Did not accept the fast before the yahrzeit and began fasting, should he finish

Someone who was not accustomed to fasting on the day of his mother's yahrzeit and began fasting, and at noon found out that it is not a fast since he did not accept the fast, can he eat at noon?

He should finish the day fasting, but it is not considered a fast. However, there are those who are lenient and say that he does not need to finish the day fasting.

It is written in the Shulchan Aruch, Orach Chaim, Siman 562, Seif 5: "Any fast that an individual did not accept upon himself beforehand is not a fast. Note: to pray 'Aneinu' and not regarding whether he is obligated to fast simply, and if he fasted like this, he has not fulfilled his vow." And it is written in the Mishnah Berurah, Siman 562, Seif Katan 23: "Nevertheless, if he began to fast on this day, he must fulfill his vow with what he accepted now, even if he did not accept it beforehand, and it is also an atonement for sin, not worse than one who abstains from meat and wine, as will be said later in Seif 11. So wrote the PM"G and the GRA in explaining the words of the RMA, and in 'Yad Ephraim', as well as in 'Shiurei Leket', they question this, as he thought to fulfill his previous vow with this, therefore it was an erroneous acceptance, and he can reconsider and eat on that day."