Fatigue; failed navigation systems and non-routine ATC handling on a complex arrival to LAX resulted in an unstabilized raw data approach; a go around and eventual safe landing for an air carrier flt crew.

Narrative

On final to RWY 24R at LAX the pilot flying (Captain) lost the flight director and the autopilot in IMC. The approach was continued using raw data; but the aircraft descended below the glideslope. The pilot not flying (first officer) called sink rate and the EGPWS said 'too low' and we initiated the missed approach at approximately 500 AGL. On the subsequent approach; flown on autopilot B; the Captain's flight director was lost again and the First Officer continued the approach on his FD and landed safely. Other factors that contributed the pilot's action in this incident are extended duty and flight time due to weather on the previous leg and multiple changes to the RNAV arrival and runway assignment by ATC. These factors did not attribute directly to the loss of the FD; but degraded the crew's performance in dealing with the loss of flight instruments. Fatigue and time compression distracted from the task of flying the approach and my late decision to abandon the approach or relinquish control to the other pilot. Two changes to the RNAV arrival; vectors off the RNAV arrival and several unpublished speed restrictions task saturated the crew inside KONZL. The bottom line is the crew recognized an unstable approach; made appropriate calls to safely execute a missed approach and return for a safe landing at LAX.