The Genealogy of Jesus

The genealogy of Jesus in Mathew and Luke are written opposite to each other, descending vs. ascending, but they do not match. How to explain that?

According to Mathew, Jesus the son of Joseph, the son of *Jacob, the son of Matthan, *Eleazar, *Eliud, *Akim, *Zadok, *Azor, *Eliakim, *Abiud, *Zerubbabel, *Shealtiel, *Jeconiah, *Josiah, *Amon, *Manasseh, *Hezekiah,*Ahaz, *Jotham, *Uzziah, the son of *Jehoramin.....

However, According to Luke, Jesus the son of Joseph, the son of *Heli, the son of Matthat, *Levi, *Melki, *Jannai, *Joseph, *Mattathias, *Amos, *Nahum, *Esli, *Naggai, *Maath  *Mattathias, *Semin, *Joseph, *Joda, *Joanan, *Rhesa, *Zerubbabel, *Shealtiel, *Neri, the son of *Meliki.......

Some apologists claim Mathew listed the paternal genealogy of Joseph, the husband of Mary. So, Mathew stated that Jacob has begotten Joseph. But, Luke recorded the maternal genealogy of Joseph to prove Jesus was the son of David on Mary's side too. Luke named Joseph the son of Heli while actually, Mary was the daughter of Heli, not Joseph.

However, there is no mention in the bible of Luke that the genealogy is according to Mary. Also, there was no mention of Mary was the daughter of Heli. Moreover, wasn't mary the daughter of Joachim and Anne, not Heli. 

According to the story, Joachim, and Anne, the parents of Mary were virtuous and childless longing for a child. They were ashamed of being childless so, they promised to give their offspring to god if their prayer for a child was answered. The angel appeared and told Joachim about Mary's birth. According to the Catholic Church, Mary had an immaculate birth without sex like Jesus too. This story, Mathew, and Luke do not match each other about the geology of Jesus.

I guess they were trying to double prove Jesus descent is from David. But Something is confusing here.