There is another version of the Breath of Fire that is mentioned along with the normal one in the text: Kumbhaka Paddhati.
Note a version of this is taught in Kundalini Yoga as well but not as strong as this one.
From the Kumbhaka Paddhati:
Close the left nostril and rapidly and forcefully exhale and inhale several times. Then inhale through the right nostril hold the breath to capacity and exhale through the left nostril. After this close the right nostril and rapidly exhale and inhale several times through the left nostril. Then inhale and hold the breath to capacity and exhale through the right nostril.
The purpose of Kundalini Yoga as stated by the texts is the purification of the three main nadi's of the spine. This technique relates to the sun and moon channels were the main method with both nostrils open relates to the center channel. Fire is the element of purification.
Kumbhaka Paddhati of Raghuvira