B737-300 FLT CREW HAS A TRACK AND HDG DEV DURING CIVET ARR TO LAX FOLLOWING DUAL FMC FAILURES.

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WE BRIEFED THE CIVET 5 TO RWY 25R DURING CRUISE. DURING DSCNT ON THE CIVET 5 APCHING RUSTT WE WERE GIVEN A CLRNC TO DSND VIA THE CIVET 5 FOR RWY 24R. THE CAPT WAS PF AND I AS PNF SELECTED 15000 FT IN THE MCP. I THEN PROCEEDED TO PROGRAM THE FMC FOR RWY 24R. IT WAS INITIALLY SET UP FOR RWY 25R. WE WERE APPROX 1-2 MI FROM RUSTT AND NOW SHOWED A DISCONTINUITY ON THE FMC. THE CAPT TURNED OFF THE AUTOPLT. I THEN TOLD THE CAPT WE NEEDED TO GO TO MANUAL WITH A 226 DEG COURSE TO PARADISE VOR; WHICH WAS SET. AS WE WERE ESTABLISHING OURSELVES ON THE COURSE; BOTH FMC'S SCREENS SAID 'FMC' AT WHICH TIME I REQUESTED A HDG WITH SOCAL AND EXPLAINED THAT WE WERE HAVING FMC PROBS. SOCAL GAVE US A HDG OF 260 DEGS TO JOIN THE RWY 24R LOC. THE CTLR GAVE US THE LOC FREQ OF 108.5; WHICH WE SET MANUALLY ON BOTH SIDES. VOR/LOC WAS ARMED AND CAPTURED AND WE WERE GIVEN A DSCNT TO 10000 FT. THERE WAS A B737 JOINING THE RWY 25R LOC WHILE WE WERE SHOWING VOR/LOC CAPTURE FOR RWY 24R. THE CAPT THEN TURNED TO JOIN THE RWY 24R LOC. THE TURN TO TRACK INBOUND ON THE RWY 24R LOC POINTED US TOWARD THE OTHER B737 JET. I EXPLAINED THAT WE NEEDED TO COME R; AND THE CAPT CAME R; ONCE AGAIN THE VOR/LOC SHOWED CAPTURE WITH RWY 24R FREQ SELECTED. THE CAPT TURNED TO TRACK THE RWY 24R LOC AGAIN. ATC THEN ASKED US IF WE WERE SET UP FOR RWY 24R; I RESPONDED THAT WE HAD 108.5 SELECTED. ATC THEN ASKED THE OTHER B737 TO OFFSET TO THE S. WE THEN GOT A TCAS TFC ALERT FOLLOWED BY A MONITOR VERT SPD RA. I THEN TOLD THE CAPT THAT WE NEEDED TO COME R AS THE LOC WAS ERRATIC AND APPEARED TO BE GIVING US FALSE GUIDANCE. ATC THEN GAVE US A 290 DEG HDG AND THE CAPT TURNED TO 290 DEGS. I THEN LOOKED AT THE COURSE WINDOW AND APCH CHART AND NOTICE THAT WE STILL HAD THE 226 DEG COURSE SET. I THEN SET BOTH COURSE WINDOWS TO THE 249 DEG COURSE FOR THE LOC; AND WE WERE GIVEN A 250 DEG HDG TO JOIN RWY 24R. WE REARMED VOR/LOC AND JOINED THE RWY 24R LOC AND WERE GIVEN A DSCNT TO 7000 FT. WE WERE THEN GIVEN A CLRNC FOR THE ILS RWY 24R AND LANDED WITH NO OTHER EVENTS.