Dark Souls | Laurentius of the Great Swamp, pyromancy vendor in Firelink Shrine, upgrading pyromancy

Dark Souls Remastered

Once you free Laurentius of the Great Swamp (see link), you will see him at Firelink Shrine. He is the main pyromancy vendor, where you can acquire and upgrade the Pyromancy Flame, allowing you the ability to use pyromancies. Pyromancer is also an optional starting class that will start the character with a Pyromancy Flame as well as a basic set of fire spells.

https://www.youtube.com/watch?v=nXElUGw_BVY

The items that Laurentius sells:

Item, Cost (in souls)
-=-=-=-=-=-=-
Fire Orb, 8000
Combustion, 500
Iron Flesh, 2000
Flash Sweat, 2000
Fireball, 800

You need a total of 340,500 souls to fully upgrade the Pyromancy Flame.

In order to max out the upgrade with Laurentius (to +15), you need a total of 149,500 souls.
Once maxed at with Laurentius, you need to visit Quelana of Izalith to upgrade further.
In order to max out with Quelana of Izalith, you need an additional 191,000 souls.

Upgrade level, Magic Adjustment, Cost (in souls), Total cost so far (in souls)
-=-=-=-=-=-=-
Pyromancy Flame, 100, N/A, 0
Pyromancy Flame +1, 105, 500, 500
Pyromancy Flame +2, 110, 1,000, 1,500
Pyromancy Flame +3, 115, 2,000, 3,500
Pyromancy Flame +4, 120, 3,000, 6,500
Pyromancy Flame +5, 125, 4,000, 10,500
Pyromancy Flame +6, 131, 5,500, 16,000
Pyromancy Flame +7, 137 7,000, 23,000
Pyromancy Flame +8, 143, 8,500, 31,500
Pyromancy Flame +9, 149, 10,000, 41,500
Pyromancy Flame +10, 155, 12,000, 53,500
Pyromancy Flame +11, 162, 14,000, 67,500
Pyromancy Flame +12, 169, 16,000, 83,500
Pyromancy Flame +13, 176, 19,000, 102,500
Pyromancy Flame +14, 183, 22,000, 124,500
Pyromancy Flame +15, 190, 25,000, 149,500

Upgrades past +15 requires Quelana of Izalith (along with 191,000 more souls):

Upgrade level, Magic Adjustment, Cost (in souls), Total cost so far (in souls)
Pyromancy Flame, 196, 25,000, 174,500
Pyromancy Flame +1, 202, 27,000, 201,500
Pyromancy Flame +2, 208, 30,000, 231,500
Pyromancy Flame +3, 214, 33,000, 264,500
Pyromancy Flame +4, 220, 36,000, 300,500
Pyromancy Flame +5, 230, 40,000, 340,500