EMB135 FLT CREW HAS A TRACK HDG DEV WHILE ATTEMPTING TO NAV TO GRENE INTXN ON THE LGA ILS RWY 4 APCH.

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WE WERE ENRTE TO THE GRENE FIX AS DIRECTED BY NEW YORK APCH FOR THE ILS RWY 4 AT LGA; WHEN NEW YORK APCH ADVISED US WE WERE R OF COURSE AND GAVE US A 020 DEG HDG TO INTERCEPT THE LOC. OUR FLT ORIGINATED FROM RDU AND SHORTLY AFTER TKOF OUR ACFT EXPERIENCED DUAL FMS/GPS PROBS. ACFT POS WAS NOT RELIABLE SO ENRTE TO LGA WE NAV'ED USING VOR NAV. AT ONE POINT NEAR PXT; WE ADVISED ZDC OF OUR SITUATION. FOLLOWING THE DSCNT ON THE KORRY 3; WE WERE DIRECTED TO PROCEED DIRECT GRENE. PER CHART PAGE 11-1; FOR NEW YORK; ILS 4; GRENE IS LISTED AS BEING LOCATED AT D11.1 LGA (FREQ 113.1). ACCORDINGLY; A FIX-TO-FIX WAS BEING EXECUTED FROM OUR POS OF APPROX LGA D50.0 TO D11.1. THE SELECTED NAV RADIAL WAS SET AS 044 AND MY PFD INDICATED WE WERE W OF THE 044 DEG RADIAL AT APPROX 25 DME WHEN WE WERE ADVISED BY NEW YORK APCH THAT WE WERE R OF COURSE AND GOT THE SUBSEQUENT VECTOR TO INTERCEPT THE LOC. LOC SELECTION WAS BEING DELAYED UNTIL APPROX 19 MI TO ENSURE LOC SIGNAL RELIABILITY PARAMETERS (18.1 MI). IN MY OPINION; I WAS ON TRACK TO INTERCEPT THE LGA 044/11.1 FIX TO INTERCEPT GRENE AS LISTED ON PAGE 11-1. SUSPECT THAT GRENE IS LISTED INCORRECTLY ON NEW YORK PAGE 11-1 AS D11.1 LGA WHEN IT SHOULD BE D11.1 ILGA. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: RPTR STATED THAT THE FMC/GPS WAS SHOWING A MESSAGE 'POS NOT KNOWN' THROUGHOUT THE FLT AND THAT THE FLT CREW WAS FLYING FIX-TO-FIX USING THE VOR/DME METHOD. AFTER APCHING KORRY INTXN ON THE KORRY 3 ARR ATC DIRECTED THE FLT CREW TO FLY DIRECT GRENE INTXN. THE RPTR STATED THAT HE TUNED THE LGA VOR (113.1) AND THE 044 DEG COURSE INBOUND AND FLEW TO INTERCEPT THE COURSE. ATC INDICATED THAT THE ACFT WAS R OF COURSE AND GAVE A 020 DEG HDG TO INTERCEPT THE 044 LOC COURSE. RPTR INDICATED THAT HE SHOWED THE ACFT L OF COURSE. RPTR BELIEVED THAT GRENE SHOULD BE DEPICTED BY THE ILGA (110.5) 044/11.1 NOT THE LGA (113.1) 044/11.1.