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Murray was aiming to reach his second quarter-final in as many weeks as he continues to work his way back from hip problems.
Rudd defeated Murray 7-5, 6-4 and afterwards paid tribute to his comeback.
“On the opposite side of the net from me was a legend of the game for the past decade or so, even what he’s done now for the past year, fighting his way back from very serious injury has been an inspiration for me,” said Rudd.
The first set was looked at 5-5 before Rudd stepped up a gear to take the lead.
Murray then fell behind early in the second set, dropping five consecutive games during a run that included a stretch of seven consecutive unforced errors.
There was good news for Cameron Norrie, who made it into the next round after he defeated fellow Brit Dan Evans 7-6 (7-3), 6-3.
Murray has been awarded a wildcard into the Indian Wells main draw, which starts on Monday, and he is due to make his first appearance there since 2017.