Shalom and Hiya! 
the discrepencies of the genealogies is not too difficult. there are no real contradictions involved, merely misunderstandings of things. first, though it doesn not directly involve the thread, one thing must be considered. that the Aramaic Peshitta could very well be the primary language in which the New Testament writings were written in. i say this only because of research into these things. both the Aramaic and Greek manuscripts all end up as the oldest around the same date, approximately 1-2 hundred years after the original autograph. either way, despite which was primary, one would certainly shed light on the other as far as meaning and even detail. there are some "mess-ups" in the Greek that the Aramaic seems to answer, and there are also in the Aramaic that the Greek seems to answer.
now concerning the genealogy of Messiah. in short, Luke's is Joseph's and Matthew's is Mary's. now, i realise that women are not mentioned as the main progenitor in a genealogy, but Mary is unique. she was a virgin, no earthly father to the kid. the Aramaic Matthew seems to indicate this:
"Jacob fathered Joseph, the father of Mary, of whom was born Yeshua who is called Mshikha." ~1:16
in Aramaic, Matthew 1:16 and 1:19, when speaking of Joseph as "husband", use two different words, even the Greek does as well but the differences are too subtle to tell. Aramaic, however, in verse 16 says, "Joseph, the Gav'rah (father) of Mary...", and verse 19 says, "Joseph her B'alah (husband)...". Gav'rah means Guardian, Kinsman even father. b'alah means Lord, Master even husband. in seeing Joseph, the father of Mary, we see her as added to the genealogy, as Tamar Rachav and Ruth are as well, even the wife of Uriyah. this explanation also explaines the discrepency of Matthew's genealogy as having 14 from Abraham to David, 14 from David to Babylon, and 14... errr... ooops... 13 from Babylon to Yeshua/Jesus.
the explanation of Luke's being Joseph's line, is the simple statement:
"And Jesus Himself was beginning to be about thirty years old, being, as was supposed, the son of Joseph, the son of Heli...etc" ~3:23
there are more interesting things about the genealogy that i have discovered concerning the interpretations of the names, but time and space wont allow it at this time. in part two i will have listed the genealogy...
many blessings,
John