOR RWY 24R; MAKE A CHANGE TO RWY 25L; AND THEN WAS UNABLE TO SWITCH BACK TO RWY 24R. ATTEMPTS TO ENTER FIXES MANUALLY WERE UNSUCCESSFUL UNTIL APCHING THE FINAL APCH FIX FOR RWY 24R. BECAUSE THE ARR IS STRICTLY RNAV WITH NO BACKUP RADIAL AND DME INFO WITH WHICH TO IDENT THE POINTS; WE WERE UNABLE TO CONTINUE ON THE ARR WITH A FAILURE OF THE FMC. I THINK FIXES SHOULD HAVE GND NAVAID INFO AS BACKUPS; OR ALLOW FOR SOME KIND OF INSTANT COM TO A CTLR IF THE ABILITY TO NAV ON THE RNAV APCH BECOMES IMPOSSIBLE. WHEN THE FREQ IS BUSY (A NORMAL OCCURRENCE IN THE LAX TERMINAL AREA) WE ARE DEAD IN THE WATER WITH A FMC PROB. FIGURE OUT WHY THE FMC PROB OCCURRED AND PROVIDE A FIX TO PREVENT IT FROM HAPPENING AGAIN.