I found the qiblah direction at my location thanks to an application off the Internet but since it doesn't seem to be very sophisticated I was wondering what the tolerance is for the direction of the qiblah. I've heard that it is 45%; is that correct?
What must be understood about the Qibla is that you don't really need to be 100% accurate, or even within N% tolerance. Islam is a religion and way of life and not engineering. Remember that during the days of the Prophet they did not have the means for accurate direction finding that we have today, but that did not stop them from praying. So all you need to do is face Makkah to the best of your ability. If you are with a group agree together upon a direction and pray together. That is what's really important.
"It is not righteousness that you turn your faces towards the East and the West, but righteousness is this that one should believe in Allah and the last day and the angels and the Book and the prophets, and give away wealth out of love for Him to the near of kin and the orphans and the needy and the wayfarer and the beggars and for (the emancipation of) the captives, and keep up prayer and pay the poor-rate; and the performers of their promise when they make a promise, and the patient in distress and affliction and in time of conflicts-- these are they who are true (to themselves) and these are they who guard (against evil)."
Al Baqara 2:177
The theory of tolerance here is technically incorrect for the following reasons:
It's irrelevant at the antipode (-21.422441174957708,-140.1738229393959) of kaaba (21.422441174957708, 39.826177060604095). Meaning that at that point you have a 100% tolerance for choosing the direction of qiblah (kaaba) along the curvature of the Earth.
Same as with the previous point in regards to the anitpode it's irrelevant inside the kaaba
AFAIK qibla is not the, zero dimensional, center of the kaaba but the kaaba itself, therefore anything in between it's edge can be faced as qibla and therefore the tolerance would be bound to a continously varying graph (would appreciate addition of mathematically accurate info here) and not a fixed 100%.
So strictly speaking the mentioned tolerance level seems technically illogical, or at least one should present a derivation for the conclusion. And since you mention that it is not very sophisticated then you can rely on it
Actions are dependent upon their Intentions
And the some of the relevant fatwa from Ayatollah al-Sayyid 'Ali al-Husayni al-Sistani:
... a person who is far, would stand in such a manner that people would say that he is praying facing the Qibla, and that would suffice.
... acquire such information as may amount to certainty, like testimony of two reliable persons.
... form an idea from the Niche (Mehrab) of the Masjid or from the graves of the Muslims, or by other ways, and act accordingly.
... if a non-Muslim who can determine Qibla by scientific method, indicates Qibla satisfactorily, he can be relied upon.
If a person, who has a mere surmise about Qibla, and is in a position to have a better idea, he should not act on that guess work.
In your case sophisticate should suffice.
Sophisticated in itself is a relative term. That of today was military-grade/very sophisticated of the past.
When facing the Qiblah, it is ok to face a little to the right or left because you do not have to face the Qiblah exactly unless you are in the Masjidul Haram in Makkah. The Prophet (peace and blessings be upon him) said:
مَا بَيْنَ الْمَشْرِقِ وَالْمَغْرِبِ قِبْلَةٌ
What is between the east and the west is Qiblah
Deviation from direction of Qibla can be intentional or unintentional:
1- If it is unintentional, for example a Muslim is entered a new home and some one tells him the direction for Qibla falsely and he pray for one year in a false direction. In this case when he understand the correct direction, if the deviation has been less than 90 degree his past prayers are accepted and there no need to repeat them. But if the deviation has been more than 90 degree the past prayers with false Qibla are void and needed to be repeated.
2- If one know the direction of Qibla correct and with certainty no intentional deviation is accepted and an intentional deviation more than 5 degree makes the prayer void.
- The persian part of QA section of official website of Ayatollah Khamenei the superm leader of Shia Muslims.